Finite Group Databasebeta

← #10690721

Subgroups — #10690721

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78601C211yesmaximal, characteristic, complemented
2#73569C211yesmaximal, characteristic
3#69919C211yesmaximal, characteristic, complemented
4C2×#4468C211yesmaximal, characteristic, complemented
5C2×#4468C211yesmaximal, characteristic, complemented
6C3×#1282C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9107—31nomaximal, Sylow
9C2×C4×#67C2211yescharacteristic
10C22×(C2×D3)⋊C4C2211yescharacteristic
11C2×#882C411yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C2×#830C2211yescharacteristic
16C2×#830C2211yescharacteristic
17C8×#67—22no—
18#4958C411yescharacteristic
19#4955C411yescharacteristic
20#4950C411yescharacteristic
21#4940—21no—
22#4939—21no—
23#67⋊C8—22no—
24D3×C22⋊C8—21no—
25#4718—21no—
26#4670—21no—
27#4665—21no—
28#4469—21no—
29#4468—21no—
30#4468—22no—
31#4468—22no—
32#2972—31no—
33#2321—31no—
34#1459—31no—
35C2×#323—31no—
36C2×#323—31no—
37#1282—31no—
38#1282D311yescharacteristic, complemented
39C24×D3C2×C411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×#67C2×C411yescharacteristic
42C23×Dic3C2×C411yescharacteristic
43C22×D12—21no—
44C22×C4×D3D411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C2×C3×C22⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4C2×C411yescharacteristic
49C2×C3×C42—21no—
50C2×#217C2×C411yescharacteristic
51C4×#67—21no—
52C4×#67—21no—
53C4×#67—21no—
54C4×#67—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4D411yescharacteristic
66C2×C4⋊Dic3—21no—
67C2×Dic3⋊C4D411yescharacteristic
68C2×C4×Dic3D411yescharacteristic
69C22×C3⋊C8—21no—
70C22×C3⋊C8—21no—
71C2×D3×C8—22no—
72#882—22no—
73#882C814yescomplemented
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8D412yescomplemented
78C3×#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830D412yescomplemented
85(C2×D3)⋊C8—22no—
86Dic3⋊C8—22no—
87C8×Dic3—22no—
88C2×#120—31no—
89C2×C4×D4—31no—
90C22×C22⋊C4—31no—
91C8×D4—62no—
92C8×D4—62no—
93#431—61no—
94C2×C22⋊C8—31no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8—61no—
97C2×C22⋊C8C2×D311yescharacteristic
98C2×C22⋊C8C2×D311yescharacteristic
99#390—61no—
100#388—61no—
101#388—61no—
102#386—31no—
103#386—61no—
104#385—31no—
105#379—31no—
106C4×C22⋊C4C2×D311yescharacteristic
107#324—61no—
108#323—62no—
109#323—61no—
110#323—62no—
111C24×C3C22×C411yescharacteristic
112C23×D3—21no—
113C23×D3—21no—
114C23×D3—21no—
115C23×D3—21no—
116C23×D3—21no—
117C23×D3—21no—
118C23×D3—81no—
119C23×D3—81no—
120C23×D3C22⋊C411yescharacteristic
121C23×D3C2×C812yescomplemented
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—81no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×#67—21no—
132C2×#67—21no—
133C2×#67—21no—
134C2×#67—21no—
135C2×#67C2×C812yes—
136C2×#67M1612yes—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—81no—
141C22×Dic3C22⋊C411yescharacteristic
142C22×Dic3C22⋊C411yescharacteristic
143C22×Dic3C22⋊C411yescharacteristic
144C2×D12—24no—
145C2×D12—81no—
146C2×D12—81no—
147C2×C4×D3—21no—
148C2×C4×D3—21no—
149C2×C4×D3—81no—
150C2×C3×C8—22no—
151C2×C3×C8—41no—
152C2×C3×C8—81no—
153C2×C3×C8—81no—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—21no—
156C3×C22⋊C4—21no—
157C3×C22⋊C4C2×C812yescomplemented
158C3×C42—81no—
159#217—21no—
160#217—21no—
161(C2×D3)⋊C4—21no—
162(C2×D3)⋊C4—21no—
163(C2×D3)⋊C4—24no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—41no—
168(C2×D3)⋊C4—81no—
169(C2×D3)⋊C4—81no—
170(C2×D3)⋊C4—81no—
171C4⋊Dic3—81no—
172Dic3⋊C4—21no—
173Dic3⋊C4—21no—
174C4×Dic3—21no—
175C4×Dic3—21no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—81no—
179C2×C3⋊C8—81no—
180D3×C8—82no—
181C25—31no—
182C22×D4—31no—
183C22×D4—61no—
184C23×C4—31no—
185C23×C4—31no—
186C23×C4C22×D311yescharacteristic
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—62no—
190C22×C8—61no—
191C22×C8—61no—
192#120—34no—
193#120—62no—
194C4×D4—61no—
195C4×D4—61no—
196C4×D4—61no—
197C4×D4—61no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—61no—
200C2×C22⋊C4—31no—
201C2×C22⋊C4—31no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4C4×D311yescharacteristic
213C2×C22⋊C4C4×D311yescharacteristic
214C2×C42—21no—
215C2×C42—31no—
216C4⋊C8—62no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220C22⋊C8—32no—
221C22⋊C8—62no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225C22⋊C8D1212yescomplemented
226C4×C8—62no—
227#95—21no—
228#95—61no—
229#95—61no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3C2×C22⋊C411yescharacteristic
233C23×C3C4×D411yescharacteristic
234C23×C3C4×D411yescharacteristic
235C23×C3C22×C811yescharacteristic
236C23×C3C2×M1611yescharacteristic
237C23×C3#13111yescharacteristic
238C23×C3#13111yescharacteristic
239C22×D3—21no—
240C22×D3—21no—
241C22×D3—22no—
242C22×D3—22no—
243C22×D3—22no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—41no—
252C22×D3—41no—
253C22×D3—41no—
254C22×D3—82no—
255C22×D3—82no—
256C22×D3—81no—
257C22×D3—81no—
258C22×D3—81no—
259C22×D3—81no—
260C22×D3—82no—
261C22×D3—82no—
262C22×D3C22⋊C812yes—
263C2×C3×C4—22no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4#12011yescharacteristic
276C2×C3×C4#12111yescharacteristic
277C2×C3×C4#12111yescharacteristic
278C2×C3×C4#12311yescharacteristic
279#67—22no—
280#67—22no—
281#67—22no—
282#67—22no—
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—81no—
293C2×Dic3—81no—
294C2×Dic3C22⋊C812yes—
295D12—84no—
296C4×D3—42no—
297C4×D3—81no—
298C3×C8—42no—
299C3×C8—81no—
300C3⋊C8—42no—
301C3⋊C8—81no—
302C24—31no—
303C24—32no—
304C24—61no—
305C24—61no—
306C24—61no—
307C24—61no—
308C24—61no—
309C24—61no—
310C24—241no—
311C24—241no—
312C24C2×C4×D311yescharacteristic
313C2×D4—32no—
314C2×D4—32no—
315C2×D4—64no—
316C2×D4—61no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—241no—
321C2×D4—241no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—61no—
328C22×C4—61no—
329C22×C4—61no—
330C22×C4—61no—
331C22×C4—61no—
332C22×C4—81no—
333C22×C4—241no—
334C22×C4—241no—
335C22×C4C2×D1211yescharacteristic
336C22×C4#23511yescharacteristic
337C22×C4D3×D411yescharacteristic
338C22×C4D3×D411yescharacteristic
339C22×C4D3×D411yescharacteristic
340C22×C4#23911yescharacteristic
341C2×C8—22no—
342C2×C8—41no—
343C2×C8—62no—
344C2×C8—81no—
345C2×C8—81no—
346C2×C8—121no—
347C2×C8—242no—
348C2×C8—241no—
349C2×C8—241no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—241no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—64no—
357C22⋊C4—61no—
358C22⋊C4—61no—
359C22⋊C4—61no—
360C22⋊C4—61no—
361C22⋊C4—121no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C22⋊C4—121no—
365C22⋊C4—241no—
366C22⋊C4—241no—
367C22⋊C4—241no—
368C22⋊C4D3×C812yescomplemented
369C42—61no—
370C42—61no—
371C42—81no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3#38611yescharacteristic
383C22×C3C2×C22⋊C811yescharacteristic
384C22×C3#40811yescharacteristic
385C22×C3C8×D411yescharacteristic
386C22×C3C8×D411yescharacteristic
387C22×C3#43511yescharacteristic
388C22×C3#43511yescharacteristic
389C2×D3—22no—
390C2×D3—41no—
391C2×D3—41no—
392C2×D3—42no—
393C2×D3—42no—
394C2×D3—42no—
395C2×D3—42no—
396C2×D3—81no—
397C2×D3—81no—
398C2×D3—81no—
399C2×D3—82no—
400C2×D3—82no—
401C2×D3—81no—
402C2×D3—81no—
403C2×D3—82no—
404C2×D3—82no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—42no—
410C3×C4—81no—
411Dic3—22no—
412Dic3—41no—
413Dic3—41no—
414Dic3—81no—
415C23—32no—
416C23—41no—
417C23—41no—
418C23—61no—
419C23—61no—
420C23—62no—
421C23—62no—
422C23—62no—
423C23—121no—
424C23—121no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23—121no—
429C23—121no—
430C23—121no—
431C23—121no—
432C23—121no—
433C23—241no—
434C23—241no—
435C23—241no—
436C23—241no—
437C23—242no—
438C23—242no—
439C23—242no—
440C23—242no—
441C23C4×D1211yescharacteristic
442C23D3×C22⋊C411yescharacteristic
443C23#89311yescharacteristic
444C23C2×D3×C811yescharacteristic
445C23#90111yescharacteristic
446C23D3×M1611yescharacteristic
447C23#90711yescharacteristic
448D4—62no—
449D4—62no—
450D4—62no—
451D4—62no—
452D4—244no—
453C2×C4—22no—
454C2×C4—32no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—241no—
477C2×C4—241no—
478C2×C4—241no—
479C2×C4#88211yescharacteristic
480C2×C4#88411yescharacteristic
481C2×C4#89411yescharacteristic
482C2×C4#89511yescharacteristic
483C8—42no—
484C8—81no—
485C8—122no—
486C8—241no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3#186911yescharacteristic
494C2×C3#191311yescharacteristic
495C2×C3#191411yescharacteristic
496D3—42no—
497D3—82no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—62no—
509C22—122no—
510C22—122no—
511C22—122no—
512C22—121no—
513C22—121no—
514C22—122no—
515C22—241no—
516C22—242no—
517C22—242no—
518C22—241no—
519C22—241no—
520C22—242no—
521C22—242no—
522C22—241no—
523C22—241no—
524C22#466511yescharacteristic
525C22C8×D1211yescharacteristic
526C22#470611yescharacteristic
527C22D3×C22⋊C811yescharacteristic
528C22#472711yescharacteristic
529C22#483411yescharacteristic
530C22#483611yescharacteristic
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—42no—
536C4—62no—
537C4—81no—
538C4—121no—
539C4—121no—
540C4—241no—
541C3#910711yescharacteristic, complemented, Sylow
542C2—21no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2—21no—
547C2—21no—
548C2—122no—
549C2—242no—
550C2#6866811yescharacteristic
551C2#6896411yescharacteristic
552C2#6904011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC23×C3C22×C8
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#386
prime radicalC2×#882C4
cosocleC23×C3×C4C23
hypercenter#1282D3
hypocenterC3#9107
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
1. #78601C2semidirect
3. #69919C2semidirect
4. C2×#4468C2semidirect
5. C2×#4468C2semidirect
6. C3×#1282C2semidirect
7. #67617C2semidirect
38. #1282D3semidirect
73. #882C8semidirect
77. C3×C22⋊C8D4semidirect
84. #830D4semidirect
121. C23×D3C2×C8semidirect
157. C3×C22⋊C4C2×C8semidirect
225. C22⋊C8D12semidirect
368. C22⋊C4D3×C8semidirect
541. C3#9107semidirect

Maximal subgroups

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

subgroupindex
#786012
#735692
#699192
C2×#44682
C3×#12822
#676172
#91073