Finite Group Databasebeta

← #10948645

Subgroups — #10948645

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82009C211yesmaximal, characteristic
2#79249C211yesmaximal, characteristic, complemented
3#79106C211yesmaximal, characteristic, complemented
4C2×C3×#365C211yesmaximal, characteristic, complemented
5#71425C211yesmaximal, characteristic, complemented
6#70687C211yesmaximal, characteristic, complemented
7C2×#4488C211yesmaximal, characteristic, complemented
8#69310C214yesmaximal, complemented
9#69309C214yesmaximal, complemented
10#33469—31nomaximal, Sylow
11#5745—21no—
12C2×#904C2211yescharacteristic
13C2×#904C2211yescharacteristic
14#5504—21no—
15C2×C3×M32C2211yescharacteristic
16C2×C3×#108C2211yescharacteristic
17C2×C3×#106C2211yescharacteristic
18#5139—21no—
19#5101C2211yescharacteristic
20C2×#812C2211yescharacteristic
21#4966—21no—
22#4909C412yescomplemented
23#4909C2212yes—
24#4908C412yescomplemented
25#4908C2212yes—
26#4904C414yescomplemented
27#4895—22no—
28#4894C2214yes—
29#4869C2212yescomplemented
30#4853C2212yescomplemented
31#4852—22no—
32C3×#365C2214yescomplemented
33#4512C2214yescomplemented
34#4496C2212yescomplemented
35#4495C2212yescomplemented
36#4488C2214yes—
37#3414—31no—
38#2957—31no—
39#2952—31no—
40AGL(1,ℤ/16)—34no—
41#2180—34no—
42C2×#365D311yescharacteristic, complemented
43#2148—31no—
44C2×#360—31no—
45#2110—31no—
46C2×#235—21no—
47C2×#235—21no—
48C2×C3×M16—21no—
49C22×C3×C8C2311yescharacteristic
50C2×C3×C4⋊C4—21no—
51C4×#67—42no—
52#926D411yescharacteristic
53C2×#208D411yescharacteristic
54#907—24no—
55D3×M16—42no—
56C2×Dic12C2×C411yescharacteristic, complemented
57C2×Dic12C2311yescharacteristic
58#904—21no—
59#904—21no—
60#904—21no—
61#904—21no—
62#904C2×C412yes—
63#904C2×C412yes—
64#904C2×C412yescomplemented
65#904C2312yes—
66C2×D24C2×C411yescharacteristic, complemented
67C2×D24C2311yescharacteristic
68C2×#204—21no—
69C2×#204—21no—
70#901—42no—
71#893—22no—
72#892—41no—
73D3×C4⋊C4—41no—
74#887—22no—
75C3×M32C2×C412yescomplemented
76C3×M32C2312yes—
77C2×C3×C16C2×C412yescomplemented
78C3×#108—21no—
79C3×#108C2312yes—
80C3×#106—21no—
81C3×#106C2312yes—
82#825—24no—
83#815C2312yes—
84C8×Dic3D412yes—
85#812—21no—
86#812C2312yes—
87C2×C3⋊C16—21no—
88#810—22no—
89#809—22no—
90Dic24C2×C414yescomplemented
91#800C2×C414yescomplemented
92#800C2×C414yescomplemented
93D48C2×C414yescomplemented
94C2×#135—31no—
95C2×#135—31no—
96#568—61no—
97#520—61no—
98#510—32no—
99#510—32no—
100#509—32no—
101#509—32no—
102#508—34no—
103C2×M32—31no—
104C2×M32C2×D311yescharacteristic
105#444—34no—
106#443—62no—
107#442—32no—
108#441—32no—
109C4×SD16—62no—
110C2×#108C2×D311yescharacteristic
111C2×#106C2×D311yescharacteristic
112#421—61no—
113#417—61no—
114#405—31no—
115#365C2×D314yescomplemented
116#362—32no—
117#361—32no—
118#360—34no—
119#347—34no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C2×#67—41no—
123C2×#67—41no—
124#235—22no—
125#235—22no—
126#235—22no—
127#235—22no—
128#235—41no—
129#235—41no—
130#235—41no—
131#235—41no—
132C2×D12—21no—
133C2×D12—21no—
134C2×C4×D3—41no—
135C2×C4×D3—41no—
136C2×C4×D3—42no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C3×M16—22no—
140C3×M16—41no—
141C2×C3×C8—41no—
142C2×C3×C8C22×C411yescharacteristic
143C2×C3×C8C22×C411yescharacteristic
144C2×C3×C8C2×D411yescharacteristic
145C2×C3×C8#4111yescharacteristic
146C2×C3×C8#4111yescharacteristic
147C2×C3×C8C2411yescharacteristic
148C3×C4⋊C4—22no—
149C3×C4⋊C4—41no—
150#217—21no—
151(C2×D3)⋊C4—42no—
152C4⋊Dic3—21no—
153Dic3⋊C4—42no—
154C4×Dic3—42no—
155C4×Dic3C2×D412yes—
156#208—21no—
157#208—21no—
158C2×C3⋊C8C2×D412yes—
159Dic12—21no—
160Dic12—21no—
161Dic12C22×C412yes—
162Dic12C22×C412yes—
163D24—21no—
164D24—21no—
165D24C22×C412yes—
166D24C22×C412yes—
167#204—22no—
168#204—22no—
169#204—22no—
170#204—22no—
171ΓL(1,𝔽25)—44no—
172D3×C8—44no—
173C3×C16C22×C414yes—
174C3⋊C16—22no—
175C2×#41—61no—
176C2×#41—61no—
177#135—32no—
178#135—32no—
179#135—32no—
180#135—32no—
181#135—61no—
182#135—61no—
183#135—61no—
184#135—61no—
185C2×Q16—31no—
186C2×Q16—31no—
187C2×SD16—61no—
188C2×SD16—61no—
189C2×D8—31no—
190C2×D8—31no—
191#131—64no—
192#131—122no—
193C2×M16—21no—
194C2×M16—31no—
195C2×M16—122no—
196C22×C8C22×D311yescharacteristic
197C4×Q8—62no—
198C4×D4—62no—
199C4×D4—122no—
200#117—31no—
201#117—121no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—121no—
204Q32—34no—
205SD32—34no—
206SD32—34no—
207D16—34no—
208M32—32no—
209M32—61no—
210M32C4×D312yescomplemented
211M32C22×D312yes—
212C2×C16—61no—
213C2×C16C4×D312yescomplemented
214#108—21no—
215#108C22×D312yes—
216#106—21no—
217#106C22×D312yes—
218#104—64no—
219#103—62no—
220#102—62no—
221#97—32no—
222C4×C8—32no—
223C23×C3AGL(1,ℤ/8)11yescharacteristic
224C22×D3—41no—
225C22×D3—41no—
226C2×C3×C4—42no—
227C2×C3×C4—161no—
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4C22×D411yescharacteristic
234#67—42no—
235#67—42no—
236#67—42no—
237#67—42no—
238C2×Dic3—22no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241D12—22no—
242D12—22no—
243D12—41no—
244D12—41no—
245C4×D3—42no—
246C4×D3—42no—
247C4×D3—42no—
248C4×D3—42no—
249C4×D3—162no—
250Dic6—22no—
251Dic6—22no—
252Dic6—41no—
253Dic6—41no—
254C3×C8—42no—
255C3×C8C4×D411yescharacteristic
256C3×C8C4×D411yescharacteristic
257C3×C8C23×C411yescharacteristic
258C3×C8C2×#4111yescharacteristic
259C3⋊C8—22no—
260#41—62no—
261#41—62no—
262#41—62no—
263#41—62no—
264#41—121no—
265#41—121no—
266#41—121no—
267#41—121no—
268C2×Q8—61no—
269C2×Q8—61no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—121no—
273C2×D4—121no—
274C22×C4—41no—
275C22×C4—122no—
276C22×C4—121no—
277C22×C4—121no—
278C22×C4D3×D411yescharacteristic
279Q16—32no—
280Q16—32no—
281Q16—61no—
282Q16—61no—
283SD16—62no—
284SD16—62no—
285SD16—62no—
286SD16—62no—
287D8—32no—
288D8—32no—
289D8—61no—
290D8—61no—
291M16—22no—
292M16—41no—
293M16—61no—
294M16—61no—
295M16—124no—
296C2×C8—32no—
297C2×C8—41no—
298C2×C8—124no—
299C2×C8C2×C4×D311yescharacteristic
300C2×C8C2×C4×D311yescharacteristic
301C2×C8D3×D411yescharacteristic
302C2×C8#23911yescharacteristic
303C2×C8#23911yescharacteristic
304C2×C8C23×D311yescharacteristic
305C4⋊C4—22no—
306C4⋊C4—41no—
307C4⋊C4—61no—
308C4⋊C4—122no—
309C22⋊C4—61no—
310C22⋊C4—122no—
311C42—32no—
312C42—122no—
313C16—62no—
314C16C2×C4×D314yes—
315C22×C3—41no—
316C22×C3#44211yescharacteristic
317C22×C3#44211yescharacteristic
318C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
319C2×D3—42no—
320C2×D3—42no—
321C2×D3—81no—
322C2×D3—81no—
323C2×D3—161no—
324C3×C4—82no—
325C3×C4#44211yescharacteristic
326C3×C4#44211yescharacteristic
327C3×C4C2×C4×D411yescharacteristic
328C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
329Dic3—42no—
330Dic3—42no—
331Dic3—42no—
332C23—121no—
333C23—121no—
334C23#91411yescharacteristic
335Q8—62no—
336Q8—62no—
337Q8—121no—
338Q8—121no—
339D4—62no—
340D4—62no—
341D4—122no—
342D4—122no—
343D4—122no—
344D4—122no—
345D4—121no—
346D4—121no—
347C2×C4—42no—
348C2×C4—62no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—161no—
356C2×C4—482no—
357C2×C4#89311yescharacteristic
358C2×C4#89311yescharacteristic
359C2×C4#91411yescharacteristic
360C2×C4#91411yescharacteristic
361C2×C4#91411yescharacteristic
362C2×C4C2×D3×D411yescharacteristic
363C8—42no—
364C8—62no—
365C8#89311yescharacteristic
366C8#89311yescharacteristic
367C8C22×C4×D311yescharacteristic
368C8C2×#23911yescharacteristic
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3C2×#44211yescharacteristic
373D3—82no—
374D3—82no—
375C22—41no—
376C22—122no—
377C22—122no—
378C22—241no—
379C22—241no—
380C22—481no—
381C22#486911yescharacteristic
382C22#486911yescharacteristic
383C22C2×#91411yescharacteristic
384C4—82no—
385C4—122no—
386C4—122no—
387C4—122no—
388C4#486911yescharacteristic
389C4#486911yescharacteristic
390C4C2×#89311yescharacteristic
391C4C2×#91411yescharacteristic
392C3#3346911yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—21no—
396C2—242no—
397C2—242no—
398C2C2×#486911yescharacteristic
subgroupG/H
centerC2C2×#4869
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC2×C3×#365C2
socleC2×C3C2×#442
prime radical#82009C2
cosocleC2×C3×C8C24
hypercenterC2×#365D3
hypocenterC3#33469
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#365

Complements

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

NH ≅ G/Ntype
2. #79249C2semidirect
3. #79106C2semidirect
4. C2×C3×#365C2semidirect
5. #71425C2semidirect
6. #70687C2semidirect
7. C2×#4488C2semidirect
8. #69310C2semidirect
9. #69309C2semidirect
22. #4909C4semidirect
24. #4908C4semidirect
26. #4904C4semidirect
29. #4869C22semidirect
30. #4853C22semidirect
32. C3×#365C22semidirect
33. #4512C22semidirect
34. #4496C22semidirect
35. #4495C22semidirect
42. C2×#365D3semidirect
56. C2×Dic12C2×C4semidirect
64. #904C2×C4semidirect
66. C2×D24C2×C4semidirect
75. C3×M32C2×C4semidirect
77. C2×C3×C16C2×C4semidirect
90. Dic24C2×C4semidirect
91. #800C2×C4semidirect
92. #800C2×C4semidirect
93. D48C2×C4semidirect
115. #365C2×D3semidirect
210. M32C4×D3semidirect
213. C2×C16C4×D3semidirect
392. C3#33469semidirect

Maximal subgroups

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

subgroupindex
#820092
#792492
#791062
C2×C3×#3652
#714252
#706872
C2×#44882
#693102
#693092
#334693