Finite Group Databasebeta

← #10702931

Subgroups — #10702931

Order 768.

804 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83768C211yesmaximal, characteristic
2C2×#5132C211yesmaximal, characteristic
3C2×#5005C211yesmaximal, characteristic
4C2×C3×#344C211yesmaximal, characteristic, complemented
5C2×C3⋊#338C211yesmaximal, characteristic, complemented
6C2×#4556C211yesmaximal, characteristic, complemented
7#71574C211yesmaximal, characteristic, complemented
8#11179—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C2×#824—21no—
17#5133—22no—
18#5132—22no—
19#5132—22no—
20#5131—21no—
21C2×#823—21no—
22#5120—21no—
23#5113C414yescomplemented
24#5005—22no—
25#5005—22no—
26C2×#805—22no—
27C3×#344—22no—
28C3⋊#338—22no—
29#4556—22no—
30#4532—22no—
31C2×#421—31no—
32#2896—31no—
33C2×#413—31no—
34C2×#344—31no—
35C2×#344D311yescharacteristic, complemented
36#1733—31no—
37C2×#338—31no—
38C23×C3×C4C2311yescharacteristic
39C22×#67—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×#235—21no—
50C2×#235C2×C412yes—
51C2×#235D412yes—
52C22×D12D411yescharacteristic
53C22×C4×D3—21no—
54C22×Dic6D411yescharacteristic
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×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16D412yes—
67C22×C3×C8—21no—
68C22×C3×C8C2×C412yes—
69C3×#117—21no—
70C3×#117—21no—
71C3×#117—21no—
72C3×#117D412yes—
73C2×C3×C4⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—21no—
76C2×#217—21no—
77#926—21no—
78#926—21no—
79#926—21no—
80#926C2×C412yes—
81C2×C4⋊Dic3—21no—
82C2×C4×Dic3—21no—
83C2×#208—21no—
84C2×#208—21no—
85C2×#208—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208D412yes—
94C2×#208D412yes—
95C22×C3⋊C8—21no—
96C22×C3⋊C8—21no—
97#824—24no—
98#823—24no—
99#821—24no—
100#816—24no—
101#805—28no—
102C22×#41—31no—
103C22×M16—31no—
104C22×M16—31no—
105C22×M16C2×D311yescharacteristic
106C22×M16C2×D311yescharacteristic
107C2×#117—31no—
108C2×#117C2×D311yescharacteristic
109#421—62no—
110#421—62no—
111C2×#104—62no—
112#419—61no—
113#418—61no—
114#417—62no—
115#416—34no—
116#413—62no—
117#413—62no—
118C2×#101—61no—
119C2×#100—61no—
120#344—22no—
121#344—62no—
122#339—62no—
123#338—62no—
124C24×C3#4111yescharacteristic
125C23×D3—21no—
126C22×C3×C4—41no—
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C2×#67—22no—
142C2×#67—22no—
143C2×#67—41no—
144C2×#67—41no—
145C2×#67—41no—
146C2×#67—41no—
147C22×Dic3—21no—
148C22×Dic3—41no—
149#235—22no—
150#235—22no—
151#235—22no—
152#235—22no—
153#235—22no—
154#235—22no—
155#235—22no—
156#235—22no—
157#235—41no—
158#235—41no—
159#235—41no—
160#235—41no—
161#235—41no—
162#235—41no—
163#235—41no—
164#235—41no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12C22⋊C412yes—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—41no—
174C2×C4×D3—41no—
175C2×C4×D3—41no—
176C2×C4×D3—41no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6C22⋊C412yes—
183C3×M16—22no—
184C3×M16—22no—
185C3×M16—41no—
186C3×M16—41no—
187C3×M16—41no—
188C3×M16—41no—
189C3×M16—41no—
190C3×M16—41no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—41no—
196C2×C3×C8—41no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—41no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—41no—
201C3×C42—22no—
202C3×C42—41no—
203#217—22no—
204#217—41no—
205C4⋊Dic3—22no—
206C4⋊Dic3—41no—
207C4×Dic3—22no—
208C4×Dic3—41no—
209#208—22no—
210#208—22no—
211#208—22no—
212#208—22no—
213#208—41no—
214#208—41no—
215#208—41no—
216#208—41no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—41no—
220C2×C3⋊C8—41no—
221C2×C3⋊C8—41no—
222C2×C3⋊C8—41no—
223C2×#41—32no—
224C2×#41—32no—
225C2×#41—61no—
226C2×#41—61no—
227C2×#41—61no—
228C2×#41—61no—
229C2×#41—61no—
230C2×#41—61no—
231C2×#41—61no—
232C2×#41—61no—
233C2×#41—61no—
234C2×#41—61no—
235C22×Q8—31no—
236C22×D4—31no—
237C22×D4—61no—
238C23×C4—61no—
239C23×C4C22×D311yescharacteristic
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—32no—
252C2×M16—32no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16—61no—
259C2×M16—61no—
260C2×M16—61no—
261C2×M16—61no—
262C2×M16—61no—
263C2×M16D1212yes—
264C22×C8—21no—
265C22×C8—61no—
266C22×C8—61no—
267C22×C8C4×D312yes—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—32no—
272#117—61no—
273#117—61no—
274#117—61no—
275#117#6712yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—61no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—61no—
280C2×C42—21no—
281C2×C42—61no—
282#104—68no—
283#103—64no—
284#102—64no—
285#101—64no—
286#100—64no—
287C23×C3—81no—
288C23×C3C4×D411yescharacteristic
289C23×C3#12111yescharacteristic
290C23×C3#12311yescharacteristic
291C23×C3#12311yescharacteristic
292C23×C3#12411yescharacteristic
293C23×C3#12611yescharacteristic
294C23×C3#12611yescharacteristic
295C22×D3—22no—
296C22×D3—41no—
297C22×D3—41no—
298C22×D3—82no—
299C2×C3×C4—42no—
300C2×C3×C4—82no—
301C2×C3×C4C2×C22⋊C411yescharacteristic
302C2×C3×C4#11711yescharacteristic
303C2×C3×C4#11711yescharacteristic
304C2×C3×C4C4×D411yescharacteristic
305C2×C3×C4C4×D411yescharacteristic
306C2×C3×C4C4×D411yescharacteristic
307C2×C3×C4#12011yescharacteristic
308C2×C3×C4#12011yescharacteristic
309C2×C3×C4#12111yescharacteristic
310C2×C3×C4#12111yescharacteristic
311C2×C3×C4#12111yescharacteristic
312C2×C3×C4#12111yescharacteristic
313C2×C3×C4#12111yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4#12111yescharacteristic
316C2×C3×C4#12311yescharacteristic
317C2×C3×C4#12311yescharacteristic
318C2×C3×C4#12311yescharacteristic
319C2×C3×C4#12311yescharacteristic
320C2×C3×C4#12311yescharacteristic
321C2×C3×C4#12311yescharacteristic
322C2×C3×C4#12311yescharacteristic
323C2×C3×C4#12311yescharacteristic
324C2×C3×C4#12411yescharacteristic
325C2×C3×C4#12411yescharacteristic
326C2×C3×C4#12411yescharacteristic
327C2×C3×C4#12611yescharacteristic
328C2×C3×C4#12611yescharacteristic
329#67—42no—
330#67—42no—
331#67—42no—
332#67—42no—
333C2×Dic3—22no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—42no—
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—
352C3⋊C8—42no—
353C3⋊C8—42no—
354C24—61no—
355C24#23511yescharacteristic
356#41—62no—
357#41—62no—
358#41—62no—
359#41—62no—
360#41—62no—
361#41—62no—
362#41—62no—
363#41—62no—
364#41—121no—
365#41—121no—
366#41—121no—
367#41—121no—
368#41—121no—
369#41—121no—
370#41—121no—
371#41—121no—
372C2×Q8—32no—
373C2×Q8—61no—
374C2×Q8—61no—
375C2×Q8—61no—
376C2×Q8—61no—
377C2×Q8—61no—
378C2×D4—32no—
379C2×D4—62no—
380C2×D4—62no—
381C2×D4—61no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—61no—
385C2×D4—61no—
386C2×D4—121no—
387C2×D4—121no—
388C2×D4—121no—
389C2×D4—121no—
390C22×C4—41no—
391C22×C4—61no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—121no—
395C22×C4—121no—
396C22×C4—121no—
397C22×C4—121no—
398C22×C4—121no—
399C22×C4C2×C4×D311yescharacteristic
400C22×C4C2×D1211yescharacteristic
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—41no—
416M16—41no—
417M16—41no—
418M16—41no—
419M16—41no—
420M16—41no—
421M16—62no—
422M16—62no—
423M16—62no—
424M16—62no—
425M16—121no—
426M16—121no—
427M16—121no—
428M16—121no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—62no—
436C2×C8—62no—
437C2×C8—121no—
438C2×C8—121no—
439C2×C8—121no—
440C2×C8—121no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—62no—
444C4⋊C4—121no—
445C22⋊C4—22no—
446C22⋊C4—41no—
447C22⋊C4—62no—
448C22⋊C4—121no—
449C42—22no—
450C42—41no—
451C42—62no—
452C42—121no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3#38611yescharacteristic
461C22×C3#38811yescharacteristic
462C22×C3#38811yescharacteristic
463C22×C3#39411yescharacteristic
464C22×C3#39711yescharacteristic
465C22×C3#39711yescharacteristic
466C22×C3#39711yescharacteristic
467C2×D3—42no—
468C2×D3—82no—
469C2×D3—82no—
470C2×D3—82no—
471C3×C4—82no—
472C3×C4#38111yescharacteristic
473C3×C4#38611yescharacteristic
474C3×C4#38811yescharacteristic
475C3×C4#38811yescharacteristic
476C3×C4#39411yescharacteristic
477C3×C4#39411yescharacteristic
478C3×C4#39411yescharacteristic
479C3×C4#39711yescharacteristic
480Dic3—42no—
481Dic3—82no—
482C23—62no—
483C23—81no—
484C23—121no—
485C23—121no—
486C23—242no—
487C23C4×D1211yescharacteristic
488C23#87611yescharacteristic
489C23#88411yescharacteristic
490C23#88511yescharacteristic
491C23#89411yescharacteristic
492C23#89811yescharacteristic
493C23#92911yescharacteristic
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—242no—
517C2×C4#87211yescharacteristic
518C2×C4#87511yescharacteristic
519C2×C4D3×C22⋊C411yescharacteristic
520C2×C4#67⋊C411yescharacteristic
521C2×C4#88211yescharacteristic
522C2×C4#88311yescharacteristic
523C2×C4#88311yescharacteristic
524C2×C4#88311yescharacteristic
525C2×C4#88411yescharacteristic
526C2×C4#88411yescharacteristic
527C2×C4#88511yescharacteristic
528C2×C4#88611yescharacteristic
529C2×C4#89211yescharacteristic
530C2×C4#89311yescharacteristic
531C2×C4#89411yescharacteristic
532C2×C4#89411yescharacteristic
533C2×C4#89411yescharacteristic
534C2×C4#89411yescharacteristic
535C2×C4#89511yescharacteristic
536C2×C4#89511yescharacteristic
537C2×C4#89811yescharacteristic
538C2×C4#89811yescharacteristic
539C2×C4C4×#6711yescharacteristic
540C2×C4#93011yescharacteristic
541C2×C4#93711yescharacteristic
542C2×C4#93811yescharacteristic
543C2×C4#93911yescharacteristic
544C2×C4#94711yescharacteristic
545C8—42no—
546C8—42no—
547C8—122no—
548C8—122no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3#144011yescharacteristic
557D3—82no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—122no—
566C22—242no—
567C22—242no—
568C22—242no—
569C22#466511yescharacteristic
570C22#467011yescharacteristic
571C22#467511yescharacteristic
572C22#467511yescharacteristic
573C22#494011yescharacteristic
574C22#495911yescharacteristic
575C22#499211yescharacteristic
576C4—82no—
577C4—122no—
578C4—242no—
579C4#466411yescharacteristic
580C4#466911yescharacteristic
581C4#467211yescharacteristic
582C4#467411yescharacteristic
583C4#495611yescharacteristic
584C4#496111yescharacteristic
585C4#499011yescharacteristic
586C4#499111yescharacteristic
587C3#1117911yescharacteristic, complemented, Sylow
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—21no—
595C2—242no—
596C2#6942911yescharacteristic
subgroupG/H
centerC4#4669
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#344C2
socleC2×C3#1440
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#344D3
hypocenterC3#11179
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
4. C2×C3×#344C2semidirect
5. C2×C3⋊#338C2semidirect
6. C2×#4556C2semidirect
7. #71574C2semidirect
23. #5113C4semidirect
35. C2×#344D3semidirect
587. C3#11179semidirect

Maximal subgroups

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

subgroupindex
#837682
C2×#51322
C2×#50052
C2×C3×#3442
C2×C3⋊#3382
C2×#45562
#715742
#111793