Finite Group Databasebeta

← #10948539

Subgroups — #10948539

Order 768.

660 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4918C211yesmaximal, characteristic
2#79721C211yesmaximal, characteristic
3#79720C211yesmaximal, characteristic, complemented
4C3×#2176C211yesmaximal, characteristic, complemented
5Dic3×Q32C211yesmaximal, characteristic, complemented
6C4×#829C211yesmaximal, characteristic, complemented
7C4×#828C211yesmaximal, characteristic, complemented
8#69346C211yesmaximal, characteristic, complemented
9#69337C211yesmaximal, characteristic, complemented
10#69220C211yesmaximal, characteristic
11D3×#358C211yesmaximal, characteristic, complemented
12#69209C211yesmaximal, characteristic, complemented
13#69208C211yesmaximal, characteristic, complemented
14#69077C211yesmaximal, characteristic, complemented
15#69072C211yesmaximal, characteristic, complemented
16#33455—31nomaximal, Sylow
17C2×#919C2211yescharacteristic
18C2×D3×Q16C2211yescharacteristic
19#5574—21no—
20#5572—21no—
21C2×C3×Q32C2211yescharacteristic
22C3×C4×Q16C2211yescharacteristic
23C3×C4×Q16C2211yescharacteristic
24Dic3×Q16C2211yescharacteristic
25Dic3×Q16C2211yescharacteristic
26C2×#829C2211yescharacteristic
27C2×#828C2211yescharacteristic
28#5030—21no—
29#5026—21no—
30#4918C418yescomplemented
31C2×#800C2211yescharacteristic
32C2×#798C2211yescharacteristic
33C2×D3×C16C2211yescharacteristic
34#4880C2211yescharacteristic
35D3×#107C2211yescharacteristic
36#4873C2211yescharacteristic
37#4872C2211yescharacteristic
38#4817—21no—
39#4803—21no—
40#4802—21no—
41#4789—21no—
42C4×Dic12C2211yescharacteristic
43C4×D24C2211yescharacteristic
44#4689C2211yescharacteristic
45C3×#367C2211yescharacteristic
46C3×#358C2211yescharacteristic
47C3×#358C2211yescharacteristic
48C3×#346C2211yescharacteristic
49#4564C2211yescharacteristic
50#4564C2211yescharacteristic
51#4509C2211yescharacteristic
52#4504C2211yescharacteristic
53#4503C2211yescharacteristic
54#4502C2211yescharacteristic
55Dic3×C16C2211yescharacteristic
56#4492C2211yescharacteristic
57#4491C2211yescharacteristic
58#4490C2211yescharacteristic
59#4489C2211yescharacteristic
60C4×C3⋊C16C2211yescharacteristic
61C2×#510—31no—
62#2951—31no—
63#2950—31no—
64#2176—31no—
65#2176D311yescharacteristic, complemented
66#2175—31no—
67#2175—31no—
68C4×Q32—31no—
69C4×Q32—31no—
70C4×SD32—31no—
71C4×SD32—31no—
72#2159—31no—
73#2138—31no—
74C2×#358—31no—
75#2108—31no—
76C2×#239—21no—
77C2×D3×Q8—21no—
78C2×C3×Q16C2311yescharacteristic
79C2×C3×Q16C2311yescharacteristic
80C3×C4×Q8—21no—
81C3×C4×Q8—21no—
82Q8×Dic3—21no—
83Q8×Dic3—21no—
84C2×#216—21no—
85C2×#215—21no—
86#919—22no—
87#919C2×C414yes—
88D3×Q16—22no—
89D3×Q16C2×C414yescomplemented
90C2×Dic12C2311yescharacteristic
91C2×D24C2311yescharacteristic
92C2×ΓL(1,𝔽25)D411yescharacteristic
93C2×D3×C8C2311yescharacteristic
94#893—21no—
95#893—41no—
96#892—21no—
97#892—41no—
98D3×C4⋊C4—21no—
99D3×C4⋊C4—41no—
100#887—21no—
101#887—41no—
102C4×D12—21no—
103C4×D12—41no—
104#872—41no—
105#872—41no—
106#872D411yescharacteristic
107C4×Dic6—21no—
108C4×Dic6—41no—
109C3×Q32C2×C414yescomplemented
110C2×C3×C16C2311yescharacteristic
111C2×C3×C16C2311yescharacteristic
112C3×#107C2311yescharacteristic
113C3×#107C2311yescharacteristic
114C3×#103—21no—
115C3×#103—21no—
116C3×C4×C8C2311yescharacteristic
117#835—21no—
118#835—21no—
119#829C2×C414yescomplemented
120#828C2×C414yescomplemented
121#821—21no—
122#818C2311yescharacteristic
123#818C2311yescharacteristic
124#816—21no—
125#815D411yescharacteristic
126C8×Dic3C2311yescharacteristic
127C2×C3⋊C16C2311yescharacteristic
128C2×C3⋊C16C2311yescharacteristic
129#810—21no—
130#809—21no—
131#808—21no—
132#807—21no—
133#803D411yescharacteristic
134#800C2×C414yescomplemented
135#798C2×C414yescomplemented
136D3×C16—22no—
137C2×#135—31no—
138C22×Q16—31no—
139#520—61no—
140#519—61no—
141#510—38no—
142C2×Q32—31no—
143C2×Q32C2×D311yescharacteristic
144C2×SD32—31no—
145C2×SD32—31no—
146C2×M32—31no—
147C22×C16—31no—
148#441—61no—
149#441—61no—
150#440—61no—
151#440—61no—
152C4×Q16—31no—
153C4×Q16—31no—
154C4×Q16—31no—
155C4×Q16—31no—
156C4×Q16C2×D311yescharacteristic
157C4×Q16C2×D311yescharacteristic
158C4×D8—31no—
159C4×D8—31no—
160#427—31no—
161C2×#107—31no—
162#419—61no—
163#417—61no—
164#405—31no—
165#367—31no—
166#367C2×D311yescharacteristic
167#366—31no—
168#366—31no—
169#358—31no—
170#358—31no—
171#358—31no—
172#358—31no—
173#358C2×D311yescharacteristic
174#358C2×D311yescharacteristic
175#357—31no—
176#357—31no—
177#346—31no—
178#346C2×D311yescharacteristic
179C4×C16—31no—
180C4×C16—31no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183#239—24no—
184#239—42no—
185D3×Q8—24no—
186D3×Q8—42no—
187C2×D12—21no—
188C2×D12—41no—
189C2×C4×D3—41no—
190C2×C4×D3—41no—
191C2×C4×D3—41no—
192C2×C4×D3—41no—
193C2×C4×D3—41no—
194C2×C4×D3C2×D411yescharacteristic
195C2×Dic6—21no—
196C2×Dic6—41no—
197C3×Q16—21no—
198C3×Q16—21no—
199C3×Q16C22×C412yes—
200C3×Q16C22×C412yes—
201C2×C3×C8C2×D411yescharacteristic
202C2×C3×C8C2411yescharacteristic
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—41no—
207C3×C42—41no—
208C3×C42—41no—
209C3×C42C2×D411yescharacteristic
210#216—24no—
211#215—24no—
212(C2×D3)⋊C4—41no—
213(C2×D3)⋊C4—41no—
214(C2×D3)⋊C4—41no—
215(C2×D3)⋊C4D811yescharacteristic
216(C2×D3)⋊C4D811yescharacteristic
217C4⋊Dic3—21no—
218C4⋊Dic3—21no—
219C4⋊Dic3—41no—
220C4⋊Dic3—41no—
221Dic3⋊C4—41no—
222Dic3⋊C4—41no—
223Dic3⋊C4—41no—
224Dic3⋊C4D811yescharacteristic
225Dic3⋊C4D811yescharacteristic
226C4×Dic3—41no—
227C4×Dic3—41no—
228C4×Dic3—41no—
229C4×Dic3—41no—
230C4×Dic3—41no—
231C4×Dic3C2×D411yescharacteristic
232C2×C3⋊C8C2×D411yescharacteristic
233C2×C3⋊C8C2×D411yescharacteristic
234Dic12—21no—
235Dic12C22×C412yes—
236D24—21no—
237D24C22×C412yes—
238ΓL(1,𝔽25)—22no—
239D3×C8C22×C412yes—
240D3×C8#4112yes—
241C3×C16—21no—
242C3×C16C22×C412yes—
243C3⋊C16—21no—
244C3⋊C16C22×C412yes—
245C2×#41—61no—
246C22×Q8—61no—
247#135—34no—
248#135—62no—
249C2×Q16—34no—
250C2×Q16—31no—
251C2×Q16—62no—
252C2×Q16—61no—
253C2×Q16C22×D311yescharacteristic
254C2×Q16C22×D311yescharacteristic
255C2×SD16—61no—
256C2×D8—31no—
257C2×M16—31no—
258C22×C8—31no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—61no—
262C4×Q8—61no—
263C4×Q8—61no—
264C4×Q8—61no—
265C4×Q8—121no—
266C4×Q8—121no—
267C4×D4—61no—
268C4×D4—61no—
269C4×D4—121no—
270C4×D4—121no—
271#117—31no—
272#117—61no—
273#117—121no—
274#117—121no—
275#117—121no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—121no—
278Q32—34no—
279Q32C4×D314yescomplemented
280SD32—34no—
281SD32—34no—
282M32—34no—
283C2×C16—31no—
284C2×C16—31no—
285C2×C16—62no—
286C2×C16C22×D311yescharacteristic
287C2×C16C22×D311yescharacteristic
288#107—31no—
289#107—31no—
290#107—61no—
291#107C22×D311yescharacteristic
292#107C22×D311yescharacteristic
293#106—61no—
294#103—21no—
295#103—21no—
296#103—61no—
297#103—61no—
298#103—61no—
299#103—61no—
300#102—61no—
301#102—61no—
302#97—31no—
303#97—31no—
304C4×C8—31no—
305C4×C8C22×D311yescharacteristic
306C22×D3—41no—
307C22×D3C2×D811yescharacteristic
308C3×Q8—22no—
309C3×Q8—22no—
310C3×Q8—41no—
311C3×Q8—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4C2×D811yescharacteristic
317C2×C3×C4C2×D811yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3C2×D811yescharacteristic
323C2×Dic3C2×D811yescharacteristic
324C2×Dic3C2×D811yescharacteristic
325D12—22no—
326D12—44no—
327D12—41no—
328C4×D3—44no—
329C4×D3—44no—
330C4×D3—161no—
331C4×D3—161no—
332C4×D3—161no—
333C4×D3C4×D412yes—
334C4×D3#13512yes—
335Dic6—22no—
336Dic6—41no—
337Dic6—44no—
338C3×C8—21no—
339C3×C8C23×C411yescharacteristic
340C3×C8C2×#4111yescharacteristic
341C3⋊C8—21no—
342C3⋊C8C4×D412yes—
343#41—64no—
344#41—122no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—64no—
348C2×Q8—61no—
349C2×Q8—121no—
350C2×Q8—122no—
351C2×D4—61no—
352C2×D4—121no—
353C22×C4—31no—
354C22×C4—121no—
355C22×C4—121no—
356C22×C4—121no—
357C22×C4—121no—
358C22×C4—121no—
359Q16—21no—
360Q16—21no—
361Q16—32no—
362Q16—64no—
363Q16—61no—
364Q16C2×C4×D312yes—
365Q16C2×C4×D312yes—
366SD16—64no—
367D8—32no—
368D8—61no—
369M16—62no—
370C2×C8—31no—
371C2×C8—32no—
372C2×C8—32no—
373C2×C8—31no—
374C2×C8D3×D411yescharacteristic
375C2×C8C23×D311yescharacteristic
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—31no—
379C4⋊C4—31no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—61no—
383C4⋊C4—61no—
384C4⋊C4—121no—
385C4⋊C4—121no—
386C4⋊C4—121no—
387C4⋊C4—121no—
388C4⋊C4—121no—
389C22⋊C4—31no—
390C22⋊C4—31no—
391C22⋊C4—121no—
392C22⋊C4—121no—
393C22⋊C4—121no—
394C42—31no—
395C42—41no—
396C42—41no—
397C42—121no—
398C42—121no—
399C42—121no—
400C42—121no—
401C42—121no—
402C42D3×D411yescharacteristic
403C16—21no—
404C16—32no—
405C16—61no—
406C16C2×C4×D312yes—
407C22×C3C22×D811yescharacteristic
408C2×D3—21no—
409C2×D3—21no—
410C2×D3—42no—
411C2×D3—81no—
412C2×D3—81no—
413C2×D3C4×D812yes—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4—81no—
419C3×C4—81no—
420C3×C4C2×C4×D411yescharacteristic
421C3×C4C2×#13511yescharacteristic
422Dic3—21no—
423Dic3—21no—
424Dic3—42no—
425Dic3—81no—
426Dic3—81no—
427Dic3C4×D812yes—
428C23—31no—
429C23—121no—
430Q8—22no—
431Q8—22no—
432Q8—41no—
433Q8—41no—
434Q8—62no—
435Q8—124no—
436Q8—121no—
437D4—62no—
438D4—124no—
439D4—121no—
440C2×C4—31no—
441C2×C4—32no—
442C2×C4—32no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—124no—
450C2×C4—124no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—481no—
455C2×C4—481no—
456C2×C4—481no—
457C2×C4D3×D811yescharacteristic
458C2×C4D3×D811yescharacteristic
459C2×C4C2×D3×D411yescharacteristic
460C8—21no—
461C8—32no—
462C8—61no—
463C8C22×C4×D311yescharacteristic
464C8D3×#4111yescharacteristic
465C2×C3C2×C4×D811yescharacteristic
466C2×C3#341611yescharacteristic
467C2×C3#341711yescharacteristic
468D3—22no—
469D3—82no—
470C22—32no—
471C22—61no—
472C22—61no—
473C22—122no—
474C22—241no—
475C22—241no—
476C22C2×D3×D811yescharacteristic
477C4—21no—
478C4—21no—
479C4—32no—
480C4—42no—
481C4—42no—
482C4—61no—
483C4—61no—
484C4—81no—
485C4—81no—
486C4—122no—
487C4—241no—
488C4—241no—
489C4C4×D3×D411yescharacteristic
490C4D3×#13511yescharacteristic
491C3#3345511yescharacteristic, complemented, Sylow
492C2—62no—
493C2—242no—
494C2C4×D3×D811yescharacteristic
495C2#8204011yescharacteristic
496C2#8204111yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC3×#2176C2
socleC22×C3C22×D8
prime radicalC2×#919C22
cosocleC2×C3×C8C24
hypercenter#2176D3
hypocenterC3#33455
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2176

Complements

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

NH ≅ G/Ntype
3. #79720C2semidirect
4. C3×#2176C2semidirect
5. Dic3×Q32C2semidirect
6. C4×#829C2semidirect
7. C4×#828C2semidirect
8. #69346C2semidirect
9. #69337C2semidirect
11. D3×#358C2semidirect
12. #69209C2semidirect
13. #69208C2semidirect
14. #69077C2semidirect
15. #69072C2semidirect
30. #4918C4semidirect
65. #2176D3semidirect
89. D3×Q16C2×C4semidirect
109. C3×Q32C2×C4semidirect
119. #829C2×C4semidirect
120. #828C2×C4semidirect
134. #800C2×C4semidirect
135. #798C2×C4semidirect
279. Q32C4×D3semidirect
491. C3#33455semidirect

Maximal subgroups

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

subgroupindex
C2×#49182
#797212
#797202
C3×#21762
Dic3×Q322
C4×#8292
C4×#8282
#693462
#693372
#692202
D3×#3582
#692092
#692082
#690772
#690722
#334553