Finite Group Databasebeta

← #10702939

Subgroups — #10702939

Order 768.

820 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5139C211yesmaximal, characteristic
2#83768C211yesmaximal, characteristic
3C2×#5006C211yesmaximal, characteristic
4C3×#1742C211yesmaximal, characteristic, complemented
5C2×#4557C211yesmaximal, characteristic, complemented
6#71982C211yesmaximal, characteristic, complemented
7#71593C211yesmaximal, characteristic, complemented
8#11180—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C2×#926C2211yescharacteristic
16#5139—22no—
17#5139—22no—
18C2×#825—22no—
19#5133—22no—
20#5131C412yescomplemented
21#5120C412yescomplemented
22#5113—22no—
23#5006—22no—
24#5006—22no—
25C2×#806—22no—
26C3×#343—22no—
27#4557—22no—
28#4555—22no—
29#4527—22no—
30C2×#421—31no—
31#2896—31no—
32C2×#410—31no—
33#1742D311yescharacteristic, complemented
34#1740—31no—
35#1733—31no—
36C2×H(3,ℤ/4)—31no—
37C23×C3×C4C2311yescharacteristic
38C22×#67—21no—
39C2×#235—21no—
40C2×#235—21no—
41C2×#235—21no—
42C2×#235—21no—
43C2×#235—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235D412yes—
50C2×#235D412yes—
51C22×D12C2×C411yescharacteristic
52C22×C4×D3—21no—
53C22×Dic6C2×C411yescharacteristic
54C2×C3×M16—21no—
55C2×C3×M16—21no—
56C2×C3×M16—21no—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16C2×C412yes—
65C2×C3×M16D412yes—
66C22×C3×C8—21no—
67C22×C3×C8—21no—
68C3×#117—21no—
69C3×#117—21no—
70C3×#117—21no—
71C3×#117—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C22⋊C4D412yes—
74C2×C3×C42—21no—
75C2×#217—21no—
76C2×#217—21no—
77C2×#217—21no—
78#926—21no—
79#926—21no—
80#926—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84#926—21no—
85#926—21no—
86#926—21no—
87#926—21no—
88#926C2×C412yes—
89#926D412yes—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—21no—
95C2×C4×Dic3D412yes—
96#825—28no—
97#821—24no—
98#816—24no—
99#806—28no—
100C22×#41—31no—
101C22×M16C2×D311yescharacteristic
102C22×M16C2×D311yescharacteristic
103C2×#117—31no—
104C2×#117—31no—
105C2×#117—31no—
106C2×#117C2×D311yescharacteristic
107#421—62no—
108#421—62no—
109C2×#104—62no—
110#419—32no—
111#418—32no—
112#417—62no—
113#416—62no—
114#410—62no—
115#410—62no—
116C2×#99—62no—
117#343—22no—
118#342—62no—
119#339—62no—
120H(3,ℤ/4)—62no—
121C24×C3C2×D411yescharacteristic
122C23×D3—21no—
123C22×C3×C4—22no—
124C22×C3×C4C22×C411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C2×#67—22no—
139C2×#67—22no—
140C2×#67—41no—
141C2×#67—41no—
142C2×#67—41no—
143C2×#67—41no—
144C22×Dic3—21no—
145C22×Dic3—22no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148#235—22no—
149#235—22no—
150#235—22no—
151#235—22no—
152#235—22no—
153#235—22no—
154#235—22no—
155#235—22no—
156#235—41no—
157#235—41no—
158#235—41no—
159#235—41no—
160#235—41no—
161#235—41no—
162#235—41no—
163#235—41no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12C22⋊C412yes—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—41no—
173C2×C4×D3—41no—
174C2×C4×D3—41no—
175C2×C4×D3—41no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6C22⋊C412yes—
182C3×M16—22no—
183C3×M16—22no—
184C3×M16—22no—
185C3×M16—22no—
186C3×M16—41no—
187C3×M16—41no—
188C3×M16—41no—
189C3×M16—41no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—41no—
193C2×C3×C8—41no—
194C2×C3×C8—41no—
195C2×C3×C8—41no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—41no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C42—41no—
201C3×C42—41no—
202#217—22no—
203#217—22no—
204#217—41no—
205#217—41no—
206#217—41no—
207#217—41no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3—41no—
211C4⋊Dic3—41no—
212C4⋊Dic3—41no—
213C4⋊Dic3—41no—
214C4×Dic3—22no—
215C4×Dic3—22no—
216C4×Dic3—22no—
217C4×Dic3—22no—
218C4×Dic3—41no—
219C4×Dic3—41no—
220C2×#41—32no—
221C2×#41—32no—
222C2×#41—61no—
223C2×#41—61no—
224C2×#41—61no—
225C2×#41—61no—
226C2×#41—61no—
227C2×#41—61no—
228C2×#41—61no—
229C2×#41—61no—
230C2×#41—61no—
231C2×#41—61no—
232C22×Q8—31no—
233C22×D4—31no—
234C22×D4—61no—
235C23×C4—61no—
236C23×C4C22×D311yescharacteristic
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16C4×D312yes—
248C2×M16#6712yes—
249C22×C8—21no—
250C22×C8—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—32no—
256#117—32no—
257#117—61no—
258#117—61no—
259#117—61no—
260#117—61no—
261#117—61no—
262#117—61no—
263#117—61no—
264#117—61no—
265#117—61no—
266#117—61no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4D1212yes—
275C2×C42—21no—
276C2×C42—32no—
277C2×C42—61no—
278C2×C42—61no—
279#104—68no—
280#103—64no—
281#102—64no—
282#99—68no—
283C23×C3—81no—
284C23×C3C4×D411yescharacteristic
285C23×C3#12011yescharacteristic
286C23×C3#12111yescharacteristic
287C23×C3#12111yescharacteristic
288C23×C3#12111yescharacteristic
289C23×C3#12311yescharacteristic
290C23×C3#12411yescharacteristic
291C22×D3—22no—
292C22×D3—41no—
293C22×D3—41no—
294C22×D3—82no—
295C2×C3×C4—42no—
296C2×C3×C4—82no—
297C2×C3×C4C2×C22⋊C411yescharacteristic
298C2×C3×C4#11711yescharacteristic
299C2×C3×C4#11711yescharacteristic
300C2×C3×C4C4×D411yescharacteristic
301C2×C3×C4C4×D411yescharacteristic
302C2×C3×C4C4×D411yescharacteristic
303C2×C3×C4#12011yescharacteristic
304C2×C3×C4#12111yescharacteristic
305C2×C3×C4#12111yescharacteristic
306C2×C3×C4#12111yescharacteristic
307C2×C3×C4#12111yescharacteristic
308C2×C3×C4#12111yescharacteristic
309C2×C3×C4#12311yescharacteristic
310C2×C3×C4#12311yescharacteristic
311C2×C3×C4#12311yescharacteristic
312C2×C3×C4#12311yescharacteristic
313C2×C3×C4#12311yescharacteristic
314C2×C3×C4#12311yescharacteristic
315C2×C3×C4#12311yescharacteristic
316C2×C3×C4#12311yescharacteristic
317C2×C3×C4#12311yescharacteristic
318C2×C3×C4#12411yescharacteristic
319C2×C3×C4#12411yescharacteristic
320C2×C3×C4#12411yescharacteristic
321C2×C3×C4#12611yescharacteristic
322C2×C3×C4#12611yescharacteristic
323C2×C3×C4#12611yescharacteristic
324C2×C3×C4#12611yescharacteristic
325#67—42no—
326#67—42no—
327#67—42no—
328#67—42no—
329C2×Dic3—22no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—82no—
336C2×Dic3—82no—
337C2×Dic3—82no—
338D12—22no—
339D12—22no—
340D12—41no—
341D12—41no—
342C4×D3—42no—
343C4×D3—42no—
344C4×D3—42no—
345C4×D3—42no—
346Dic6—22no—
347Dic6—22no—
348Dic6—41no—
349Dic6—41no—
350C3×C8—42no—
351C3×C8—42no—
352C24—61no—
353C24C2×D1211yescharacteristic
354#41—62no—
355#41—62no—
356#41—62no—
357#41—62no—
358#41—62no—
359#41—62no—
360#41—62no—
361#41—62no—
362#41—121no—
363#41—121no—
364#41—121no—
365#41—121no—
366#41—121no—
367#41—121no—
368#41—121no—
369#41—121no—
370C2×Q8—32no—
371C2×Q8—61no—
372C2×Q8—61no—
373C2×Q8—61no—
374C2×Q8—61no—
375C2×Q8—61no—
376C2×D4—32no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—121no—
385C2×D4—121no—
386C2×D4—121no—
387C2×D4—121no—
388C22×C4—22no—
389C22×C4—61no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—121no—
394C22×C4—121no—
395C22×C4—121no—
396C22×C4—121no—
397C22×C4—121no—
398C22×C4—121no—
399C22×C4C2×C4×D311yescharacteristic
400C22×C4#23511yescharacteristic
401C22×C4#23511yescharacteristic
402C22×C4#23511yescharacteristic
403C22×C4#23511yescharacteristic
404C22×C4D3×D411yescharacteristic
405C22×C4D3×D411yescharacteristic
406C22×C4D3×D411yescharacteristic
407C22×C4D3×D411yescharacteristic
408C22×C4#23711yescharacteristic
409C22×C4#23711yescharacteristic
410C22×C4#23911yescharacteristic
411C22×C4#23911yescharacteristic
412C22×C4C2×#6711yescharacteristic
413M16—22no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—41no—
418M16—41no—
419M16—41no—
420M16—41no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—62no—
430C4⋊C4—62no—
431C4⋊C4—121no—
432C4⋊C4—121no—
433C4⋊C4—121no—
434C4⋊C4—121no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—62no—
438C22⋊C4—62no—
439C22⋊C4—121no—
440C22⋊C4—121no—
441C22⋊C4—121no—
442C22⋊C4—121no—
443C42—41no—
444C42—41no—
445C42—62no—
446C42—62no—
447C42—62no—
448C42—62no—
449C42—121no—
450C42—121no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3#38611yescharacteristic
459C22×C3#38811yescharacteristic
460C22×C3#38811yescharacteristic
461C22×C3#39411yescharacteristic
462C22×C3#39411yescharacteristic
463C22×C3#39411yescharacteristic
464C22×C3#39711yescharacteristic
465C2×D3—42no—
466C2×D3—82no—
467C2×D3—82no—
468C2×D3—82no—
469C3×C4—82no—
470C3×C4#38111yescharacteristic
471C3×C4#38611yescharacteristic
472C3×C4#38811yescharacteristic
473C3×C4#38811yescharacteristic
474C3×C4#39411yescharacteristic
475C3×C4#39711yescharacteristic
476C3×C4#39711yescharacteristic
477C3×C4#39711yescharacteristic
478Dic3—42no—
479Dic3—82no—
480Dic3—82no—
481Dic3—82no—
482C23—62no—
483C23—81no—
484C23—121no—
485C23—121no—
486C23—242no—
487C23C4×D1211yescharacteristic
488C23#87511yescharacteristic
489C23#88211yescharacteristic
490C23#88611yescharacteristic
491C23#89511yescharacteristic
492C23#89511yescharacteristic
493C23#93011yescharacteristic
494Q8—62no—
495Q8—62no—
496Q8—121no—
497Q8—121no—
498D4—62no—
499D4—62no—
500D4—122no—
501D4—122no—
502D4—122no—
503D4—122no—
504D4—121no—
505D4—121no—
506C2×C4—42no—
507C2×C4—62no—
508C2×C4—82no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—242no—
519C2×C4—242no—
520C2×C4—242no—
521C2×C4#87211yescharacteristic
522C2×C4#87611yescharacteristic
523C2×C4D3×C22⋊C411yescharacteristic
524C2×C4#67⋊C411yescharacteristic
525C2×C4#88311yescharacteristic
526C2×C4#88311yescharacteristic
527C2×C4#88311yescharacteristic
528C2×C4#88411yescharacteristic
529C2×C4#88411yescharacteristic
530C2×C4#88411yescharacteristic
531C2×C4#88511yescharacteristic
532C2×C4#88511yescharacteristic
533C2×C4#89211yescharacteristic
534C2×C4#89311yescharacteristic
535C2×C4#89411yescharacteristic
536C2×C4#89411yescharacteristic
537C2×C4#89411yescharacteristic
538C2×C4#89411yescharacteristic
539C2×C4#89411yescharacteristic
540C2×C4#89811yescharacteristic
541C2×C4#89811yescharacteristic
542C2×C4#89811yescharacteristic
543C2×C4C4×#6711yescharacteristic
544C2×C4#92911yescharacteristic
545C2×C4#93711yescharacteristic
546C2×C4#93811yescharacteristic
547C2×C4#93911yescharacteristic
548C2×C4#94711yescharacteristic
549C8—42no—
550C8—42no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3#144011yescharacteristic
559D3—82no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—122no—
568C22—242no—
569C22—242no—
570C22—242no—
571C22#466511yescharacteristic
572C22#467011yescharacteristic
573C22#467211yescharacteristic
574C22#467411yescharacteristic
575C22#494011yescharacteristic
576C22#496111yescharacteristic
577C22#499111yescharacteristic
578C4—82no—
579C4—122no—
580C4—242no—
581C4—242no—
582C4—242no—
583C4#466411yescharacteristic
584C4#466911yescharacteristic
585C4#467511yescharacteristic
586C4#467511yescharacteristic
587C4#495611yescharacteristic
588C4#495911yescharacteristic
589C4#499011yescharacteristic
590C4#499211yescharacteristic
591C3#1118011yescharacteristic, complemented, Sylow
592C2—21no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—242no—
600C2#6942911yescharacteristic
subgroupG/H
centerC4#4675
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1742C2
socleC2×C3#1440
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1742D3
hypocenterC3#11180
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
4. C3×#1742C2semidirect
5. C2×#4557C2semidirect
6. #71982C2semidirect
7. #71593C2semidirect
20. #5131C4semidirect
21. #5120C4semidirect
33. #1742D3semidirect
591. C3#11180semidirect

Maximal subgroups

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

subgroupindex
C2×#51392
#837682
C2×#50062
C3×#17422
C2×#45572
#719822
#715932
#111803