Finite Group Databasebeta

← #10949593

Subgroups — #10949593

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3156C211yesmaximal, characteristic, complemented
2#83894C211yesmaximal, characteristic
3C2×#5179C211yesmaximal, characteristic, complemented
4#78580C211yesmaximal, characteristic, complemented
5#72081C211yesmaximal, characteristic, complemented
6C2×#4565C211yesmaximal, characteristic, complemented
7#71325C214yesmaximal, complemented
8#71317C214yesmaximal, complemented
9#69885C211yesmaximal, characteristic, complemented
10#33596—31nomaximal, Sylow
11C2×C3×#137—21no—
12C2×C3×#135C2211yescharacteristic
13C22×C3×Q16C2211yescharacteristic
14C3×#536—21no—
15#5745—21no—
16C2×#904C2211yescharacteristic
17C3×#497—22no—
18C3×#495C2212yescomplemented
19C3×#494C2212yescomplemented
20C3×#405C2211yescharacteristic
21#5195—21no—
22#5194—22no—
23#5193—21no—
24C2×#829—21no—
25#5179—22no—
26#5179C2212yes—
27#5179C2214yes—
28C2×#828—21no—
29#5147C2212yes—
30C2×#819C2211yescharacteristic
31C2×#812C2211yescharacteristic
32C2×#812C2211yescharacteristic
33#5005—21no—
34#4717—22no—
35#4700C2214yes—
36#4565C2214yescomplemented
37#4565C2212yescomplemented
38#4563C2212yescomplemented
39#4463C2214yescomplemented
40#3414—31no—
41C2×#510—31no—
42#3156D311yescharacteristic, complemented
43#2957—31no—
44#2214—34no—
45#2212—34no—
46#2153—31no—
47#2148—31no—
48C2×#362—31no—
49C2×C3×#41—21no—
50C22×C3×Q8—21no—
51C2×#235—21no—
52C3×#137—42no—
53C3×#137—42no—
54C3×#135—21no—
55C3×#135—21no—
56C3×#135—21no—
57C3×#135—21no—
58C2×C3×Q16—21no—
59C2×C3×Q16—21no—
60C2×C3×Q16—21no—
61C2×C3×Q16—22no—
62C2×C3×Q16—21no—
63C2×C3×Q16—21no—
64C2×C3×Q16C2312yes—
65C2×C3×Q16C2311yescharacteristic
66C2×C3×SD16—21no—
67C2×C3×SD16—22no—
68C2×C3×D8C2311yescharacteristic
69C2×C3×M16D411yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×C4⋊Q8—22no—
72C3×#124—22no—
73C3×#123—42no—
74C3×C22⋊Q8—42no—
75C3×#117D411yescharacteristic
76C2×#208—21no—
77#907—42no—
78D3×M16—42no—
79C2×Dic12D411yescharacteristic
80#904—21no—
81#904—21no—
82#904D412yes—
83#904C2312yes—
84C2×D24D411yescharacteristic
85C2×#204—21no—
86#901—24no—
87C3×#97D412yes—
88C3×C4×C8C2312yes—
89#829—24no—
90#829—22no—
91#828—24no—
92#828—22no—
93#827—22no—
94#826—22no—
95#819—21no—
96#819C2312yes—
97#812—21no—
98#812—21no—
99#812C2312yes—
100#812C2312yes—
101C2×C3⋊C16—21no—
102C2×C3⋊C16—21no—
103#805—24no—
104C2×#137—21no—
105C2×#135—31no—
106C2×#135C2×D311yescharacteristic
107C22×Q16C2×D311yescharacteristic
108#568—61no—
109#536—21no—
110#510—34no—
111#510—32no—
112#510—62no—
113#510—61no—
114#509—32no—
115#509—61no—
116#508—62no—
117C2×Q32—61no—
118C2×SD32—61no—
119C2×M32—31no—
120C2×M32—31no—
121#497—22no—
122#495C2×D312yescomplemented
123#494C2×D312yescomplemented
124#444—62no—
125#443—34no—
126C2×#108—31no—
127#421—61no—
128#405C2×D311yescharacteristic
129#364—34no—
130#362—34no—
131#362—32no—
132#361—32no—
133C3×#41—41no—
134C3×#41—41no—
135C3×#41—41no—
136C3×#41—41no—
137C2×C3×Q8—22no—
138C2×C3×Q8—21no—
139C2×C3×Q8—41no—
140C2×C3×Q8—41no—
141C2×C3×Q8—41no—
142C2×C3×Q8—41no—
143C2×C3×Q8—41no—
144C2×C3×D4—21no—
145C2×C3×D4—41no—
146C22×C3×C4—41no—
147C22×C3×C4—41no—
148C22×C3×C4C2×D411yescharacteristic
149C2×#67—41no—
150#235—22no—
151#235—22no—
152#235—41no—
153#235—41no—
154C2×D12—21no—
155C2×C4×D3—41no—
156C2×Dic6—21no—
157C3×Q16—22no—
158C3×Q16—21no—
159C3×Q16—22no—
160C3×Q16—21no—
161C3×Q16—41no—
162C3×Q16—41no—
163C3×Q16—44no—
164C3×SD16—41no—
165C3×SD16—41no—
166C3×SD16—44no—
167C3×D8—21no—
168C3×D8—21no—
169C3×M16D812yes—
170C3×M16D812yes—
171C2×C3×C8C2×D411yescharacteristic
172C2×C3×C8C2×D411yescharacteristic
173C2×C3×C8C2×D411yescharacteristic
174C2×C3×C8C2×D412yes—
175C2×C3×C8#4111yescharacteristic
176C2×C3×C8#4111yescharacteristic
177C2×C3×C8C2411yescharacteristic
178C3×C4⋊C4—21no—
179C3×C4⋊C4—44no—
180C3×C22⋊C4—21no—
181C3×C22⋊C4—44no—
182C3×C42C2×D412yes—
183#208—22no—
184#208—41no—
185C2×C3⋊C8—41no—
186Dic12—21no—
187Dic12C2×D412yes—
188D24—21no—
189D24C2×D412yes—
190#204—22no—
191#204—22no—
192ΓL(1,𝔽25)—44no—
193D3×C8—44no—
194C3⋊C16—22no—
195C3⋊C16—22no—
196C2×#41—21no—
197C2×#41—61no—
198C22×Q8—21no—
199#137—42no—
200#137—42no—
201#135—21no—
202#135—21no—
203#135—21no—
204#135—21no—
205#135—32no—
206#135—32no—
207#135—61no—
208#135—61no—
209C2×Q16—21no—
210C2×Q16—21no—
211C2×Q16—21no—
212C2×Q16—21no—
213C2×Q16—21no—
214C2×Q16—22no—
215C2×Q16—31no—
216C2×Q16C22×D312yes—
217C2×Q16C22×D311yescharacteristic
218C2×SD16—21no—
219C2×SD16—22no—
220C2×SD16—61no—
221C2×D8—31no—
222C2×D8C22×D311yescharacteristic
223#131—64no—
224#131—122no—
225C2×M16—61no—
226C2×M16—122no—
227C2×M16#6711yescharacteristic
228C22×C8C22×D311yescharacteristic
229C4⋊Q8—22no—
230#124—22no—
231#123—42no—
232C22⋊Q8—42no—
233#117#6711yescharacteristic
234Q32—64no—
235Q32—62no—
236SD32—62no—
237SD32—64no—
238SD32—62no—
239D16—62no—
240M32—32no—
241M32—32no—
242M32—61no—
243M32—61no—
244C2×C16—61no—
245C2×C16—61no—
246#108—32no—
247#108—61no—
248#104—64no—
249#97#6712yes—
250C4×C8C22×D312yes—
251C23×C3—41no—
252C23×C3AGL(1,ℤ/8)11yescharacteristic
253C22×D3—41no—
254C3×Q8—41no—
255C3×Q8—42no—
256C3×Q8—42no—
257C3×Q8—41no—
258C3×Q8—81no—
259C3×Q8—81no—
260C3×D4—41no—
261C3×D4—41no—
262C3×D4—81no—
263C3×D4—81no—
264C2×C3×C4—22no—
265C2×C3×C4—42no—
266C2×C3×C4—41no—
267C2×C3×C4—81no—
268C2×C3×C4—81no—
269C2×C3×C4—81no—
270C2×C3×C4—81no—
271C2×C3×C4#12111yescharacteristic
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4C2×D811yescharacteristic
274C2×C3×C4C2×D811yescharacteristic
275C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
276C2×C3×C4C22×D411yescharacteristic
277#67—42no—
278#67—42no—
279C2×Dic3—41no—
280D12—22no—
281D12—41no—
282C4×D3—42no—
283C4×D3—42no—
284Dic6—22no—
285Dic6—41no—
286C3×C8#12111yescharacteristic
287C3×C8#12111yescharacteristic
288C3×C8C2×D814yes—
289C3×C8C22×D411yescharacteristic
290C3×C8C2×#4111yescharacteristic
291C3⋊C8—42no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296#41—62no—
297#41—62no—
298#41—121no—
299#41—121no—
300C2×Q8—22no—
301C2×Q8—21no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—61no—
308C2×D4—21no—
309C2×D4—41no—
310C2×D4—61no—
311C2×D4—121no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—121no—
315C22×C4C2×#6711yescharacteristic
316Q16—22no—
317Q16—21no—
318Q16—22no—
319Q16—21no—
320Q16—32no—
321Q16—44no—
322Q16—41no—
323Q16—41no—
324Q16—61no—
325SD16—44no—
326SD16—41no—
327SD16—41no—
328SD16—62no—
329SD16—62no—
330D8—21no—
331D8—21no—
332D8—32no—
333D8—61no—
334M16—62no—
335M16—121no—
336M16—124no—
337M16#21312yes—
338M16#21312yes—
339C2×C8—121no—
340C2×C8—124no—
341C2×C8D3×D411yescharacteristic
342C2×C8D3×D411yescharacteristic
343C2×C8#23711yescharacteristic
344C2×C8#23711yescharacteristic
345C2×C8C2×#6712yes—
346C2×C8C2×#6711yescharacteristic
347C2×C8C23×D311yescharacteristic
348C4⋊C4—21no—
349C4⋊C4—44no—
350C22⋊C4—21no—
351C22⋊C4—44no—
352C42C2×#6712yes—
353C16—62no—
354C16—62no—
355C22×C3—41no—
356C22×C3—82no—
357C22×C3—81no—
358C22×C3#45911yescharacteristic
359C22×C3#45911yescharacteristic
360C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
361C2×D3—42no—
362C2×D3—161no—
363C3×C4—42no—
364C3×C4—81no—
365C3×C4—82no—
366C3×C4#45911yescharacteristic
367C3×C4#45911yescharacteristic
368C3×C4C2×#12111yescharacteristic
369C3×C4C22×D811yescharacteristic
370Dic3—42no—
371C23—41no—
372C23—121no—
373C23#93211yescharacteristic
374Q8—42no—
375Q8—41no—
376Q8—41no—
377Q8—42no—
378Q8—62no—
379Q8—81no—
380Q8—81no—
381Q8—121no—
382D4—41no—
383D4—41no—
384D4—62no—
385D4—81no—
386D4—81no—
387D4—122no—
388D4—122no—
389D4—121no—
390C2×C4—22no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—121no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4C2×#21311yescharacteristic
401C2×C4C2×#21311yescharacteristic
402C2×C4#93211yescharacteristic
403C2×C4#93811yescharacteristic
404C2×C4#93811yescharacteristic
405C2×C4C22×#6711yescharacteristic
406C8—122no—
407C8C2×#21314yes—
408C8#93811yescharacteristic
409C8#93811yescharacteristic
410C8C2×D3×D411yescharacteristic
411C8C2×#23711yescharacteristic
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—161no—
416C2×C3C2×#45911yescharacteristic
417D3—82no—
418C22—41no—
419C22—82no—
420C22—81no—
421C22—122no—
422C22—481no—
423C22#507211yescharacteristic
424C22#507211yescharacteristic
425C22C2×#93211yescharacteristic
426C4—42no—
427C4—81no—
428C4—82no—
429C4—122no—
430C4#507211yescharacteristic
431C4#507211yescharacteristic
432C4C22×#21311yescharacteristic
433C4C2×#93811yescharacteristic
434C3#3359611yescharacteristic, complemented, Sylow
435C2—21no—
436C2—21no—
437C2—21no—
438C2—161no—
439C2—242no—
440C2C2×#507211yescharacteristic
subgroupG/H
centerC2C2×#5072
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3156C2
socleC2×C3C2×#459
prime radical#83894C2
hypercenter#3156D3
hypocenterC3#33596
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #3156

Complements

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

NH ≅ G/Ntype
1. C3×#3156C2semidirect
3. C2×#5179C2semidirect
4. #78580C2semidirect
5. #72081C2semidirect
6. C2×#4565C2semidirect
7. #71325C2semidirect
8. #71317C2semidirect
9. #69885C2semidirect
18. C3×#495C22semidirect
19. C3×#494C22semidirect
36. #4565C22semidirect
37. #4565C22semidirect
38. #4563C22semidirect
39. #4463C22semidirect
42. #3156D3semidirect
122. #495C2×D3semidirect
123. #494C2×D3semidirect
434. C3#33596semidirect

Maximal subgroups

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

subgroupindex
C3×#31562
#838942
C2×#51792
#785802
#720812
C2×#45652
#713252
#713172
#698852
#335963