Finite Group Databasebeta

← #33108008

Subgroups — #33108008

Order 18688.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988867C211yesmaximal, characteristic
2#22613020C211yesmaximal, characteristic, complemented
3#22610903C211yesmaximal, characteristic, complemented
4#22610697C211yesmaximal, characteristic, complemented
5#22610666C211yesmaximal, characteristic, complemented
6#22610661C211yesmaximal, characteristic, complemented
7#22610520C211yesmaximal, characteristic, complemented
8#22610499C211yesmaximal, characteristic, complemented
9#22610192C211yesmaximal, characteristic
10#22610178C211yesmaximal, characteristic
11#22609806C211yesmaximal, characteristic, complemented
12#22609756C211yesmaximal, characteristic
13#22609756C211yesmaximal, characteristic
14C4×#14556512C211yesmaximal, characteristic
15C73×#2349C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17#17988867C418yescomplemented
18C2×#14556524C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556502C2211yescharacteristic
28C2×#14556502C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30#17988491—21no—
31#17988491—21no—
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988483—21no—
35C4⋊C4×Dic73—21no—
36C4×C4⋊Dic73C2211yescharacteristic
37C4×C4⋊Dic73C2211yescharacteristic
38C4×C4⋊Dic73C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49C2×C4×D4×C73C2211yescharacteristic
50C2×C4×D4×C73C2211yescharacteristic
51C2×C4×D4×C73C2211yescharacteristic
52#389×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#385×C73—21no—
55#385×C73—21no—
56#382×C73C2211yescharacteristic
57#380×C73—21no—
58C4×C4⋊C4×C73C2211yescharacteristic
59C2×#95×C73—21no—
60C22×Dic146—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556591—22no—
66#14556591C2×C414yescomplemented
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×C4×Dic73—21no—
75C2×C4×Dic73—21no—
76C2×C4×Dic73—21no—
77#14556524—41no—
78#14556524C2×C414yescomplemented
79#14556524C2×C414yescomplemented
80#14556522—41no—
81C4⋊#10596831C2×C414yescomplemented
82#14556512—22no—
83#14556512C2×C414yes—
84#14556512C2×C414yes—
85#14556512Q814yescomplemented
86#14556502—22no—
87#14556502—41no—
88C4×#10596831—41no—
89#14556498—21no—
90#14556498—21no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94C22×D4×C73C2311yescharacteristic
95C23×C4×C73—21no—
96C23×C4×C73—21no—
97C23×C4×C73—21no—
98C4×D4×C73—22no—
99C4×D4×C73—22no—
100C4×D4×C73—22no—
101C4×D4×C73—41no—
102C4×D4×C73—41no—
103C4×D4×C73C2×C414yes—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73D411yescharacteristic
107C2×C4⋊C4×C73D411yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C42×C73C2311yescharacteristic
115C2×C42×C73C2311yescharacteristic
116C2×C42×C73C2311yescharacteristic
117#95×C73—21no—
118#95×C73—21no—
119#95×C73—21no—
120#95×C73—21no—
121#95×C73D411yescharacteristic
122#95×C73D411yescharacteristic
123C2×Dic146—24no—
124C2×Dic146—81no—
125C2×Dic146—81no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831C22×C412yes—
140C2×#10596831C22×C412yes—
141C2×#10596831C2×Q812yes—
142C4⋊Dic73—24no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73C22×C412yes—
149C4⋊Dic73C22×C412yes—
150C4⋊Dic73C2×Q812yes—
151#11924007—81no—
152#11924007—81no—
153C4×Dic73—42no—
154C4×Dic73—81no—
155C4×Dic73—81no—
156C24×C73—21no—
157C2×D4×C73—21no—
158C2×D4×C73—21no—
159C2×D4×C73—21no—
160C2×D4×C73C22×C412yes—
161C2×D4×C73C2×Q812yes—
162C2×D4×C73#4112yes—
163C22×C4×C73—21no—
164C22×C4×C73—24no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—42no—
168C22×C4×C73—42no—
169C22×C4×C73—42no—
170C22×C4×C73—42no—
171C22×C4×C73—81no—
172C22×C4×C73—81no—
173C22×C4×C73—81no—
174C22×C4×C73—81no—
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2411yescharacteristic
182C4⋊C4×C73—22no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—41no—
189C4⋊C4×C73—81no—
190C4⋊C4×C73—81no—
191C4⋊C4×C73C22×C412yes—
192C22⋊C4×C73—24no—
193C22⋊C4×C73—42no—
194C22⋊C4×C73—42no—
195C42×C73—21no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73C22×C412yes—
200C42×C73#4112yes—
201C23×C73—22no—
202C23×C73—22no—
203C23×C73—22no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73C22×D411yescharacteristic
209D4×C73—22no—
210D4×C73—22no—
211D4×C73—22no—
212D4×C73C4×Q814yes—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—22no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—84no—
238C2×C4×C73—84no—
239C2×C4×C73—84no—
240C2×C4×C73C4×D412yes—
241C2×C4×C73C4×D412yes—
242C2×C4×C73#13712yes—
243C2×C4×C73C23×C411yescharacteristic
244C2×C4×C73C22×Q811yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73H(5,𝔽2)11yescharacteristic
248C2×C4×C732−1+411yescharacteristic
249C2×Dic73—22no—
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261Dic146—84no—
262#10596831—22no—
263#10596831—22no—
264#10596831—22no—
265#10596831—41no—
266C22×C73—24no—
267C22×C73—42no—
268C22×C73—42no—
269C22×C73—42no—
270C22×C73—42no—
271C22×C73—42no—
272C22×C73—42no—
273C22×C73C2×C4×D411yescharacteristic
274C22×C73#54611yescharacteristic
275C22×C73D4×Q811yescharacteristic
276C22×C73C2×#13711yescharacteristic
277C22×C73#57511yescharacteristic
278C22×C73#57611yescharacteristic
279C22×C73#57611yescharacteristic
280C4×C73—22no—
281C4×C73—22no—
282C4×C73—22no—
283C4×C73—42no—
284C4×C73—41no—
285C4×C73—41no—
286C4×C73—41no—
287C4×C73—81no—
288C4×C73—81no—
289C4×C73C2×C4×Q811yescharacteristic
290C4×C73#52011yescharacteristic
291C4×C73#55211yescharacteristic
292C4×C73#55411yescharacteristic
293Dic73—42no—
294Dic73—42no—
295Dic73—42no—
296Dic73—81no—
297#22979—731nomaximal, Sylow
298C2×C73—44no—
299C2×C73#236011yescharacteristic
300C2×C73C4×#13711yescharacteristic
301C2×C73#294811yescharacteristic
302C2×C73#331411yescharacteristic
303C2×C73#331911yescharacteristic
304C2×C73#338511yescharacteristic
305C2×C73#338911yescharacteristic
306C2×#474—731no—
307#2369—731no—
308#2349D7311yescharacteristic
309#1993—731no—
310#1949—731no—
311#1943—731no—
312#1942—731no—
313#1921—731no—
314#1918—731no—
315#1864—731no—
316#1862—731no—
317#1771—731no—
318#1762—731no—
319#1762—731no—
320C4×#102—731no—
321C73#2297911yescharacteristic, complemented, Sylow
322C2×C4⋊Q8—731no—
323C2×C4×D4C2×D7311yescharacteristic
324C2×C4×D4C2×D7311yescharacteristic
325C2×C4×D4C2×D7311yescharacteristic
326#474—738no—
327C2×#107—731no—
328C2×#107—731no—
329C2×#107—731no—
330C2×#106—731no—
331C2×C4⋊C8—731no—
332C2×#102—731no—
333C2×#102—731no—
334C2×#102—731no—
335C2×#102—731no—
336C2×#97—731no—
337C2×#97—731no—
338C2×C4×C8—731no—
339#391—1461no—
340#391—1461no—
341#389—731no—
342#389—731no—
343#389—1461no—
344#389C2×D7311yescharacteristic
345#389C2×D7311yescharacteristic
346#385—21no—
347#385—21no—
348#382C2×D7311yescharacteristic
349#380—21no—
350C4×C4⋊C4—731no—
351C4×C4⋊C4—731no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—1461no—
354C4×C4⋊C4C2×D7311yescharacteristic
355C2×#95—21no—
356#340—731no—
357#340—731no—
358#340—731no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#336—731no—
365#336—731no—
366C22×Q8—1461no—
367C22×D4C22×D7311yescharacteristic
368C23×C4—21no—
369C23×C4—21no—
370C23×C4—21no—
371C22×C8—731no—
372C22×C8—731no—
373C22×C8—731no—
374C22×C8—731no—
375C4⋊Q8—734no—
376C4⋊Q8—1462no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—41no—
381C4×D4—41no—
382C4×D4C4×D7314yes—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—1461no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4C22⋊D7311yescharacteristic
393C2×C4⋊C4C22⋊D7311yescharacteristic
394C2×C4⋊C4C22×D7311yescharacteristic
395C2×C4⋊C4C22×D7311yescharacteristic
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—1461no—
401C2×C42—1461no—
402C2×C42—1461no—
403C2×C42C22×D7311yescharacteristic
404C2×C42C22×D7311yescharacteristic
405C2×C42C22×D7311yescharacteristic
406#107—2921no—
407#107—734no—
408#107—734no—
409#106—2921no—
410C4⋊C8—734no—
411#102—734no—
412#102—734no—
413#102—734no—
414#102—1462no—
415#97—2921no—
416#97—1462no—
417C4×C8—2921no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427#95C22⋊D7311yescharacteristic
428#95C22⋊D7311yescharacteristic
429C24—21no—
430C2×Q8—5841no—
431C2×Q8—5841no—
432C2×Q8—1464no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4C2×C4×D7312yes—
437C2×D4Q8×D7312yes—
438C2×D4#1192402712yes—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4C2×C22⋊D7311yescharacteristic
456C22×C4C2×C22⋊D7311yescharacteristic
457C22×C4C2×C22⋊D7311yescharacteristic
458C22×C4C2×C22⋊D7311yescharacteristic
459C22×C4C2×C22⋊D7311yescharacteristic
460C22×C4C2×C22⋊D7311yescharacteristic
461C22×C4C23×D7311yescharacteristic
462C2×C8—732no—
463C2×C8—732no—
464C2×C8—732no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—5841no—
484C4⋊C4—5841no—
485C4⋊C4—732no—
486C4⋊C4—732no—
487C4⋊C4—732no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1464no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4C2×C4×D7312yes—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—2922no—
503C42—5841no—
504C42—5841no—
505C42C2×C4×D7312yes—
506C42#1192402812yes—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C22×C22⋊D7311yescharacteristic
515Q8—5844no—
516D4—22no—
517D4—22no—
518D4—22no—
519D4#1455655614yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—1462no—
557C2×C4—1462no—
558C2×C4—1462no—
559C2×C4C4×C22⋊D7312yes—
560C2×C4C4×C22⋊D7312yes—
561C2×C4#1455662112yes—
562C2×C4C22×C4×D7311yescharacteristic
563C2×C4C2×Q8×D7311yescharacteristic
564C2×C4C2×#1192402711yescharacteristic
565C2×C4C2×#1192402811yescharacteristic
566C2×C4#1455663611yescharacteristic
567C2×C4#1455663911yescharacteristic
568C8—2921no—
569C8—1462no—
570C8—1462no—
571C8—1462no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×C4×C22⋊D7311yescharacteristic
580C22#1798923111yescharacteristic
581C22Q8×C22⋊D7311yescharacteristic
582C22C2×#1455662111yescharacteristic
583C22#1798938511yescharacteristic
584C22#1798939011yescharacteristic
585C22#1798939011yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—2922no—
596C4—2922no—
597C4—2922no—
598C4—5841no—
599C4C2×#1455655611yescharacteristic
600C4#1798906811yescharacteristic
601C4#1798927211yescharacteristic
602C4#1798929211yescharacteristic
603C2—44no—
604C2#2261293611yescharacteristic
605C2C4×#1455662111yescharacteristic
606C2#2261878111yescharacteristic
607C2#2262201211yescharacteristic
608C2#2262208711yescharacteristic
609C2#2262284111yescharacteristic
610C2#2262288511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2349C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#2349D73
hypocenterC73#22979
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #22613020C2semidirect
3. #22610903C2semidirect
4. #22610697C2semidirect
5. #22610666C2semidirect
6. #22610661C2semidirect
7. #22610520C2semidirect
8. #22610499C2semidirect
11. #22609806C2semidirect
17. #17988867C4semidirect
66. #14556591C2×C4semidirect
78. #14556524C2×C4semidirect
79. #14556524C2×C4semidirect
81. C4⋊#10596831C2×C4semidirect
85. #14556512Q8semidirect
321. C73#22979semidirect

Maximal subgroups

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

subgroupindex
C2×#179888672
#226130202
#226109032
#226106972
#226106662
#226106612
#226105202
#226104992
#226101922
#226101782
#226098062
#226097562
C4×#145565122
C73×#23492
#2297973