Finite Group Databasebeta

← #33122630

Subgroups — #33122630

Order 18688.

1072 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988891C211yesmaximal, characteristic, complemented
2C2×#17988687C211yesmaximal, characteristic
3#22615301C211yesmaximal, characteristic, complemented
4#22611266C211yesmaximal, characteristic, complemented
5#22611256C211yesmaximal, characteristic, complemented
6#22611079C211yesmaximal, characteristic, complemented
7#22611069C211yesmaximal, characteristic, complemented
8#22610901C211yesmaximal, characteristic, complemented
9#22610872C211yesmaximal, characteristic, complemented
10#22610265C211yesmaximal, characteristic
11#22610265C211yesmaximal, characteristic
12#22610192C211yesmaximal, characteristic
13#22610178C211yesmaximal, characteristic
14#22609931C211yesmaximal, characteristic
15C2×#554×C73C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18#17988891—24no—
19#17988687—24no—
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556512C2211yescharacteristic
29C2×#14556512C2211yescharacteristic
30C2×#14556502C2211yescharacteristic
31#17988520—21no—
32#17988520—21no—
33#17988494—21no—
34#17988493C2211yescharacteristic
35#17988493C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988485C2211yescharacteristic
38#17988483—21no—
39#17988457C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50#554×C73—24no—
51#554×C73—24no—
52C2×C4⋊Q8×C73C2211yescharacteristic
53C2×#125×C73C2211yescharacteristic
54C2×#125×C73C2211yescharacteristic
55C2×C22⋊Q8×C73—21no—
56C2×C22⋊Q8×C73—21no—
57C2×C22⋊Q8×C73—21no—
58C2×C4×Q8×C73C2211yescharacteristic
59C2×C4×D4×C73C2211yescharacteristic
60C2×C4×D4×C73C2211yescharacteristic
61C2×C4×D4×C73C2211yescharacteristic
62C22×C4⋊C4×C73—21no—
63C22×Dic146C2311yescharacteristic
64C2×#11924016—22no—
65C2×#11924016—22no—
66C2×#11924016Q814yescomplemented
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C4×Dic146—22no—
72C4×Dic146—22no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×C4×Dic73—21no—
82#14556524—22no—
83#14556524—22no—
84#14556522—22no—
85C4⋊#10596831—22no—
86#14556514—22no—
87#14556514—22no—
88#14556514—22no—
89#14556512—22no—
90#14556512—22no—
91#14556512—22no—
92#14556502—22no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99C22×Q8×C73—21no—
100C22×Q8×C73D411yescharacteristic
101C22×D4×C73C2311yescharacteristic
102C23×C4×C73—21no—
103C23×C4×C73—21no—
104C23×C4×C73—21no—
105C4⋊Q8×C73—22no—
106C4⋊Q8×C73—22no—
107#125×C73—22no—
108#125×C73—22no—
109#125×C73—22no—
110#125×C73D414yes—
111C22⋊Q8×C73—28no—
112C22⋊Q8×C73—28no—
113C22⋊Q8×C73—28no—
114C4×Q8×C73—22no—
115C4×Q8×C73—22no—
116C4×D4×C73—22no—
117C4×D4×C73—22no—
118C4×D4×C73—22no—
119C4×D4×C73—22no—
120C4×D4×C73—22no—
121C4×D4×C73D414yes—
122C2×C4⋊C4×C73—28no—
123C2×C4⋊C4×C73—21no—
124C2×C4⋊C4×C73—21no—
125C2×C4⋊C4×C73—21no—
126C2×C4⋊C4×C73—21no—
127C2×C4⋊C4×C73—21no—
128C2×C4⋊C4×C73D411yescharacteristic
129C2×C4⋊C4×C73D411yescharacteristic
130C2×C4⋊C4×C73D411yescharacteristic
131C2×C4⋊C4×C73C2311yescharacteristic
132C2×C4⋊C4×C73C2311yescharacteristic
133C2×C4⋊C4×C73C2311yescharacteristic
134C2×C22⋊C4×C73—21no—
135C2×C22⋊C4×C73—21no—
136C2×C22⋊C4×C73—21no—
137C2×C42×C73C2311yescharacteristic
138C2×C42×C73C2311yescharacteristic
139C2×C42×C73C2311yescharacteristic
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146—21no—
144C2×Dic146—21no—
145C2×Dic146C2×Q812yes—
146C22×Dic73—21no—
147C22×Dic73—21no—
148C22×Dic73—21no—
149C22×Dic73—21no—
150#11924016—28no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831C2×Q812yes—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—42no—
170C4⋊Dic73—81no—
171#11924007—42no—
172#11924007—81no—
173#11924007—81no—
174C4×Dic73—42no—
175C24×C73—21no—
176C2×Q8×C73—24no—
177C2×Q8×C73—22no—
178C2×Q8×C73—22no—
179C2×Q8×C73—41no—
180C2×Q8×C73—81no—
181C2×Q8×C73—81no—
182C2×D4×C73—21no—
183C2×D4×C73—21no—
184C2×D4×C73—21no—
185C2×D4×C73C2×D412yes—
186C2×D4×C73C2×Q812yes—
187C2×D4×C73#4112yes—
188C22×C4×C73—24no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73—24no—
193C22×C4×C73—24no—
194C22×C4×C73—42no—
195C22×C4×C73—42no—
196C22×C4×C73—81no—
197C22×C4×C73—81no—
198C22×C4×C73—81no—
199C22×C4×C73—81no—
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73C2×D411yescharacteristic
202C22×C4×C73C2×D411yescharacteristic
203C22×C4×C73C2×D411yescharacteristic
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73C2×D411yescharacteristic
206C22×C4×C73C2411yescharacteristic
207C4⋊C4×C73—24no—
208C4⋊C4×C73—24no—
209C4⋊C4×C73—24no—
210C4⋊C4×C73—22no—
211C4⋊C4×C73—24no—
212C4⋊C4×C73—24no—
213C4⋊C4×C73—22no—
214C4⋊C4×C73—22no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—21no—
217C4⋊C4×C73—21no—
218C4⋊C4×C73C2×D412yes—
219C4⋊C4×C73C2×D412yes—
220C4⋊C4×C73C2×D412yes—
221C22⋊C4×C73—24no—
222C22⋊C4×C73—24no—
223C22⋊C4×C73—24no—
224C42×C73—21no—
225C42×C73—21no—
226C42×C73—21no—
227C42×C73—21no—
228C42×C73C2×D412yes—
229C42×C73C2×D412yes—
230C23×C73—22no—
231C23×C73—22no—
232C23×C73—22no—
233C23×C73—41no—
234C23×C73—41no—
235C23×C73—41no—
236C23×C73—41no—
237C23×C73C22×D411yescharacteristic
238Q8×C73—44no—
239Q8×C73—84no—
240D4×C73—22no—
241D4×C73—22no—
242D4×C73—22no—
243D4×C73C22⋊Q814yes—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—21no—
252C2×C4×C73—21no—
253C2×C4×C73—22no—
254C2×C4×C73—22no—
255C2×C4×C73—22no—
256C2×C4×C73—21no—
257C2×C4×C73—21no—
258C2×C4×C73—21no—
259C2×C4×C73—21no—
260C2×C4×C73—21no—
261C2×C4×C73—21no—
262C2×C4×C73—48no—
263C2×C4×C73—41no—
264C2×C4×C73—41no—
265C2×C4×C73—41no—
266C2×C4×C73—41no—
267C2×C4×C73—41no—
268C2×C4×C73—41no—
269C2×C4×C73—84no—
270C2×C4×C73—84no—
271C2×C4×C73#12012yes—
272C2×C4×C73#12012yes—
273C2×C4×C73#13712yes—
274C2×C4×C73C22×D411yescharacteristic
275C2×C4×C73C22×D411yescharacteristic
276C2×C4×C73C22×Q811yescharacteristic
277C2×C4×C73C2×#4111yescharacteristic
278C2×C4×C732−1+411yescharacteristic
279C2×C4×C732−1+411yescharacteristic
280C2×Dic73—22no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—41no—
288C2×Dic73—41no—
289C2×Dic73—41no—
290C2×Dic73—41no—
291C2×Dic73—41no—
292Dic146—22no—
293Dic146—22no—
294Dic146—41no—
295Dic146—41no—
296#10596831—22no—
297#10596831—41no—
298#10596831—41no—
299#10596831—41no—
300C22×C73—24no—
301C22×C73—42no—
302C22×C73—42no—
303C22×C73—42no—
304C22×C73—42no—
305C22×C73—42no—
306C22×C73—42no—
307C22×C73C2×#12011yescharacteristic
308C22×C73#54811yescharacteristic
309C22×C73D4×Q811yescharacteristic
310C22×C73C2×#13711yescharacteristic
311C22×C73#57511yescharacteristic
312C22×C73#57611yescharacteristic
313C22×C73#57711yescharacteristic
314C4×C73—22no—
315C4×C73—22no—
316C4×C73—22no—
317C4×C73—41no—
318C4×C73—42no—
319C4×C73—41no—
320C4×C73—41no—
321C4×C73—42no—
322C4×C73—42no—
323C4×C73C2×C22⋊Q811yescharacteristic
324C4×C73#53611yescharacteristic
325C4×C73#54811yescharacteristic
326C4×C73D4×Q811yescharacteristic
327Dic73—42no—
328Dic73—81no—
329Dic73—81no—
330Dic73—81no—
331#24349—731nomaximal, Sylow
332C2×C73—44no—
333C2×C73#240911yescharacteristic
334C2×C73#300711yescharacteristic
335C2×C73#301211yescharacteristic
336C2×C73#337011yescharacteristic
337C2×C73#337211yescharacteristic
338C2×C73#338511yescharacteristic
339C2×C73#338811yescharacteristic
340C2×#554D7311yescharacteristic
341C2×#479—731no—
342C2×#440—731no—
343#2582—731no—
344#2059—731no—
345#2057—731no—
346#2026—731no—
347#2024—731no—
348#1992—731no—
349#1985—731no—
350#1877—731no—
351#1877—731no—
352#1864—731no—
353#1862—731no—
354#1802—731no—
355C73#2434911yescharacteristic, complemented, Sylow
356C22×SD16—731no—
357#554—24no—
358#554—24no—
359C2×C4⋊Q8C2×D7311yescharacteristic
360C2×#125C2×D7311yescharacteristic
361C2×#125C2×D7311yescharacteristic
362C2×C22⋊Q8—21no—
363C2×C22⋊Q8—21no—
364C2×C22⋊Q8—21no—
365C2×C4×Q8—731no—
366C2×C4×Q8C2×D7311yescharacteristic
367C2×C4×D4C2×D7311yescharacteristic
368C2×C4×D4C2×D7311yescharacteristic
369C2×C4×D4C2×D7311yescharacteristic
370C22×C4⋊C4—21no—
371#479—1464no—
372#440—1464no—
373C2×#107—731no—
374C2×#107—731no—
375C2×#106—731no—
376C2×C4⋊C8—731no—
377C2×#103—731no—
378C2×#103—731no—
379C2×#103—731no—
380C2×#102—731no—
381C2×#102—731no—
382C2×#102—731no—
383C2×#97—731no—
384#398—1461no—
385#398—1461no—
386#391—731no—
387#391—731no—
388#391—1461no—
389#389—731no—
390#389—731no—
391#389—1461no—
392#384—731no—
393#340—731no—
394#340—731no—
395#340—731no—
396#340—731no—
397#340—731no—
398#340—731no—
399#336—731no—
400#336—731no—
401#336—731no—
402#336—731no—
403C22×Q8—21no—
404C22×Q8—731no—
405C22×Q8C22⋊D7311yescharacteristic
406C22×D4C22×D7311yescharacteristic
407C23×C4—21no—
408C23×C4—21no—
409C23×C4—21no—
410C2×SD16—734no—
411C2×SD16—1462no—
412C2×SD16—1462no—
413C22×C8—731no—
414C22×C8—731no—
415C22×C8—731no—
416C22×C8—731no—
417C4⋊Q8—22no—
418C4⋊Q8—22no—
419#125—22no—
420#125—22no—
421#125—22no—
422#125C22⋊D7314yes—
423C22⋊Q8—28no—
424C22⋊Q8—28no—
425C22⋊Q8—28no—
426C4×Q8—22no—
427C4×Q8—22no—
428C4×Q8—1462no—
429C4×Q8—1462no—
430C4×D4—22no—
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—22no—
434C4×D4—22no—
435C4×D4C22⋊D7314yes—
436C2×C4⋊C4—28no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—731no—
443C2×C4⋊C4—731no—
444C2×C4⋊C4—731no—
445C2×C4⋊C4—1461no—
446C2×C4⋊C4—1461no—
447C2×C4⋊C4—1461no—
448C2×C4⋊C4—1461no—
449C2×C4⋊C4—1461no—
450C2×C4⋊C4C22⋊D7311yescharacteristic
451C2×C4⋊C4C22⋊D7311yescharacteristic
452C2×C4⋊C4C22⋊D7311yescharacteristic
453C2×C4⋊C4C22×D7311yescharacteristic
454C2×C4⋊C4C22×D7311yescharacteristic
455C2×C4⋊C4C22×D7311yescharacteristic
456C2×C22⋊C4—21no—
457C2×C22⋊C4—21no—
458C2×C22⋊C4—21no—
459C2×C42—1461no—
460C2×C42C22×D7311yescharacteristic
461C2×C42C22×D7311yescharacteristic
462C2×C42C22×D7311yescharacteristic
463#107—1462no—
464#107—1462no—
465#106—1462no—
466C4⋊C8—1462no—
467#103—1462no—
468#103—1462no—
469#103—1462no—
470#102—1462no—
471#102—1462no—
472#102—1462no—
473#97—1462no—
474#95—1461no—
475#95—1461no—
476#95—1461no—
477#95—1461no—
478#95—1461no—
479#95—1461no—
480C24—21no—
481C2×Q8—24no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—41no—
485C2×Q8—81no—
486C2×Q8—81no—
487C2×Q8—732no—
488C2×Q8—1461no—
489C2×Q8—1461no—
490C2×Q8—1461no—
491C2×Q8—1461no—
492C2×Q8—1461no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4C2×C22⋊D7312yes—
497C2×D4Q8×D7312yes—
498C2×D4#1192402912yes—
499C22×C4—24no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—81no—
510C22×C4—81no—
511C22×C4—1461no—
512C22×C4—1461no—
513C22×C4—1461no—
514C22×C4—1461no—
515C22×C4C2×C22⋊D7311yescharacteristic
516C22×C4C2×C22⋊D7311yescharacteristic
517C22×C4C2×C22⋊D7311yescharacteristic
518C22×C4C2×C22⋊D7311yescharacteristic
519C22×C4C2×C22⋊D7311yescharacteristic
520C22×C4C2×C22⋊D7311yescharacteristic
521C22×C4C23×D7311yescharacteristic
522SD16—1468no—
523C2×C8—732no—
524C2×C8—1461no—
525C2×C8—1461no—
526C2×C8—1461no—
527C2×C8—1461no—
528C2×C8—1461no—
529C2×C8—1461no—
530C2×C8—1461no—
531C2×C8—1461no—
532C2×C8—1461no—
533C2×C8—1461no—
534C2×C8—1461no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—2922no—
547C4⋊C4—2922no—
548C4⋊C4—5841no—
549C4⋊C4—5841no—
550C4⋊C4—5841no—
551C4⋊C4—1461no—
552C4⋊C4—1461no—
553C4⋊C4—1461no—
554C4⋊C4—1461no—
555C4⋊C4—1461no—
556C4⋊C4—1461no—
557C4⋊C4C2×C22⋊D7312yes—
558C4⋊C4C2×C22⋊D7312yes—
559C4⋊C4C2×C22⋊D7312yes—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—2922no—
568C42C2×C22⋊D7312yes—
569C42C2×C22⋊D7312yes—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23C22×C22⋊D7311yescharacteristic
578Q8—44no—
579Q8—84no—
580Q8—2921no—
581Q8—2921no—
582Q8—1462no—
583Q8—1462no—
584D4—22no—
585D4—22no—
586D4—22no—
587D4#1455656714yes—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—48no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—2921no—
616C2×C4—2921no—
617C2×C4—2921no—
618C2×C4—2921no—
619C2×C4—2921no—
620C2×C4—2921no—
621C2×C4—2921no—
622C2×C4—2921no—
623C2×C4—2921no—
624C2×C4—2921no—
625C2×C4—2921no—
626C2×C4—1462no—
627C2×C4#1455656012yes—
628C2×C4#1455656012yes—
629C2×C4#1455662112yes—
630C2×C4C22×C22⋊D7311yescharacteristic
631C2×C4C22×C22⋊D7311yescharacteristic
632C2×C4C2×Q8×D7311yescharacteristic
633C2×C4C2×#1192402911yescharacteristic
634C2×C4#1455663911yescharacteristic
635C2×C4#1455663911yescharacteristic
636C8—2921no—
637C8—2921no—
638C8—2921no—
639C8—1462no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22C2×#1455656011yescharacteristic
648C22#1798924811yescharacteristic
649C22Q8×C22⋊D7311yescharacteristic
650C22C2×#1455662111yescharacteristic
651C22#1798938511yescharacteristic
652C22#1798939011yescharacteristic
653C22#1798939811yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—41no—
658C4—42no—
659C4—41no—
660C4—41no—
661C4—42no—
662C4—42no—
663C4—2922no—
664C4—5841no—
665C4—5841no—
666C4—5841no—
667C4C2×#1455656711yescharacteristic
668C4#1798916111yescharacteristic
669C4#1798924811yescharacteristic
670C4Q8×C22⋊D7311yescharacteristic
671C2—44no—
672C2#2261329011yescharacteristic
673C2#2261919111yescharacteristic
674C2#2261923511yescharacteristic
675C2#2262268411yescharacteristic
676C2#2262271211yescharacteristic
677C2#2262284111yescharacteristic
678C2#2262287611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#554×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenterC2×#554D73
hypocenterC73#24349
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#17988891C2semidirect
3. #22615301C2semidirect
4. #22611266C2semidirect
5. #22611256C2semidirect
6. #22611079C2semidirect
7. #22611069C2semidirect
8. #22610901C2semidirect
9. #22610872C2semidirect
66. C2×#11924016Q8semidirect
355. C73#24349semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179888912
C2×#179886872
#226153012
#226112662
#226112562
#226110792
#226110692
#226109012
#226108722
#226102652
#226101922
#226101782
#226099312
C2×#554×C732
#2434973