Finite Group Databasebeta

← #10819916

Subgroups — #10819916

Order 768.

946 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5175C211yesmaximal, characteristic
2#83843C211yesmaximal, characteristic, complemented
3#78947C211yesmaximal, characteristic, complemented
4#78916C211yesmaximal, characteristic, complemented
5#78572C211yesmaximal, characteristic, complemented
6C3×#1977C211yesmaximal, characteristic, complemented
7#70708C211yesmaximal, characteristic, complemented
8#70702C211yesmaximal, characteristic, complemented
9#70345C211yesmaximal, characteristic, complemented
10#70323C211yesmaximal, characteristic
11#70107C211yesmaximal, characteristic, complemented
12#70103C211yesmaximal, characteristic, complemented
13#70073C211yesmaximal, characteristic, complemented
14#70070C211yesmaximal, characteristic
15#69889C211yesmaximal, characteristic, complemented
16#23198—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18#5812C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#940C2211yescharacteristic
22#5798C2211yescharacteristic
23C22×#214C2211yescharacteristic
24C22×#204C2211yescharacteristic
25#5521C2211yescharacteristic
26#5516C2211yescharacteristic
27C2×C3×#108C2211yescharacteristic
28C3×#419C2211yescharacteristic
29C3×#418C2211yescharacteristic
30C2×C3×#101C2211yescharacteristic
31C2×C3×#100C2211yescharacteristic
32C3×#405C2211yescharacteristic
33#5225C2211yescharacteristic
34#5215C2211yescharacteristic
35#5175C418yescomplemented
36#5164—22no—
37Dic3×SD16—22no—
38C2×#824C2211yescharacteristic
39#5131C2211yescharacteristic
40#5131C2211yescharacteristic
41C2×#823C2211yescharacteristic
42C2×#822C2211yescharacteristic
43#5122C2211yescharacteristic
44#5122C2211yescharacteristic
45#5120C2211yescharacteristic
46#5120C2211yescharacteristic
47#5106C2211yescharacteristic
48#5101C2211yescharacteristic
49C2×C8×Dic3C2211yescharacteristic
50#4998C2211yescharacteristic
51#4817—22no—
52#4788—22no—
53#4758—22no—
54#4757—22no—
55#4710—22no—
56C4×#204—22no—
57#4555C2211yescharacteristic
58#4555C2211yescharacteristic
59#4555C2211yescharacteristic
60#4532C2211yescharacteristic
61C2×#492—31no—
62#2949—31no—
63#2949—31no—
64#2949—31no—
65#2949—31no—
66#1977—31no—
67#1977—31no—
68#1977—31no—
69#1977D311yescharacteristic, complemented
70#1887—31no—
71#1887—31no—
72#1886—31no—
73#1886—31no—
74#1772—31no—
75#1772—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C22×#67—41no—
79C22×D12C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C2×C3×SD16—21no—
82C2×C3×SD16—21no—
83C2×C3×SD16—22no—
84C2×C3×SD16C2×C414yescomplemented
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#117C2311yescharacteristic
90C2×#217—41no—
91Q8×Dic3—22no—
92Q8×Dic3—22no—
93#944—41no—
94#944C2×C414yescomplemented
95C2×#215—21no—
96C2×#215—21no—
97C2×#215—22no—
98C2×#215C2×C414yescomplemented
99#940—41no—
100#940C2×C414yes—
101D4×Dic3—22no—
102D4×Dic3—22no—
103C2×#214—21no—
104C2×#214—21no—
105C2×#214—22no—
106C2×#214C2×C414yescomplemented
107#926—41no—
108#926C2311yescharacteristic
109#926C2311yescharacteristic
110#926C2311yescharacteristic
111C2×Dic3⋊C4—41no—
112C2×C4×Dic3C2311yescharacteristic
113C2×#208C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C2×#204—21no—
116C2×#204—21no—
117C2×#204—22no—
118C2×#204C2×C414yescomplemented
119#893—22no—
120#887—22no—
121D3×C22⋊C4—41no—
122#877—41no—
123C4×D12—22no—
124C4×Dic6—22no—
125C3×#108—22no—
126C3×#103—22no—
127C3×#102—22no—
128C3×#101—41no—
129C3×#100—41no—
130C3×#97—21no—
131C3×C4×C8—21no—
132#835—22no—
133#832—22no—
134#824—41no—
135#823—41no—
136#822—22no—
137#821—22no—
138#821—22no—
139#818D412yescomplemented
140#817D412yescomplemented
141#816—22no—
142#816—22no—
143#815—21no—
144#815—21no—
145#815—21no—
146C8×Dic3—21no—
147C8×Dic3—21no—
148C8×Dic3—21no—
149C8×Dic3C2×C414yescomplemented
150#808D412yescomplemented
151#807D412yescomplemented
152C22×SD16—31no—
153C22×SD16—31no—
154C22×SD16—31no—
155C22×SD16C2×D311yescharacteristic
156C2×C4⋊Q8—31no—
157C2×#127—31no—
158#519—31no—
159#519—31no—
160#518—31no—
161#518—31no—
162#492—38no—
163#440—62no—
164#440—62no—
165#440—62no—
166#440—62no—
167C4×SD16—62no—
168C4×SD16—62no—
169C4×SD16—62no—
170C4×SD16—62no—
171C2×#108—31no—
172C2×#108C2×D311yescharacteristic
173#427—31no—
174#427—31no—
175#419—31no—
176#419—31no—
177#419—31no—
178#419C2×D311yescharacteristic
179#418—31no—
180#418—31no—
181#418—31no—
182#418C2×D311yescharacteristic
183C2×#101—31no—
184C2×#101C2×D311yescharacteristic
185C2×#100—31no—
186C2×#100C2×D311yescharacteristic
187#405—31no—
188#405—31no—
189#405—31no—
190#405C2×D311yescharacteristic
191C2×C4×C8—31no—
192#339—31no—
193#339—31no—
194#339—31no—
195#339—31no—
196C24×C3—21no—
197C23×D3—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8C22×C412yes—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4C22×C412yes—
210C22×C3×C4—21no—
211C22×C3×C4C2411yescharacteristic
212C2×#67—44no—
213C2×#67—84no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—42no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12C22×C412yes—
223C2×C4×D3—42no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6C22×C412yes—
230C3×SD16—24no—
231C3×SD16—24no—
232C3×M16—21no—
233C3×M16—21no—
234C3×M16—21no—
235C3×M16—21no—
236C3×M16—21no—
237C3×M16—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8C22×C412yes—
242C2×C3×C8C2×D411yescharacteristic
243C2×C3×C8C2×D411yescharacteristic
244C2×C3×C8#4112yes—
245C3×C4⋊C4C2×D412yes—
246C3×C22⋊C4—21no—
247C3×C42—21no—
248#217—21no—
249#217—21no—
250#217—21no—
251#217—42no—
252#215—24no—
253#215—24no—
254#214—24no—
255#214—24no—
256(C2×D3)⋊C4—42no—
257C4⋊Dic3—42no—
258C4⋊Dic3C2×D412yes—
259C4⋊Dic3C2×D412yes—
260C4⋊Dic3C2×D412yes—
261Dic3⋊C4—44no—
262Dic3⋊C4—42no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3C22×C412yes—
270#208—21no—
271#208—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8C22×C412yes—
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8#4112yes—
277#204—24no—
278#204—24no—
279C22×Q8—31no—
280C22×Q8C22×D311yescharacteristic
281C22×D4—31no—
282C22×D4—121no—
283C22×D4C22×D311yescharacteristic
284C2×SD16—21no—
285C2×SD16—21no—
286C2×SD16—22no—
287C2×SD16—34no—
288C2×SD16—34no—
289C2×SD16—34no—
290C2×SD16—62no—
291C2×SD16—61no—
292C2×SD16—61no—
293C2×SD16—62no—
294C2×SD16—61no—
295C2×SD16—61no—
296C2×SD16—61no—
297C2×SD16—61no—
298C2×SD16—62no—
299C2×SD16C4×D314yescomplemented
300C2×M16—31no—
301C2×M16C22×D311yescharacteristic
302C2×M16C22×D311yescharacteristic
303C2×M16C22×D311yescharacteristic
304C22×C8—31no—
305C22×C8C22×D311yescharacteristic
306C4⋊Q8—34no—
307C4⋊Q8—121no—
308#127—34no—
309#127—121no—
310C4×Q8—62no—
311C4×Q8—62no—
312C4×Q8—62no—
313C4×Q8—62no—
314C4×D4—62no—
315C4×D4—62no—
316C4×D4—62no—
317C4×D4—62no—
318#117—31no—
319#117—31no—
320#117—31no—
321#117—121no—
322#117—121no—
323#117C22×D311yescharacteristic
324C2×C4⋊C4—121no—
325C2×C22⋊C4—121no—
326C2×C22⋊C4—121no—
327C2×C42—31no—
328#108—22no—
329#108—62no—
330#107—32no—
331#107—32no—
332#106—32no—
333#106—32no—
334#103—22no—
335#103—62no—
336#103—62no—
337#103—62no—
338#102—22no—
339#102—62no—
340#102—62no—
341#102—62no—
342#101—41no—
343#101—121no—
344#100—41no—
345#100—121no—
346#97—21no—
347#97—61no—
348#97—61no—
349#97—61no—
350C4×C8—21no—
351C4×C8—34no—
352C4×C8—61no—
353C4×C8—61no—
354C4×C8—61no—
355C23×C3—22no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—82no—
359C23×C3H(5,𝔽2)11yescharacteristic
360C22×D3—22no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—82no—
364C3×Q8—22no—
365C3×Q8—22no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—41no—
371C3×D4—41no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4C23×C411yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382#67—88no—
383C2×Dic3—24no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—162no—
392D12—22no—
393D12—22no—
394D12—41no—
395D12—41no—
396C4×D3—162no—
397Dic6—22no—
398Dic6—22no—
399Dic6—41no—
400Dic6—41no—
401C3×C8—41no—
402C3×C8—41no—
403C3×C8—41no—
404C3×C8C4×D412yes—
405C3×C8C4×D412yes—
406C3⋊C8—41no—
407C3⋊C8C4×D412yes—
408C3⋊C8C4×D412yes—
409C24—21no—
410C24—61no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8C2×C4×D312yes—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—32no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—124no—
435C2×D4—244no—
436C2×D4C2×C4×D312yes—
437C22×C4—21no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4C23×D311yescharacteristic
443SD16—24no—
444SD16—24no—
445SD16—64no—
446SD16—64no—
447SD16—64no—
448SD16—64no—
449SD16—64no—
450SD16—64no—
451M16—21no—
452M16—21no—
453M16—21no—
454M16—21no—
455M16—21no—
456M16—21no—
457M16—61no—
458M16—61no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—32no—
463C2×C8—31no—
464C2×C8—31no—
465C2×C8—32no—
466C2×C8—61no—
467C2×C8C2×C4×D312yes—
468C2×C8D3×D411yescharacteristic
469C2×C8D3×D411yescharacteristic
470C2×C8#23712yes—
471C4⋊C4—32no—
472C4⋊C4—32no—
473C4⋊C4—32no—
474C4⋊C4—124no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4D3×D412yes—
478C22⋊C4—21no—
479C22⋊C4—61no—
480C22⋊C4—61no—
481C22⋊C4—61no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C42—21no—
485C42—32no—
486C42—61no—
487C42—61no—
488C42—61no—
489C42—61no—
490C42—122no—
491C42—122no—
492C22×C3—22no—
493C22×C3—42no—
494C22×C3—82no—
495C22×C3—82no—
496C22×C3—161no—
497C22×C3#51811yescharacteristic
498C22×C3D4211yescharacteristic
499C22×C3#55911yescharacteristic
500C2×D3—42no—
501C2×D3—82no—
502C2×D3—82no—
503C2×D3—161no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4C2×C4×D411yescharacteristic
507C3×C4C2×C4×D411yescharacteristic
508C3×C4#55011yescharacteristic
509C3×C4#55011yescharacteristic
510Dic3—24no—
511Dic3—42no—
512Dic3—42no—
513Dic3—42no—
514Dic3—42no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—62no—
519C23—82no—
520C23—121no—
521C23—121no—
522C23—242no—
523C23#100911yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—41no—
527Q8—41no—
528Q8—62no—
529Q8—62no—
530Q8—121no—
531Q8—121no—
532D4—22no—
533D4—22no—
534D4—41no—
535D4—41no—
536D4—62no—
537D4—62no—
538D4—121no—
539D4—121no—
540D4—248no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—64no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—482no—
554C2×C4—482no—
555C2×C4C22×C4×D311yescharacteristic
556C2×C4C2×D3×D411yescharacteristic
557C2×C4C2×D3×D411yescharacteristic
558C2×C4C2×#23711yescharacteristic
559C2×C4#100411yescharacteristic
560C2×C4D3×#4111yescharacteristic
561C8—32no—
562C8—32no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—121no—
567C8#67⋊C412yes—
568C8#67⋊C412yes—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—82no—
572C2×C3#236611yescharacteristic
573C2×C3#329112yes—
574D3—82no—
575C22—22no—
576C22—42no—
577C22—82no—
578C22—82no—
579C22—122no—
580C22—161no—
581C22—242no—
582C22—242no—
583C22—481no—
584C22#554511yescharacteristic
585C22#560911yescharacteristic
586C22#568111yescharacteristic
587C4—42no—
588C4—42no—
589C4—64no—
590C4—122no—
591C4—122no—
592C4—122no—
593C4—122no—
594C4C2×#67⋊C411yescharacteristic
595C4C4×D3×D411yescharacteristic
596C4#559911yescharacteristic
597C4#571811yescharacteristic
598C3#2319811yescharacteristic, complemented, Sylow
599C2—21no—
600C2—21no—
601C2—82no—
602C2—242no—
603C2#7630611yescharacteristic
604C2#8093512yes—
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1977C2
socleC22×C3D42
prime radicalC2×#940C22
cosocleC22×C3×C4C24
hypercenter#1977D3
hypocenterC3#23198
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1977

Complements

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

NH ≅ G/Ntype
2. #83843C2semidirect
3. #78947C2semidirect
4. #78916C2semidirect
5. #78572C2semidirect
6. C3×#1977C2semidirect
7. #70708C2semidirect
8. #70702C2semidirect
9. #70345C2semidirect
11. #70107C2semidirect
12. #70103C2semidirect
13. #70073C2semidirect
15. #69889C2semidirect
35. #5175C4semidirect
69. #1977D3semidirect
84. C2×C3×SD16C2×C4semidirect
94. #944C2×C4semidirect
98. C2×#215C2×C4semidirect
106. C2×#214C2×C4semidirect
118. C2×#204C2×C4semidirect
139. #818D4semidirect
140. #817D4semidirect
149. C8×Dic3C2×C4semidirect
150. #808D4semidirect
151. #807D4semidirect
299. C2×SD16C4×D3semidirect
598. C3#23198semidirect

Maximal subgroups

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

subgroupindex
C2×#51752
#838432
#789472
#789162
#785722
C3×#19772
#707082
#707022
#703452
#703232
#701072
#701032
#700732
#700702
#698892
#231983