Finite Group Databasebeta

← #10948648

Subgroups — #10948648

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#78580C211yesmaximal, characteristic, complemented
3#78574C211yesmaximal, characteristic, complemented
4C3×#2110C211yesmaximal, characteristic, complemented
5C2×#4518C211yesmaximal, characteristic, complemented
6#71442C211yesmaximal, characteristic, complemented
7C2×#4513C211yesmaximal, characteristic, complemented
8#69091C214yesmaximal, complemented
9#69090C214yesmaximal, complemented
10#33469—31nomaximal, Sylow
11#5745—21no—
12C2×#904C2211yescharacteristic
13C2×#904C2211yescharacteristic
14#5523—21no—
15C2×C3×M32C2211yescharacteristic
16C2×C3×M32C2211yescharacteristic
17C3×#405C2211yescharacteristic
18#5133—21no—
19C2×#819C2211yescharacteristic
20C2×#817C2211yescharacteristic
21#5005—21no—
22#4909C412yescomplemented
23#4909C2212yes—
24#4908C412yescomplemented
25#4908C2212yes—
26#4904C414yescomplemented
27#4717C2214yes—
28#4716C2212yescomplemented
29#4711C2212yescomplemented
30#4700—22no—
31C4×#204—22no—
32C3×#347C2214yescomplemented
33#4518C2214yes—
34#4517C2212yescomplemented
35#4516C2212yescomplemented
36#4513C2214yescomplemented
37#3414—31no—
38#2957—31no—
39#2952—31no—
40AGL(1,ℤ/16)—34no—
41#2180—34no—
42C2×#365—31no—
43#2148—31no—
44C2×#360—31no—
45#2110D311yescharacteristic, complemented
46C2×#235—21no—
47C2×#235—21no—
48C2×C3×M16D411yescharacteristic
49C22×C3×C8C2311yescharacteristic
50C3×#117D411yescharacteristic
51C2×C4⋊Dic3—21no—
52C2×#208—21no—
53#907—42no—
54D3×M16—42no—
55C2×Dic12C2×C411yescharacteristic, complemented
56C2×Dic12C2311yescharacteristic
57#904—21no—
58#904—21no—
59#904—21no—
60#904—21no—
61#904C2×C412yes—
62#904C2×C412yes—
63#904C2×C412yescomplemented
64#904C2312yes—
65C2×D24C2×C411yescharacteristic, complemented
66C2×D24C2311yescharacteristic
67C2×#204—21no—
68C2×#204—21no—
69#901—24no—
70#892—41no—
71D3×C4⋊C4—41no—
72#67⋊C4—42no—
73C4×D12—22no—
74C4×Dic6—22no—
75C3×M32—21no—
76C3×M32C2×C412yescomplemented
77C3×M32C2312yes—
78C3×M32C2312yes—
79C2×C3×C16—21no—
80C2×C3×C16C2×C412yescomplemented
81C3×#97C2312yes—
82C3×C4×C8D412yes—
83#821—22no—
84#819—21no—
85#819C2312yes—
86#817—21no—
87#817C2312yes—
88#816—22no—
89#805—24no—
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×M32C2×D311yescharacteristic
104C2×M32C2×D311yescharacteristic
105#444—34no—
106#443—62no—
107#442—32no—
108#441—32no—
109C4×SD16—62no—
110C2×#108—31no—
111C2×#106—31no—
112#421—61no—
113#417—61no—
114#405C2×D311yescharacteristic
115#365—34no—
116#362—32no—
117#361—32no—
118#360—34no—
119#347C2×D314yescomplemented
120C22×C3×C4C2×D411yescharacteristic
121C2×#67—41no—
122C2×#67—41no—
123C22×Dic3—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—21no—
140C3×M16—21no—
141C2×C3×C8C22×C411yescharacteristic
142C2×C3×C8C22×C411yescharacteristic
143C2×C3×C8C2×D412yes—
144C2×C3×C8C2×D411yescharacteristic
145C2×C3×C8#4111yescharacteristic
146C2×C3×C8#4111yescharacteristic
147C2×C3×C8C2411yescharacteristic
148C3×C4⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C42C2×D412yes—
151(C2×D3)⋊C4—42no—
152C4⋊Dic3—22no—
153C4⋊Dic3—41no—
154Dic3⋊C4—42no—
155C4×Dic3—42no—
156#208—22no—
157#208—41no—
158C2×C3⋊C8—41no—
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×C16—22no—
174C3×C16C22×C414yes—
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—61no—
194C2×M16—122no—
195C2×M16D1211yescharacteristic
196C22×C8C22×D311yescharacteristic
197C4×Q8—62no—
198C4×D4—62no—
199C4×D4—122no—
200#117—121no—
201#117D1211yescharacteristic
202C2×C4⋊C4—61no—
203C2×C4⋊C4—121no—
204Q32—34no—
205SD32—34no—
206SD32—34no—
207D16—34no—
208M32—21no—
209M32C4×D312yescomplemented
210M32C22×D312yes—
211M32C22×D312yes—
212C2×C16—21no—
213C2×C16C4×D312yescomplemented
214#108—32no—
215#108—61no—
216#106—32no—
217#106—61no—
218#104—64no—
219#103—62no—
220#102—62no—
221#97C22×D312yes—
222C4×C8D1212yes—
223C23×C3AGL(1,ℤ/8)11yescharacteristic
224C22×D3—41no—
225C22×D3—41no—
226C2×C3×C4—22no—
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4C22×D411yescharacteristic
233#67—42no—
234#67—42no—
235#67—42no—
236#67—42no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—161no—
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—22no—
255C3×C8C4×D411yescharacteristic
256C3×C8C4×D411yescharacteristic
257C3×C8C23×C411yescharacteristic
258C3×C8C2×#4111yescharacteristic
259C3⋊C8—42no—
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—122no—
275C22×C4—121no—
276C22×C4—121no—
277C22×C4—121no—
278C22×C4C2×D1211yescharacteristic
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—21no—
292M16—21no—
293M16—62no—
294M16—124no—
295M16—121no—
296C2×C8—124no—
297C2×C8—121no—
298C2×C8C2×C4×D311yescharacteristic
299C2×C8C2×C4×D311yescharacteristic
300C2×C8C2×D1212yes—
301C2×C8C2×D1211yescharacteristic
302C2×C8#23511yescharacteristic
303C2×C8#23511yescharacteristic
304C2×C8C23×D311yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—62no—
307C4⋊C4—122no—
308C4⋊C4—121no—
309C22⋊C4—21no—
310C22⋊C4—122no—
311C42—122no—
312C42C2×D1212yes—
313C16—22no—
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—42no—
325C3×C4#44211yescharacteristic
326C3×C4#44211yescharacteristic
327C3×C4C2×C4×D411yescharacteristic
328C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
329Dic3—42no—
330Dic3—42no—
331Dic3—82no—
332C23—121no—
333C23—121no—
334C23#90811yescharacteristic
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—22no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—482no—
356C2×C4—481no—
357C2×C4C4×D1211yescharacteristic
358C2×C4C4×D1211yescharacteristic
359C2×C4#90811yescharacteristic
360C2×C4#90811yescharacteristic
361C2×C4#90811yescharacteristic
362C2×C4C22×D1211yescharacteristic
363C8—22no—
364C8—122no—
365C8C4×D1211yescharacteristic
366C8C4×D1211yescharacteristic
367C8C22×C4×D311yescharacteristic
368C8C2×#23511yescharacteristic
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#471111yescharacteristic
382C22#471111yescharacteristic
383C22C2×#90811yescharacteristic
384C4—42no—
385C4—122no—
386C4—122no—
387C4—242no—
388C4#471111yescharacteristic
389C4#471111yescharacteristic
390C4C2×C4×D1211yescharacteristic
391C4C2×#90811yescharacteristic
392C3#3346911yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—21no—
396C2—242no—
397C2—242no—
398C2C2×#471111yescharacteristic
subgroupG/H
centerC2C2×#4711
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC3×#2110C2
socleC2×C3C2×#442
prime radical#82009C2
cosocleC2×C3×C8C24
hypercenter#2110D3
hypocenterC3#33469
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. #78580C2semidirect
3. #78574C2semidirect
4. C3×#2110C2semidirect
5. C2×#4518C2semidirect
6. #71442C2semidirect
7. C2×#4513C2semidirect
8. #69091C2semidirect
9. #69090C2semidirect
22. #4909C4semidirect
24. #4908C4semidirect
26. #4904C4semidirect
28. #4716C22semidirect
29. #4711C22semidirect
32. C3×#347C22semidirect
34. #4517C22semidirect
35. #4516C22semidirect
36. #4513C22semidirect
45. #2110D3semidirect
55. C2×Dic12C2×C4semidirect
63. #904C2×C4semidirect
65. C2×D24C2×C4semidirect
76. C3×M32C2×C4semidirect
80. C2×C3×C16C2×C4semidirect
90. Dic24C2×C4semidirect
91. #800C2×C4semidirect
92. #800C2×C4semidirect
93. D48C2×C4semidirect
119. #347C2×D3semidirect
209. 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
#785802
#785742
C3×#21102
C2×#45182
#714422
C2×#45132
#690912
#690902
#334693