Finite Group Databasebeta

← #10802094

Subgroups — #10802094

Order 768.

874 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#441C211yesmaximal, characteristic, complemented
2C2×#5192C211yesmaximal, characteristic
3C4×#942C212yesmaximal
4#78486C211yesmaximal, characteristic
5#72069C211yesmaximal, characteristic, complemented
6C4×#836C212yesmaximal, complemented
7#70592C211yesmaximal, characteristic
8#70572C211yesmaximal, characteristic, complemented
9#70286C212yesmaximal
10#70263C212yesmaximal, complemented
11#69840C211yesmaximal, characteristic, complemented
12#21556—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×C4×Q8C2212yes—
15C2×#942C2212yes—
16C2×#901C2211yescharacteristic
17C2×#901C2211yescharacteristic
18C4×#235C2211yescharacteristic
19C3×#441—21no—
20C3×#441—21no—
21C3×#441—21no—
22C3×#441—21no—
23C2×C3×#106C2211yescharacteristic
24C2×C3×#103C2212yes—
25C2×C3×#97C2211yescharacteristic
26C2×#836C2212yes—
27C2×#836C2212yes—
28#5225C2212yes—
29#5192C418yescomplemented
30C2×#819C2211yescharacteristic
31C2×#819C2211yescharacteristic
32#5101C2211yescharacteristic
33#5101C2211yescharacteristic
34#5030—22no—
35C4×#216—22no—
36#5026—22no—
37C4×#215—22no—
38#4966C2212yes—
39#4964C2212yes—
40C4×#208C2211yescharacteristic
41C4×#208C2212yes—
42#4708—21no—
43#4708—21no—
44#4707—21no—
45D3×#97—21no—
46#4552C2212yes—
47#4530C2212yes—
48#4530C2212yes—
49C2×#473—31no—
50C4×#137—32no—
51C2×#441D311yescharacteristic, complemented
52#2875—31no—
53#1979—31no—
54#1951—31no—
55#1930—32no—
56#1908—32no—
57#1815—31no—
58#1778—31no—
59C4×#101—32no—
60C22×C3×Q8C2312yes—
61C2×#235C2311yescharacteristic
62C2×C3×Q16—21no—
63C2×C3×Q16—21no—
64C2×C3×Q16—22no—
65C2×C3×Q16C2×C414yescomplemented
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C4×Q8—22no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C42—22no—
75C2×C3×C42C2311yescharacteristic
76C2×#216—22no—
77#942—24no—
78#942C2×C418yes—
79C2×#215—22no—
80C4×#67—22no—
81#926C2311yescharacteristic
82C2×#208C2312yes—
83C2×#208C2312yes—
84C2×#208C2311yescharacteristic
85C2×#208C2311yescharacteristic
86#901—22no—
87#901—22no—
88#901—22no—
89#901C2×C414yes—
90C2×ΓL(1,𝔽25)—21no—
91C2×ΓL(1,𝔽25)—21no—
92C2×D3×C8—21no—
93C2×D3×C8—21no—
94C4×D12D412yes—
95#872—21no—
96C42×D3—21no—
97C4×Dic6D412yes—
98C3×#106—21no—
99C3×#106—21no—
100C3×#103—22no—
101C3×#103—22no—
102C3×#97D411yescharacteristic
103C3×#97D411yescharacteristic
104C3×#97D411yescharacteristic
105C3×#97D411yescharacteristic
106#836—24no—
107#836C2×C418yescomplemented
108#835—22no—
109#835—22no—
110#819—22no—
111#819C2×C414yescomplemented
112#815—21no—
113#815—21no—
114C8×Dic3—21no—
115C8×Dic3—21no—
116#810—22no—
117#809—22no—
118#808—22no—
119#807—22no—
120#803—21no—
121#803—22no—
122C4×C3⋊C8—21no—
123C4×C3⋊C8—22no—
124C2×#137—32no—
125C22×Q16C2×D311yescharacteristic
126C2×#131—31no—
127C2×#131—31no—
128C4×#41—31no—
129C2×C4×Q8C2×D312yes—
130#473—38no—
131#441—21no—
132#441—21no—
133#441—21no—
134#441—21no—
135#441—62no—
136#440—62no—
137C4×Q16—62no—
138C4×SD16—62no—
139C2×#108—31no—
140C2×#108—31no—
141#428—32no—
142C2×#106C2×D311yescharacteristic
143#419—32no—
144#417—32no—
145C2×#103C2×D312yes—
146C2×#101—32no—
147C2×#101—32no—
148#405—31no—
149#405—31no—
150#405—61no—
151#405—61no—
152C4×M16—31no—
153C4×M16—32no—
154C4×M16—61no—
155C2×#97—61no—
156C2×#97C2×D311yescharacteristic
157#343—32no—
158#343—32no—
159#341—32no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8C22×C414yes—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4#4112yes—
169C22×C3×C4C2411yescharacteristic
170C2×#67—21no—
171#235—21no—
172#235—21no—
173#235C22×C412yes—
174#235#4112yes—
175C2×D12C2×D411yescharacteristic
176C2×C4×D3—21no—
177C2×C4×D3—22no—
178C2×Dic6C2×D411yescharacteristic
179C3×Q16—24no—
180C3×Q16—24no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8C22×C412yes—
184C2×C3×C8C2×D411yescharacteristic
185C2×C3×C8C2×D411yescharacteristic
186C2×C3×C8C2×D411yescharacteristic
187C2×C3×C8C2×D411yescharacteristic
188C2×C3×C8#4112yes—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C42—42no—
194C3×C42—42no—
195C3×C42C2×D412yes—
196C3×C42C2×D411yescharacteristic
197C3×C42C2×D411yescharacteristic
198#217—21no—
199#216—28no—
200#215—28no—
201(C2×D3)⋊C4—22no—
202C4⋊Dic3C2×D412yes—
203Dic3⋊C4—22no—
204C4×Dic3—22no—
205C4×Dic3—21no—
206#208—21no—
207#208—21no—
208#208—22no—
209#208—22no—
210#208—22no—
211#208C22×C414yes—
212#208C22×C412yes—
213#208#4112yes—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—22no—
218ΓL(1,𝔽25)—24no—
219ΓL(1,𝔽25)—24no—
220D3×C8—24no—
221D3×C8—24no—
222C2×#41—31no—
223C22×Q8C22×D312yes—
224#137—38no—
225#137—64no—
226C2×Q16—22no—
227C2×Q16—21no—
228C2×Q16—21no—
229C2×Q16—62no—
230C2×Q16C4×D314yescomplemented
231C2×SD16—62no—
232#131—34no—
233#131—62no—
234#131—62no—
235#131—62no—
236C2×M16—31no—
237C2×M16—31no—
238C2×M16—32no—
239C2×M16—32no—
240C2×M16—61no—
241C2×M16—61no—
242C22×C8—61no—
243C22×C8—61no—
244C22×C8C22×D311yescharacteristic
245C22×C8C22×D311yescharacteristic
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—32no—
251C4×D4—32no—
252C4×D4—62no—
253#117—31no—
254#117—61no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4C22×D312yes—
257C2×C42—22no—
258C2×C42—61no—
259C2×C42C22×D311yescharacteristic
260#108—34no—
261#108—62no—
262#107—62no—
263#106—21no—
264#106—21no—
265#106—62no—
266#103—22no—
267#103—22no—
268#103—62no—
269#103—62no—
270#103—62no—
271#102—62no—
272#101—38no—
273#101—64no—
274#97—61no—
275#97—61no—
276#97—62no—
277#97—61no—
278#97#6711yescharacteristic
279#97#6711yescharacteristic
280#97#6711yescharacteristic
281#97#6711yescharacteristic
282C4×C8—61no—
283C4×C8—61no—
284C4×C8—62no—
285C4×C8—61no—
286C23×C3C2×#4111yescharacteristic
287C22×D3—21no—
288C3×Q8—24no—
289C3×Q8—24no—
290C3×Q8—42no—
291C3×Q8—42no—
292C2×C3×C4—24no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—82no—
298C2×C3×C4#12112yes—
299C2×C3×C4#12112yes—
300C2×C3×C4C23×C411yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306#67—22no—
307#67—22no—
308C2×Dic3—21no—
309C2×Dic3—22no—
310D12—21no—
311D12C4×D412yes—
312C4×D3—22no—
313C4×D3—22no—
314C4×D3—82no—
315Dic6—21no—
316Dic6C4×D412yes—
317C3×C8—21no—
318C3×C8—21no—
319C3×C8C4×D412yes—
320C3×C8C4×D412yes—
321C3⋊C8—22no—
322C3⋊C8—24no—
323C3⋊C8—22no—
324C3⋊C8—42no—
325#41—32no—
326#41—32no—
327#41—61no—
328#41—61no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—31no—
335C2×Q8C2×C4×D314yes—
336C2×D4—31no—
337C2×D4—61no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—61no—
341C22×C4—62no—
342C22×C4#23712yes—
343C22×C4C23×D311yescharacteristic
344Q16—24no—
345Q16—24no—
346Q16—68no—
347SD16—68no—
348M16—32no—
349M16—34no—
350M16—32no—
351M16—64no—
352M16—61no—
353M16—61no—
354M16—62no—
355M16—64no—
356M16—62no—
357M16—62no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—64no—
361C2×C8—62no—
362C2×C8—61no—
363C2×C8—61no—
364C2×C8—64no—
365C2×C8—62no—
366C2×C8C2×C4×D312yes—
367C2×C8#23512yes—
368C2×C8C2×#6711yescharacteristic
369C2×C8C2×#6711yescharacteristic
370C2×C8C2×#6711yescharacteristic
371C2×C8C2×#6711yescharacteristic
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—32no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—62no—
378C22⋊C4—61no—
379C22⋊C4—62no—
380C42—42no—
381C42—42no—
382C42—62no—
383C42—61no—
384C42D3×D412yes—
385C42C2×#6711yescharacteristic
386C42C2×#6711yescharacteristic
387C22×C3—22no—
388C22×C3C4×#4111yescharacteristic
389C22×C3C2×#12111yescharacteristic
390C22×C3#52911yescharacteristic
391C2×D3—22no—
392C2×D3—42no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—44no—
396C3×C4—82no—
397C3×C4C2×C4×D411yescharacteristic
398C3×C4C2×C4×D411yescharacteristic
399C3×C4#52511yescharacteristic
400C3×C4#53211yescharacteristic
401Dic3—22no—
402Dic3—42no—
403C23—61no—
404C23C2×#23711yescharacteristic
405Q8—24no—
406Q8—24no—
407Q8—32no—
408Q8—42no—
409Q8—42no—
410Q8—61no—
411D4—32no—
412D4—62no—
413D4—62no—
414D4—61no—
415C2×C4—24no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—61no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—82no—
425C2×C4—242no—
426C2×C4#93812yes—
427C2×C4#93812yes—
428C2×C4C22×C4×D311yescharacteristic
429C2×C4C2×#23511yescharacteristic
430C2×C4C2×D3×D411yescharacteristic
431C2×C4D3×#4111yescharacteristic
432C2×C4D3×#4111yescharacteristic
433C2×C4C22×#6711yescharacteristic
434C8—21no—
435C8—21no—
436C8—62no—
437C8—64no—
438C8—62no—
439C8—122no—
440C8C4×#6712yes—
441C8C4×#6712yes—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3C4×#12111yescharacteristic
445C2×C3#306912yes—
446D3—42no—
447C22—22no—
448C22—62no—
449C22—122no—
450C22C4×#23711yescharacteristic
451C22#555711yescharacteristic
452C22C2×#93811yescharacteristic
453C4—22no—
454C4—22no—
455C4—44no—
456C4—62no—
457C4—82no—
458C4—122no—
459C4C4×D3×D411yescharacteristic
460C4#554811yescharacteristic
461C4C2×C4×#6711yescharacteristic
462C4#582811yescharacteristic
463C3#2155611yescharacteristic, complemented, Sylow
464C2—21no—
465C2—21no—
466C2—122no—
467C2C4×#93811yescharacteristic
468C2#8399812yes—
subgroupG/H
centerC22C2×#938
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#441C2
socleC22×C3C2×#121
prime radicalC2×#235C23
cosocleC22×C3×C4C24
hypercenterC2×#441D3
hypocenterC3#21556
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
1. C2×C3×#441C2semidirect
5. #72069C2semidirect
6. C4×#836C2semidirect
8. #70572C2semidirect
10. #70263C2semidirect
11. #69840C2semidirect
29. #5192C4semidirect
51. C2×#441D3semidirect
65. C2×C3×Q16C2×C4semidirect
107. #836C2×C4semidirect
111. #819C2×C4semidirect
230. C2×Q16C4×D3semidirect
463. C3#21556semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4412
C2×#51922
C4×#9422
#784862
#720692
C4×#8362
#705922
#705722
#702862
#702632
#698402
#215563