Finite Group Databasebeta

← #10948513

Subgroups — #10948513

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4918C211yesmaximal, characteristic
2C4×#919C211yesmaximal, characteristic
3C4×D3×Q16C211yesmaximal, characteristic, complemented
4C3×#2176C211yesmaximal, characteristic, complemented
5Dic3×Q32C211yesmaximal, characteristic, complemented
6#70913C211yesmaximal, characteristic, complemented
7#70909C211yesmaximal, characteristic, complemented
8#69346C211yesmaximal, characteristic, complemented
9#69337C211yesmaximal, characteristic, complemented
10#69222C211yesmaximal, characteristic, complemented
11#69221C211yesmaximal, characteristic
12#69207C211yesmaximal, characteristic, complemented
13#69206C211yesmaximal, characteristic, complemented
14#69077C211yesmaximal, characteristic, complemented
15#69071C211yesmaximal, characteristic, complemented
16C4×#510—31nomaximal, Sylow
17C2×#919C2211yescharacteristic
18C2×D3×Q16C2211yescharacteristic
19C4×#239—21no—
20C4×D3×Q8—21no—
21C2×C3×Q32C2211yescharacteristic
22C3×C4×Q16C2211yescharacteristic
23C3×C4×Q16C2211yescharacteristic
24Dic3×Q16C2211yescharacteristic
25Dic3×Q16C2211yescharacteristic
26C2×#829C2211yescharacteristic
27C2×#828C2211yescharacteristic
28C4×#216—21no—
29C4×#215—21no—
30#4918C418yescomplemented
31C2×#800C2211yescharacteristic
32C2×#798C2211yescharacteristic
33C2×#798C2211yescharacteristic
34#4880C2211yescharacteristic
35D3×#107C2211yescharacteristic
36#4873C2211yescharacteristic
37#4872C2211yescharacteristic
38#4804—21no—
39D3×#103—21no—
40#4790—21no—
41#4788—21no—
42C4×Dic12C2211yescharacteristic
43C4×D24C2211yescharacteristic
44C4×D3×C8C2211yescharacteristic
45C3×#367C2211yescharacteristic
46C3×#358C2211yescharacteristic
47C3×#358C2211yescharacteristic
48C3×#346C2211yescharacteristic
49#4564C2211yescharacteristic
50#4564C2211yescharacteristic
51#4509C2211yescharacteristic
52#4504C2211yescharacteristic
53#4503C2211yescharacteristic
54Dic3×C16C2211yescharacteristic
55Dic3×C16C2211yescharacteristic
56#4492C2211yescharacteristic
57#4491C2211yescharacteristic
58#4490C2211yescharacteristic
59#4489C2211yescharacteristic
60#4461C2211yescharacteristic
61C2×#510—31no—
62C4×#135—31no—
63C2×C4×Q16—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#2141—31no—
74#2139—31no—
75C4×M32—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×D3×C8D411yescharacteristic
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—
104C42×D3—41no—
105C42×D3—41no—
106C42×D3D411yescharacteristic
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—
125C8×Dic3D411yescharacteristic
126C8×Dic3C2311yescharacteristic
127C2×C3⋊C16C2311yescharacteristic
128C2×C3⋊C16C2311yescharacteristic
129#810—21no—
130#809—21no—
131#808—21no—
132#807—21no—
133C4×C3⋊C8D411yescharacteristic
134#800C2×C414yescomplemented
135#798—22no—
136#798C2×C414yescomplemented
137C2×#135—31no—
138C22×Q16—31no—
139C4×#41—61no—
140C2×C4×Q8—61no—
141#510—38no—
142C2×Q32—31no—
143C2×Q32C2×D311yescharacteristic
144C2×SD32—31no—
145C2×SD32—31no—
146C2×M32—31no—
147C2×M32—31no—
148C4×Q16—31no—
149C4×Q16—31no—
150C4×Q16—31no—
151C4×Q16—31no—
152C4×Q16—61no—
153C4×Q16—61no—
154C4×Q16C2×D311yescharacteristic
155C4×Q16C2×D311yescharacteristic
156C4×SD16—61no—
157C4×SD16—61no—
158C4×D8—31no—
159C4×D8—31no—
160#427—31no—
161C2×#107—31no—
162#416—61no—
163C2×#103—61no—
164C2×C4×C8—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×D3D811yescharacteristic
195C2×C4×D3D811yescharacteristic
196C2×C4×D3C2×D411yescharacteristic
197C2×Dic6—21no—
198C2×Dic6—41no—
199C3×Q16—21no—
200C3×Q16—21no—
201C3×Q16C22×C412yes—
202C3×Q16C22×C412yes—
203C2×C3×C8C2×D411yescharacteristic
204C2×C3×C8C2411yescharacteristic
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—41no—
208C3×C4⋊C4—41no—
209C3×C42—41no—
210C3×C42—41no—
211C3×C42C2×D411yescharacteristic
212#216—24no—
213#215—24no—
214(C2×D3)⋊C4—41no—
215(C2×D3)⋊C4—41no—
216(C2×D3)⋊C4—41no—
217C4⋊Dic3—21no—
218C4⋊Dic3—21no—
219C4⋊Dic3—41no—
220C4⋊Dic3—41no—
221Dic3⋊C4—41no—
222Dic3⋊C4—41no—
223Dic3⋊C4—41no—
224C4×Dic3—41no—
225C4×Dic3—41no—
226C4×Dic3—41no—
227C4×Dic3—41no—
228C4×Dic3—41no—
229C4×Dic3D811yescharacteristic
230C4×Dic3D811yescharacteristic
231C4×Dic3C2×D411yescharacteristic
232C2×C3⋊C8C2×D411yescharacteristic
233C2×C3⋊C8C2×D411yescharacteristic
234Dic12—21no—
235Dic12C22×C412yes—
236D24—21no—
237D24C22×C412yes—
238D3×C8—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—
257C22×C8—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—61no—
272#117—121no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—121no—
275C2×C42—31no—
276C2×C42—121no—
277C2×C42—121no—
278Q32—34no—
279Q32C4×D314yescomplemented
280SD32—34no—
281SD32—34no—
282M32—34no—
283M32—62no—
284C2×C16—31no—
285C2×C16—31no—
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—
302C4×C8—31no—
303C4×C8—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—22no—
329C4×D3—41no—
330C4×D3—44no—
331C4×D3—44no—
332C4×D3—82no—
333C4×D3—82no—
334C4×D3—82no—
335C4×D3C4×D412yes—
336C4×D3#13512yes—
337Dic6—22no—
338Dic6—41no—
339Dic6—44no—
340C3×C8—21no—
341C3×C8C23×C411yescharacteristic
342C3×C8C2×#4111yescharacteristic
343C3⋊C8—21no—
344C3⋊C8C4×D412yes—
345#41—64no—
346#41—122no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—64no—
350C2×Q8—61no—
351C2×Q8—121no—
352C2×Q8—122no—
353C2×D4—61no—
354C2×D4—121no—
355C22×C4—31no—
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—121no—
359C22×C4—121no—
360C22×C4—121no—
361C22×C4—121no—
362C22×C4—121no—
363Q16—21no—
364Q16—21no—
365Q16—32no—
366Q16—64no—
367Q16—61no—
368Q16C2×C4×D312yes—
369Q16C2×C4×D312yes—
370SD16—64no—
371D8—32no—
372D8—61no—
373C2×C8—31no—
374C2×C8—32no—
375C2×C8—32no—
376C2×C8—31no—
377C2×C8—62no—
378C2×C8D3×D411yescharacteristic
379C2×C8C23×D311yescharacteristic
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C4⋊C4—121no—
387C4⋊C4—121no—
388C4⋊C4—121no—
389C4⋊C4—121no—
390C4⋊C4—121no—
391C22⋊C4—121no—
392C22⋊C4—121no—
393C22⋊C4—121no—
394C42—31no—
395C42—31no—
396C42—31no—
397C42—41no—
398C42—41no—
399C42—121no—
400C42—121no—
401C42—121no—
402C42—121no—
403C42—121no—
404C42D3×D411yescharacteristic
405C16—21no—
406C16—32no—
407C16—61no—
408C16C2×C4×D312yes—
409C22×C3C22×D811yescharacteristic
410C2×D3—21no—
411C2×D3—21no—
412C2×D3—42no—
413C2×D3—81no—
414C2×D3—81no—
415C2×D3C4×D812yes—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—81no—
421C3×C4—81no—
422C3×C4C2×C4×D411yescharacteristic
423C3×C4C2×#13511yescharacteristic
424Dic3—21no—
425Dic3—42no—
426Dic3—81no—
427Dic3—81no—
428Dic3C4×D812yes—
429Dic3#51012yescomplemented
430C23—31no—
431C23—121no—
432Q8—22no—
433Q8—22no—
434Q8—41no—
435Q8—41no—
436Q8—62no—
437Q8—124no—
438Q8—121no—
439D4—62no—
440D4—124no—
441D4—121no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—32no—
445C2×C4—32no—
446C2×C4—31no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—62no—
452C2×C4—121no—
453C2×C4—124no—
454C2×C4—124no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—242no—
459C2×C4—242no—
460C2×C4—242no—
461C2×C4D3×D811yescharacteristic
462C2×C4D3×D811yescharacteristic
463C2×C4C2×D3×D411yescharacteristic
464C8—21no—
465C8—32no—
466C8—61no—
467C8C22×C4×D311yescharacteristic
468C8D3×#4111yescharacteristic
469C2×C3C2×C4×D811yescharacteristic
470C2×C3C2×#51011yescharacteristic
471C2×C3#341411yescharacteristic
472D3—22no—
473D3—82no—
474C22—32no—
475C22—61no—
476C22—61no—
477C22—122no—
478C22—241no—
479C22—241no—
480C22C2×D3×D811yescharacteristic
481C4—21no—
482C4—21no—
483C4—32no—
484C4—32no—
485C4—42no—
486C4—42no—
487C4—61no—
488C4—81no—
489C4—81no—
490C4—122no—
491C4—241no—
492C4—241no—
493C4C4×D3×D411yescharacteristic
494C4D3×#13511yescharacteristic
495C3C4×#51011yescharacteristic, complemented, Sylow
496C2—62no—
497C2—242no—
498C2C4×D3×D811yescharacteristic
499C2D3×#51011yescharacteristic
500C2#8203911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC3×#2176C2
socleC22×C3C22×D8
prime radicalC2×#919C22
cosocleC2×C3×C8C24
hypercenter#2176D3
hypocenterC3C4×#510
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. C4×D3×Q16C2semidirect
4. C3×#2176C2semidirect
5. Dic3×Q32C2semidirect
6. #70913C2semidirect
7. #70909C2semidirect
8. #69346C2semidirect
9. #69337C2semidirect
10. #69222C2semidirect
12. #69207C2semidirect
13. #69206C2semidirect
14. #69077C2semidirect
15. #69071C2semidirect
30. #4918C4semidirect
65. #2176D3semidirect
89. D3×Q16C2×C4semidirect
109. C3×Q32C2×C4semidirect
119. #829C2×C4semidirect
120. #828C2×C4semidirect
134. #800C2×C4semidirect
136. #798C2×C4semidirect
279. Q32C4×D3semidirect
429. Dic3#510semidirect
495. C3C4×#510semidirect

Maximal subgroups

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

subgroupindex
C2×#49182
C4×#9192
C4×D3×Q162
C3×#21762
Dic3×Q322
#709132
#709092
#693462
#693372
#692222
#692212
#692072
#692062
#690772
#690712
C4×#5103