Finite Group Databasebeta

← #33149010

Subgroups — #33149010

Order 18688.

864 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988872C211yesmaximal, characteristic, complemented
2C2×C4×#11924015C211yesmaximal, characteristic
3#22613781C211yesmaximal, characteristic
4#22611465C211yesmaximal, characteristic
5#22611408C211yesmaximal, characteristic, complemented
6#22611302C211yesmaximal, characteristic
7#22611264C211yesmaximal, characteristic, complemented
8#22610656C211yesmaximal, characteristic, complemented
9#22610490C211yesmaximal, characteristic, complemented
10#22610286C211yesmaximal, characteristic
11#22610286C211yesmaximal, characteristic
12#22610199C211yesmaximal, characteristic, complemented
13#22610199C211yesmaximal, characteristic, complemented
14#22610191C211yesmaximal, characteristic
15Q8×C4⋊C4×C73C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×#14556591C2211yescharacteristic
18C2×#14556569—21no—
19C2×C4×D292C2211yescharacteristic
20#17988872—24no—
21C4×#11924015—24no—
22C2×#14556522C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556515C2211yescharacteristic
30C2×#14556515C2211yescharacteristic
31C2×#14556511C2211yescharacteristic
32C2×#14556511C2211yescharacteristic
33C2×#14556511C2211yescharacteristic
34C2×C4×#10596831C2211yescharacteristic
35#17988509—21no—
36#17988500—21no—
37#17988489C2211yescharacteristic
38#17988489C2211yescharacteristic
39#17988485C2211yescharacteristic
40#17988485C2211yescharacteristic
41#17988461—21no—
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×C4×Q8×C73C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52C2×C4×Q8×C73C2211yescharacteristic
53#389×C73—21no—
54#389×C73—21no—
55#389×C73C2211yescharacteristic
56#389×C73C2211yescharacteristic
57#384×C73—21no—
58#384×C73C2211yescharacteristic
59C4×C4⋊C4×C73—21no—
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×Dic146—21no—
62C22×D292C2311yescharacteristic
63C22×C4×D73—21no—
64C2×#11924015—22no—
65C2×#11924015—22no—
66C2×#11924015Q814yescomplemented
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71#14556591—22no—
72#14556591—41no—
73#14556569—44no—
74C4×D292—22no—
75C4×D292—41no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×#11924005—21no—
87#14556522—22no—
88#14556522—22no—
89#14556522—22no—
90#14556522—22no—
91#14556522—41no—
92C4⋊#10596831—22no—
93#14556515—22no—
94#14556515—22no—
95#14556515D414yes—
96#14556511—22no—
97#14556511—22no—
98#14556511—41no—
99C4×#10596831—22no—
100#14556498—21no—
101#14556498—21no—
102C22×Q8×C73C2311yescharacteristic
103C4×Q8×C73—22no—
104C4×Q8×C73—22no—
105C4×Q8×C73—22no—
106C4×Q8×C73—22no—
107C4×Q8×C73—41no—
108C4×Q8×C73—41no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73—21no—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73D411yescharacteristic
117C2×C4⋊C4×C73D411yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C42×C73—21no—
122C2×C42×C73—21no—
123C2×C42×C73—21no—
124C2×C42×C73C2311yescharacteristic
125C2×C42×C73C2311yescharacteristic
126C2×C42×C73C2311yescharacteristic
127#95×C73—21no—
128#95×C73—21no—
129#95×C73D411yescharacteristic
130#95×C73D411yescharacteristic
131C23×D73—21no—
132C2×Dic146—42no—
133C2×Dic146—42no—
134C2×Dic146—81no—
135C2×D292—21no—
136C2×D292—21no—
137C2×D292—21no—
138C2×D292—21no—
139C2×D292—21no—
140C2×D292C2×Q812yes—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C2×C4×D73—42no—
145C2×C4×D73—42no—
146C2×C4×D73—81no—
147#11924015—28no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831C2×D412yes—
159C2×#10596831C2×Q812yes—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—42no—
166C4⋊Dic73C2×D412yes—
167#11924007—42no—
168#11924007—42no—
169#11924005—42no—
170#11924005—42no—
171#11924005—42no—
172#11924005—42no—
173#11924005—81no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C2×Q8×C73C2×D412yes—
178C2×Q8×C73C2×Q812yes—
179C2×Q8×C73#4112yes—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2411yescharacteristic
191C4⋊C4×C73—24no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—41no—
198C4⋊C4×C73—42no—
199C4⋊C4×C73—42no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73—81no—
202C4⋊C4×C73—81no—
203C4⋊C4×C73—81no—
204C4⋊C4×C73SD1614yescomplemented
205C4⋊C4×C73#4112yes—
206C42×C73—24no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73—21no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—42no—
214C42×C73—42no—
215C23×C73C22×D411yescharacteristic
216C22×D73—22no—
217C22×D73—41no—
218C22×D73—41no—
219C22×D73—41no—
220C22×D73—41no—
221C22×D73—41no—
222C22×D73—41no—
223Q8×C73—22no—
224Q8×C73—22no—
225Q8×C73—22no—
226Q8×C73C22⋊Q814yescomplemented
227C2×C4×C73—21no—
228C2×C4×C73—22no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—22no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73C2×SD1612yes—
255C2×C4×C73C2×SD1612yes—
256C2×C4×C73C2×SD1612yes—
257C2×C4×C73C22×D411yescharacteristic
258C2×C4×C73C22×Q811yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C2×C4×C73C2×#4111yescharacteristic
261C2×C4×C73H(5,𝔽2)11yescharacteristic
262C2×C4×C732−1+411yescharacteristic
263C2×Dic73—22no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272D292—22no—
273D292—22no—
274D292—41no—
275D292—41no—
276C4×D73—84no—
277Dic146—84no—
278#10596831—22no—
279#10596831—22no—
280#10596831—41no—
281#10596831—41no—
282C22×C73#52511yescharacteristic
283C22×C73D4211yescharacteristic
284C22×C73D4×Q811yescharacteristic
285C22×C73C22×SD1611yescharacteristic
286C22×C73#57511yescharacteristic
287C22×C73#57711yescharacteristic
288C22×C73#57711yescharacteristic
289C2×D73—42no—
290C2×D73—42no—
291C2×D73—42no—
292C2×D73—81no—
293C2×D73—81no—
294C2×D73—81no—
295C2×D73—81no—
296C4×C73—22no—
297C4×C73—24no—
298C4×C73—22no—
299C4×C73—22no—
300C4×C73—41no—
301C4×C73—42no—
302C4×C73—41no—
303C4×C73—41no—
304C4×C73—81no—
305C4×C73—81no—
306C4×C73C2×C22⋊Q811yescharacteristic
307C4×C73#53911yescharacteristic
308C4×C73#54711yescharacteristic
309C4×C73#55411yescharacteristic
310Dic73—42no—
311Dic73—81no—
312Dic73—81no—
313#26373—731nomaximal, Sylow
314C2×C73#244911yescharacteristic
315C2×C73#310311yescharacteristic
316C2×C73#312511yescharacteristic
317C2×C73#327111yescharacteristic
318C2×C73D4×SD1611yescharacteristic
319C2×C73Q8×SD1611yescharacteristic
320C2×C73#338811yescharacteristic
321D73—82no—
322C2×#475—731no—
323C2×C4×SD16—731no—
324#2452—731no—
325Q8×C4⋊C4D7311yescharacteristic, complemented
326#2094—731no—
327#2084—731no—
328#2065—731no—
329#2058—731no—
330#1941—731no—
331#1916—731no—
332#1880—731no—
333#1880—731no—
334#1865—731no—
335#1865—731no—
336#1864—731no—
337C73#2637311yescharacteristic, complemented, Sylow
338C22×SD16—731no—
339C2×C4⋊Q8—731no—
340C2×C22⋊Q8—1461no—
341C2×C4×Q8C2×D7311yescharacteristic
342C2×C4×Q8C2×D7311yescharacteristic
343C2×C4×Q8C2×D7311yescharacteristic
344C2×C4×D4—731no—
345#475—1464no—
346C4×SD16—1464no—
347C2×#106—731no—
348C2×#106—731no—
349C2×#106—731no—
350C2×#106—731no—
351C2×#106—731no—
352C2×C4⋊C8—731no—
353C2×#103—731no—
354C2×#103—731no—
355C2×#103—731no—
356C2×#102—731no—
357C2×#102—731no—
358C2×#102—731no—
359C2×C4×C8—731no—
360#396—1461no—
361#393—1461no—
362#390—731no—
363#390—731no—
364#389—21no—
365#389—21no—
366#389—731no—
367#389—731no—
368#389C2×D7311yescharacteristic
369#389C2×D7311yescharacteristic
370#385—1461no—
371#384—21no—
372#384C2×D7311yescharacteristic
373C4×C4⋊C4—21no—
374C4×C4⋊C4C2×D7311yescharacteristic
375#340—731no—
376#340—731no—
377#340—731no—
378#340—731no—
379#336—731no—
380#336—731no—
381#336—731no—
382#336—731no—
383C22×Q8—1461no—
384C22×Q8C22×D7311yescharacteristic
385C22×D4—731no—
386C23×C4—1461no—
387C2×SD16—734no—
388C2×SD16—1462no—
389C2×SD16—1462no—
390C22×C8—731no—
391C22×C8—731no—
392C22×C8—731no—
393C22×C8—731no—
394C4⋊Q8—2921no—
395C4⋊Q8—1462no—
396C22⋊Q8—2924no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×Q8—22no—
400C4×Q8—22no—
401C4×Q8—41no—
402C4×Q8—41no—
403C4×D4—2921no—
404C4×D4—1462no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—1461no—
416C2×C4⋊C4—1461no—
417C2×C4⋊C4—1461no—
418C2×C4⋊C4C22⋊D7311yescharacteristic
419C2×C4⋊C4C22⋊D7311yescharacteristic
420C2×C4⋊C4C22×D7311yescharacteristic
421C2×C4⋊C4C22×D7311yescharacteristic
422C2×C4⋊C4C22×D7311yescharacteristic
423C2×C22⋊C4—1461no—
424C2×C22⋊C4—1461no—
425C2×C22⋊C4—1461no—
426C2×C22⋊C4—1461no—
427C2×C22⋊C4—1461no—
428C2×C42—21no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42C22×D7311yescharacteristic
432C2×C42C22×D7311yescharacteristic
433C2×C42C22×D7311yescharacteristic
434#106—2921no—
435#106—1462no—
436#106—1462no—
437#106—1462no—
438#106—1462no—
439C4⋊C8—1462no—
440#103—734no—
441#103—1462no—
442#103—1462no—
443#102—2921no—
444#102—1462no—
445#102—1462no—
446C4×C8—1462no—
447#95—21no—
448#95—21no—
449#95—1461no—
450#95—1461no—
451#95C22⋊D7311yescharacteristic
452#95C22⋊D7311yescharacteristic
453C24—1461no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—2922no—
458C2×Q8—2922no—
459C2×Q8—5841no—
460C2×Q8D4×D7312yes—
461C2×Q8Q8×D7312yes—
462C2×Q8#1192402812yes—
463C2×D4—732no—
464C2×D4—1461no—
465C2×D4—1461no—
466C2×D4—1461no—
467C2×D4—1461no—
468C2×D4—1461no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—2922no—
474C22×C4—2922no—
475C22×C4—5841no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4C2×C22⋊D7311yescharacteristic
480C22×C4C2×C22⋊D7311yescharacteristic
481C22×C4C2×C22⋊D7311yescharacteristic
482C22×C4C2×C22⋊D7311yescharacteristic
483C22×C4C2×C22⋊D7311yescharacteristic
484C22×C4C2×C22⋊D7311yescharacteristic
485C22×C4C23×D7311yescharacteristic
486SD16—1468no—
487C2×C8—732no—
488C2×C8—732no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1461no—
497C2×C8—1461no—
498C2×C8—1461no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—2922no—
513C4⋊C4—2922no—
514C4⋊C4—2922no—
515C4⋊C4—732no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4—1461no—
520C4⋊C4—1461no—
521C4⋊C4#1192401514yescomplemented
522C4⋊C4#1192402812yes—
523C22⋊C4—2922no—
524C22⋊C4—2922no—
525C22⋊C4—2922no—
526C22⋊C4—2922no—
527C22⋊C4—5841no—
528C42—24no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—42no—
537C23—2921no—
538C23—2921no—
539C23—2921no—
540C23—2921no—
541C23—2921no—
542C23—2921no—
543C23—1462no—
544C23C22×C22⋊D7311yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—5844no—
549Q8#1455656914yescomplemented
550D4—2921no—
551D4—2921no—
552D4—1462no—
553D4—1462no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—5844no—
590C2×C4—1462no—
591C2×C4C2×#1192401512yes—
592C2×C4C2×#1192401512yes—
593C2×C4C2×#1192401512yes—
594C2×C4C2×D4×D7311yescharacteristic
595C2×C4C2×Q8×D7311yescharacteristic
596C2×C4C2×#1192402811yescharacteristic
597C2×C4C2×#1192402811yescharacteristic
598C2×C4#1455663611yescharacteristic
599C2×C4#1455663911yescharacteristic
600C8—2921no—
601C8—2921no—
602C8—1462no—
603C8—1462no—
604C22—2922no—
605C22—2922no—
606C22—2922no—
607C22—5841no—
608C22—5841no—
609C22—5841no—
610C22—5841no—
611C22#1798909811yescharacteristic
612C22D4×C22⋊D7311yescharacteristic
613C22Q8×C22⋊D7311yescharacteristic
614C22C22×#1192401511yescharacteristic
615C22#1798938511yescharacteristic
616C22#1798939511yescharacteristic
617C22#1798939811yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—24no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—2922no—
629C4—5841no—
630C4—5841no—
631C4C2×#1455656911yescharacteristic
632C4#1798918411yescharacteristic
633C4#1798923811yescharacteristic
634C4#1798929211yescharacteristic
635C2—5842no—
636C2#2261374011yescharacteristic
637C2#2261996211yescharacteristic
638C2#2262020811yescharacteristic
639C2#2262151211yescharacteristic
640C2D4×#1192401511yescharacteristic
641C2Q8×#1192401511yescharacteristic
642C2#2262287011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingQ8×C4⋊C4×C73C2
socleC23×C73C22×D4
prime radicalC22×D292C23
hypercenterQ8×C4⋊C4D73
hypocenterC73#26373
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#17988872C2semidirect
5. #22611408C2semidirect
7. #22611264C2semidirect
8. #22610656C2semidirect
9. #22610490C2semidirect
12. #22610199C2semidirect
13. #22610199C2semidirect
15. Q8×C4⋊C4×C73C2semidirect
66. C2×#11924015Q8semidirect
204. C4⋊C4×C73SD16semidirect
226. Q8×C73C22⋊Q8semidirect
325. Q8×C4⋊C4D73semidirect
337. C73#26373semidirect
521. C4⋊C4#11924015semidirect
549. Q8#14556569semidirect

Maximal subgroups

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

subgroupindex
C2×#179888722
C2×C4×#119240152
#226137812
#226114652
#226114082
#226113022
#226112642
#226106562
#226104902
#226102862
#226101992
#226101912
Q8×C4⋊C4×C732
#2637373