Finite Group Databasebeta

← #10690155

Subgroups — #10690155

Order 768.

784 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78632C211yesmaximal, characteristic, complemented
2#73580C211yesmaximal, characteristic
3#69991C211yesmaximal, characteristic, complemented
4C4×#806C211yesmaximal, characteristic, complemented
5#69454C211yesmaximal, characteristic, complemented
6C3×#1297C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9009—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C4×#67C2211yescharacteristic
11C2×C4×D12—21no—
12C3×#407C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#825—22no—
18#5127—22no—
19#4961—21no—
20#4960—21no—
21C2×#806—21no—
22C4×(C2×D3)⋊C4—21no—
23C2×#805—22no—
24#4728—22no—
25D3×C22⋊C8—21no—
26#4719—21no—
27#4673C411yescharacteristic, complemented
28#4672C411yescharacteristic, complemented
29#4671C411yescharacteristic, complemented
30#4656C411yescharacteristic, complemented
31#4527—21no—
32#2973—31no—
33#2388—31no—
34#1789—31no—
35#1789—31no—
36C4×#99—31no—
37#1297—31no—
38#1297D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C22×#67D411yescharacteristic
42C2×#235—21no—
43C2×#235—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—44no—
47C22×D12—21no—
48C22×D12C2×C411yescharacteristic
49C22×C4×D3—21no—
50C22×C4×D3D411yescharacteristic
51C22×Dic6C2×C411yescharacteristic
52C2×C3×M16D412yescomplemented
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C42—21no—
56C2×C3×C42—21no—
57C2×#217—21no—
58C2×#217D411yescharacteristic
59C4×#67—21no—
60C4×#67—21no—
61C4×#67—21no—
62C4×#67—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4C2×C411yescharacteristic
68C2×C4⋊Dic3—21no—
69C2×Dic3⋊C4C2×C411yescharacteristic
70C2×C4×Dic3—21no—
71C2×C4×Dic3D411yescharacteristic
72C2×#208D412yescomplemented
73C2×D3×C8—22no—
74C4×D12—44no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×#95—21no—
78C3×#95C2×C411yescharacteristic
79C3×#95C2×C411yescharacteristic
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83#830—21no—
84#830—21no—
85#825—44no—
86(C2×D3)⋊C8—22no—
87#815—22no—
88Dic3⋊C8—22no—
89#806—24no—
90#805—44no—
91C22×#41—31no—
92C2×C4×D4—31no—
93C2×C4×D4—61no—
94#435—62no—
95#435—62no—
96#432—61no—
97C2×#104—62no—
98C2×#104—62no—
99C2×#99—61no—
100#407—31no—
101#407C2×D311yescharacteristic
102C2×C22⋊C8—61no—
103#395—31no—
104#394—31no—
105#394—61no—
106#393—31no—
107#392—31no—
108#392—61no—
109C4×C22⋊C4—31no—
110C4×C22⋊C4—61no—
111C4×C22⋊C4C2×D311yescharacteristic
112C4×C22⋊C4C2×D311yescharacteristic
113#342—61no—
114C24×C3C2×D411yescharacteristic
115C23×D3—21no—
116C23×D3C22⋊C411yescharacteristic
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—42no—
120C22×C3×C4—41no—
121C22×C3×C4C22×C411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C2×#67—21no—
128C2×#67—21no—
129C2×#67—22no—
130C2×#67—41no—
131C2×#67—42no—
132C2×#67—42no—
133C2×#67—41no—
134C2×#67—41no—
135C22×Dic3—21no—
136C22×Dic3—41no—
137C22×Dic3C22⋊C411yescharacteristic
138C22×Dic3C22⋊C411yescharacteristic
139C22×Dic3C22⋊C411yescharacteristic
140#235—24no—
141#235—24no—
142#235—42no—
143#235—42no—
144#235—42no—
145#235—42no—
146#235—42no—
147#235—42no—
148C2×D12—21no—
149C2×D12—21no—
150C2×D12—42no—
151C2×D12—42no—
152C2×D12—42no—
153C2×D12—42no—
154C2×C4×D3—24no—
155C2×C4×D3—21no—
156C2×C4×D3—21no—
157C2×C4×D3—42no—
158C2×C4×D3—41no—
159C2×C4×D3—41no—
160C2×C4×D3—42no—
161C2×Dic6—21no—
162C2×Dic6—21no—
163C2×Dic6—42no—
164C3×M16—42no—
165C2×C3×C8—22no—
166C3×C22⋊C4—22no—
167C3×C22⋊C4—42no—
168C3×C22⋊C4—41no—
169C3×C42—42no—
170C3×C42—42no—
171#217—21no—
172#217—21no—
173#217—22no—
174#217—41no—
175(C2×D3)⋊C4—21no—
176(C2×D3)⋊C4—21no—
177(C2×D3)⋊C4—42no—
178(C2×D3)⋊C4—42no—
179(C2×D3)⋊C4—42no—
180(C2×D3)⋊C4—42no—
181C4⋊Dic3—42no—
182Dic3⋊C4—21no—
183Dic3⋊C4—21no—
184C4×Dic3—21no—
185C4×Dic3—21no—
186C4×Dic3—42no—
187#208—42no—
188C2×C3⋊C8—22no—
189D3×C8—82no—
190C2×#41—61no—
191C2×#41—61no—
192C2×#41—61no—
193C2×#41—61no—
194C2×#41—124no—
195C22×Q8—31no—
196C22×D4—31no—
197C22×D4—31no—
198C22×D4—61no—
199C22×D4—61no—
200C23×C4—31no—
201C23×C4—61no—
202C23×C4C22×D311yescharacteristic
203C2×M16—32no—
204C2×M16D1212yescomplemented
205C22×C8—62no—
206C4×D4—61no—
207C4×D4—61no—
208C4×D4—61no—
209C4×D4—61no—
210C4×D4—124no—
211C2×C4⋊C4—31no—
212C2×C4⋊C4—61no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—31no—
216C2×C22⋊C4—31no—
217C2×C22⋊C4—61no—
218C2×C22⋊C4—61no—
219C2×C22⋊C4—61no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—61no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—31no—
225C2×C42—61no—
226C4⋊C8—62no—
227#104—124no—
228#104—124no—
229#99—64no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—62no—
233C22⋊C8—61no—
234C22⋊C8—61no—
235#97—62no—
236#95—21no—
237#95—61no—
238#95—61no—
239#95—61no—
240#95C4×D311yescharacteristic
241#95C4×D311yescharacteristic
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3C2×C22⋊C411yescharacteristic
247C23×C3#12111yescharacteristic
248C23×C3#12111yescharacteristic
249C22×D3—21no—
250C22×D3—21no—
251C22×D3—22no—
252C22×D3—41no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C22×D3—41no—
260C22×D3#10412yescomplemented
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
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—42no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—81no—
275C2×C3×C4—81no—
276C2×C3×C4C4×D411yescharacteristic
277C2×C3×C4C4×D411yescharacteristic
278C2×C3×C4#12011yescharacteristic
279C2×C3×C4#12311yescharacteristic
280#67—42no—
281#67—42no—
282#67—44no—
283#67—44no—
284C2×Dic3—22no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—42no—
295C2×Dic3—81no—
296C2×Dic3—81no—
297D12—22no—
298D12—22no—
299D12—41no—
300D12—41no—
301D12—48no—
302C4×D3—44no—
303C4×D3—42no—
304C4×D3—44no—
305C4×D3—81no—
306Dic6—22no—
307Dic6—22no—
308Dic6—41no—
309Dic6—41no—
310C3×C8—42no—
311C3⋊C8—42no—
312C24—31no—
313C24—61no—
314C24D3×D411yescharacteristic
315#41—64no—
316#41—64no—
317#41—122no—
318#41—122no—
319#41—122no—
320#41—122no—
321#41—122no—
322#41—122no—
323C2×Q8—61no—
324C2×Q8—61no—
325C2×Q8—122no—
326C2×D4—62no—
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—61no—
331C2×D4—122no—
332C2×D4—122no—
333C2×D4—122no—
334C2×D4—122no—
335C2×D4—121no—
336C2×D4—122no—
337C2×D4—122no—
338C2×D4—121no—
339C2×D4—121no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—64no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4—121no—
354C22×C4—121no—
355C22×C4—121no—
356C22×C4C2×C4×D311yescharacteristic
357C22×C4C2×D1211yescharacteristic
358C22×C4#23511yescharacteristic
359C22×C4D3×D411yescharacteristic
360C22×C4D3×D411yescharacteristic
361C22×C4#23911yescharacteristic
362M16—42no—
363M16—122no—
364C2×C8—22no—
365C2×C8—62no—
366C2×C8—242no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—122no—
370C22⋊C4—22no—
371C22⋊C4—42no—
372C22⋊C4—41no—
373C22⋊C4—61no—
374C22⋊C4—61no—
375C22⋊C4—61no—
376C22⋊C4—61no—
377C22⋊C4—62no—
378C22⋊C4—122no—
379C22⋊C4—122no—
380C22⋊C4—122no—
381C22⋊C4—122no—
382C22⋊C4—121no—
383C42—42no—
384C42—42no—
385C42—61no—
386C42—61no—
387C42—122no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3#38611yescharacteristic
399C22×C3C2×#10411yescharacteristic
400C22×C3#42111yescharacteristic
401C2×D3—22no—
402C2×D3—41no—
403C2×D3—41no—
404C2×D3—42no—
405C2×D3—42no—
406C2×D3—42no—
407C2×D3—42no—
408C2×D3—81no—
409C2×D3—81no—
410C2×D3—81no—
411C2×D3—81no—
412C2×D3—81no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—42no—
417C3×C4—81no—
418C3×C4—82no—
419C3×C4B(4,𝔽2)12yes—
420Dic3—41no—
421Dic3—41no—
422Dic3—42no—
423Dic3—41no—
424Dic3—82no—
425C23—21no—
426C23—21no—
427C23—32no—
428C23—41no—
429C23—41no—
430C23—62no—
431C23—61no—
432C23—61no—
433C23—121no—
434C23—121no—
435C23—121no—
436C23—121no—
437C23—121no—
438C23—121no—
439C23—121no—
440C23—121no—
441C23D3×C22⋊C411yescharacteristic
442C23#88411yescharacteristic
443C23#89511yescharacteristic
444Q8—62no—
445Q8—62no—
446Q8—121no—
447Q8—121no—
448D4—62no—
449D4—62no—
450D4—121no—
451D4—121no—
452D4—128no—
453D4—122no—
454D4—122no—
455D4—124no—
456D4—124no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—62no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—124no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—122no—
482C2×C4—124no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—122no—
486C2×C4—241no—
487C2×C4—241no—
488C2×C4—241no—
489C2×C4C4×D1211yescharacteristic
490C2×C4#88211yescharacteristic
491C2×C4#89311yescharacteristic
492C2×C4#89411yescharacteristic
493C8—42no—
494C8—122no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—41no—
498C2×C3—41no—
499C2×C3#187111yescharacteristic
500C2×C3#188411yescharacteristic
501C2×C3#188911yescharacteristic
502D3—42no—
503D3—82no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—62no—
513C22—81no—
514C22—81no—
515C22—122no—
516C22—122no—
517C22—122no—
518C22—121no—
519C22—121no—
520C22—122no—
521C22—241no—
522C22—241no—
523C22—241no—
524C22—241no—
525C22—241no—
526C22#466511yescharacteristic
527C22D3×#10411yescharacteristic
528C22#482111yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—81no—
534C4—82no—
535C4—122no—
536C4—121no—
537C4—121no—
538C4—121no—
539C4—242no—
540C4#474112yes—
541C3#900911yescharacteristic, complemented, Sylow
542C2—21no—
543C2—21no—
544C2—41no—
545C2—41no—
546C2—122no—
547C2—242no—
548C2#6875511yescharacteristic
549C2#6877711yescharacteristic
550C2#6896211yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1297C2
socleC22×C3#386
prime radical#73580C2
cosocleC23×C3×C4C23
hypercenter#1297D3
hypocenterC3#9009
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
1. #78632C2semidirect
3. #69991C2semidirect
4. C4×#806C2semidirect
5. #69454C2semidirect
6. C3×#1297C2semidirect
7. #67618C2semidirect
27. #4673C4semidirect
28. #4672C4semidirect
29. #4671C4semidirect
30. #4656C4semidirect
38. #1297D3semidirect
52. C2×C3×M16D4semidirect
72. C2×#208D4semidirect
204. C2×M16D12semidirect
260. C22×D3#104semidirect
541. C3#9009semidirect

Maximal subgroups

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

subgroupindex
#786322
#735802
#699912
C4×#8062
#694542
C3×#12972
#676182
#90093