Finite Group Databasebeta

← #10948639

Subgroups — #10948639

Order 768.

652 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
2C2×#4717C211yesmaximal, characteristic, complemented
3#78494C211yesmaximal, characteristic, complemented
4C2×C3×#347C211yesmaximal, characteristic, complemented
5#71450C211yesmaximal, characteristic, complemented
6#71442C211yesmaximal, characteristic, complemented
7#71427C211yesmaximal, characteristic, complemented
8#69091C214yesmaximal, complemented
9#69090C214yesmaximal, complemented
10#33468—31nomaximal, Sylow
11C2×#904C2211yescharacteristic
12C2×#904C2211yescharacteristic
13C2×#901—21no—
14C4×#235—21no—
15C2×C3×M32C2211yescharacteristic
16C2×C3×M32C2211yescharacteristic
17C2×C3×#97C2211yescharacteristic
18#5113—21no—
19C2×#819C2211yescharacteristic
20#5106C2211yescharacteristic
21C2×#805—21no—
22#4909C412yescomplemented
23#4909C2212yes—
24#4908C412yescomplemented
25#4908C2212yes—
26#4904C414yescomplemented
27#4717—22no—
28#4717C2214yes—
29#4716C2212yescomplemented
30#4711C2212yescomplemented
31#4710—22no—
32C3×#347C2214yescomplemented
33#4518C2214yes—
34#4517C2212yescomplemented
35#4516C2212yescomplemented
36#4513C2214yescomplemented
37#3414—31no—
38C2×#444—31no—
39#2943—31no—
40AGL(1,ℤ/16)—34no—
41#2180—34no—
42#2155—31no—
43#2148—31no—
44#2145—31no—
45C2×#347D311yescharacteristic, complemented
46C2×#235—21no—
47C2×#235—21no—
48C22×C3×C8D411yescharacteristic
49C22×C3×C8C2311yescharacteristic
50C2×C3×C42D411yescharacteristic
51C4×#67—42no—
52#926—21no—
53C2×#208—21no—
54C2×Dic12C2×C411yescharacteristic, complemented
55C2×Dic12C2311yescharacteristic
56#904—21no—
57#904—21no—
58#904—21no—
59#904—21no—
60#904C2×C412yes—
61#904C2×C412yes—
62#904C2×C412yescomplemented
63#904C2312yes—
64C2×D24C2×C411yescharacteristic, complemented
65C2×D24C2311yescharacteristic
66C2×#204—21no—
67C2×#204—21no—
68#901—24no—
69#901—42no—
70C2×ΓL(1,𝔽25)—41no—
71C2×D3×C8—41no—
72C4×D12—22no—
73#872—41no—
74C42×D3—41no—
75C4×Dic6—22no—
76C3×M32—21no—
77C3×M32C2×C412yescomplemented
78C3×M32C2312yes—
79C3×M32C2312yes—
80C2×C3×C16—21no—
81C2×C3×C16C2×C412yescomplemented
82C3×#97D412yes—
83C3×#97C2312yes—
84#821—22no—
85#819—21no—
86#819C2312yes—
87#818—21no—
88#817C2312yes—
89#816—22no—
90#805—24no—
91Dic24C2×C414yescomplemented
92#800C2×C414yescomplemented
93#800C2×C414yescomplemented
94D48C2×C414yescomplemented
95C2×#135—31no—
96C2×#135—31no—
97C2×#131—61no—
98C4×#41—61no—
99#510—32no—
100#510—32no—
101#509—32no—
102#509—32no—
103#508—34no—
104C2×M32C2×D311yescharacteristic
105C2×M32C2×D311yescharacteristic
106#444—34no—
107#444—62no—
108#442—32no—
109#441—32no—
110#440—62no—
111C2×#108—31no—
112#427—31no—
113C2×#104—61no—
114#416—61no—
115C2×#97C2×D311yescharacteristic
116#365—34no—
117#362—32no—
118#361—32no—
119#360—34no—
120#347C2×D314yescomplemented
121C22×C3×C4—21no—
122C22×C3×C4C2×D411yescharacteristic
123C2×#67—41no—
124C2×#67—41no—
125#235—22no—
126#235—22no—
127#235—22no—
128#235—22no—
129#235—41no—
130#235—41no—
131#235—41no—
132#235—41no—
133C2×D12—21no—
134C2×D12—21no—
135C2×C4×D3—41no—
136C2×C4×D3—41no—
137C2×C4×D3—42no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×C3×C8—21no—
141C2×C3×C8—21no—
142C2×C3×C8C22×C411yescharacteristic
143C2×C3×C8C22×C411yescharacteristic
144C2×C3×C8C2×D412yes—
145C2×C3×C8C2×D411yescharacteristic
146C2×C3×C8#4111yescharacteristic
147C2×C3×C8#4111yescharacteristic
148C2×C3×C8C2411yescharacteristic
149C3×C42—21no—
150C3×C42C2×D412yes—
151#217—41no—
152(C2×D3)⋊C4—42no—
153C4⋊Dic3—22no—
154Dic3⋊C4—42no—
155C4×Dic3—42no—
156C4×Dic3—41no—
157#208—22no—
158#208—41no—
159C2×C3⋊C8—41no—
160Dic12—21no—
161Dic12—21no—
162Dic12C22×C412yes—
163Dic12C22×C412yes—
164D24—21no—
165D24—21no—
166D24C22×C412yes—
167D24C22×C412yes—
168#204—22no—
169#204—22no—
170#204—22no—
171#204—22no—
172ΓL(1,𝔽25)—44no—
173D3×C8—44no—
174C3×C16—22no—
175C3×C16C22×C414yes—
176C2×#41—61no—
177C2×#41—61no—
178#135—32no—
179#135—32no—
180#135—32no—
181#135—32no—
182#135—61no—
183#135—61no—
184#135—61no—
185#135—61no—
186C2×Q16—31no—
187C2×Q16—31no—
188C2×SD16—61no—
189C2×SD16—61no—
190C2×D8—31no—
191C2×D8—31no—
192#131—64no—
193#131—122no—
194C2×M16—61no—
195C2×M16—121no—
196C22×C8—121no—
197C22×C8D1211yescharacteristic
198C22×C8C22×D311yescharacteristic
199C4×Q8—62no—
200C4×D4—62no—
201C4×D4—122no—
202#117—61no—
203#117—121no—
204C2×C42—121no—
205C2×C42D1211yescharacteristic
206Q32—34no—
207SD32—34no—
208SD32—34no—
209D16—34no—
210M32—21no—
211M32C4×D312yescomplemented
212M32C22×D312yes—
213M32C22×D312yes—
214C2×C16—21no—
215C2×C16C4×D312yescomplemented
216#108—32no—
217#108—61no—
218#107—61no—
219#106—32no—
220#104—64no—
221#103—62no—
222#102—62no—
223#97D1212yes—
224#97C22×D312yes—
225C23×C3AGL(1,ℤ/8)11yescharacteristic
226C22×D3—41no—
227C22×D3—41no—
228C2×C3×C4—22no—
229C2×C3×C4—81no—
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4C4×D411yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
235C2×C3×C4C22×D411yescharacteristic
236#67—42no—
237#67—42no—
238#67—42no—
239#67—42no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—42no—
243D12—22no—
244D12—22no—
245D12—41no—
246D12—41no—
247C4×D3—42no—
248C4×D3—42no—
249C4×D3—42no—
250C4×D3—42no—
251C4×D3—84no—
252Dic6—22no—
253Dic6—22no—
254Dic6—41no—
255Dic6—41no—
256C3×C8—22no—
257C3×C8C4×D411yescharacteristic
258C3×C8C4×D411yescharacteristic
259C3×C8C23×C411yescharacteristic
260C3×C8C2×#4111yescharacteristic
261C3⋊C8—42no—
262#41—62no—
263#41—62no—
264#41—62no—
265#41—62no—
266#41—121no—
267#41—121no—
268#41—121no—
269#41—121no—
270C2×Q8—61no—
271C2×Q8—61no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—121no—
275C2×D4—121no—
276C22×C4—21no—
277C22×C4—122no—
278C22×C4—121no—
279C22×C4—121no—
280C22×C4C2×D1211yescharacteristic
281Q16—32no—
282Q16—32no—
283Q16—61no—
284Q16—61no—
285SD16—62no—
286SD16—62no—
287SD16—62no—
288SD16—62no—
289D8—32no—
290D8—32no—
291D8—61no—
292D8—61no—
293M16—62no—
294M16—124no—
295M16—121no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—124no—
299C2×C8—121no—
300C2×C8C2×C4×D311yescharacteristic
301C2×C8C2×C4×D311yescharacteristic
302C2×C8C2×D1212yes—
303C2×C8C2×D1211yescharacteristic
304C2×C8#23511yescharacteristic
305C2×C8#23511yescharacteristic
306C2×C8C23×D311yescharacteristic
307C4⋊C4—62no—
308C4⋊C4—122no—
309C22⋊C4—122no—
310C22⋊C4—121no—
311C42—21no—
312C42—122no—
313C42—121no—
314C42C2×D1212yes—
315C16—22no—
316C16C2×C4×D314yes—
317C22×C3—41no—
318C22×C3#44211yescharacteristic
319C22×C3#44211yescharacteristic
320C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
321C2×D3—42no—
322C2×D3—42no—
323C2×D3—82no—
324C2×D3—82no—
325C3×C4—42no—
326C3×C4#44211yescharacteristic
327C3×C4#44211yescharacteristic
328C3×C4C2×C4×D411yescharacteristic
329C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
330Dic3—42no—
331Dic3—42no—
332Dic3—82no—
333C23—121no—
334C23—121no—
335C23#90811yescharacteristic
336Q8—62no—
337Q8—62no—
338Q8—121no—
339Q8—121no—
340D4—62no—
341D4—62no—
342D4—122no—
343D4—122no—
344D4—122no—
345D4—122no—
346D4—121no—
347D4—121no—
348C2×C4—22no—
349C2×C4—81no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—244no—
358C2×C4C4×D1211yescharacteristic
359C2×C4C4×D1211yescharacteristic
360C2×C4#90811yescharacteristic
361C2×C4#90811yescharacteristic
362C2×C4#90811yescharacteristic
363C2×C4C22×D1211yescharacteristic
364C8—22no—
365C8—122no—
366C8C4×D1211yescharacteristic
367C8C4×D1211yescharacteristic
368C8C22×C4×D311yescharacteristic
369C8C2×#23511yescharacteristic
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3C2×#44211yescharacteristic
374D3—82no—
375D3—82no—
376C22—41no—
377C22—122no—
378C22—122no—
379C22—242no—
380C22—242no—
381C22#471111yescharacteristic
382C22#471111yescharacteristic
383C22C2×#90811yescharacteristic
384C4—42no—
385C4—122no—
386C4—122no—
387C4—242no—
388C4#471111yescharacteristic
389C4#471111yescharacteristic
390C4C2×C4×D1211yescharacteristic
391C4C2×#90811yescharacteristic
392C3#3346811yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—21no—
396C2—242no—
397C2—242no—
398C2C2×#471111yescharacteristic
subgroupG/H
centerC4#4711
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC2×C3×#347C2
socleC2×C3C2×#442
prime radical#82009C2
cosocleC2×C3×C8C24
hypercenterC2×#347D3
hypocenterC3#33468
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ C2×#347

Complements

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

NH ≅ G/Ntype
2. C2×#4717C2semidirect
3. #78494C2semidirect
4. C2×C3×#347C2semidirect
5. #71450C2semidirect
6. #71442C2semidirect
7. #71427C2semidirect
8. #69091C2semidirect
9. #69090C2semidirect
22. #4909C4semidirect
24. #4908C4semidirect
26. #4904C4semidirect
29. #4716C22semidirect
30. #4711C22semidirect
32. C3×#347C22semidirect
34. #4517C22semidirect
35. #4516C22semidirect
36. #4513C22semidirect
45. C2×#347D3semidirect
54. C2×Dic12C2×C4semidirect
62. #904C2×C4semidirect
64. C2×D24C2×C4semidirect
77. C3×M32C2×C4semidirect
81. C2×C3×C16C2×C4semidirect
91. Dic24C2×C4semidirect
92. #800C2×C4semidirect
93. #800C2×C4semidirect
94. D48C2×C4semidirect
120. #347C2×D3semidirect
211. M32C4×D3semidirect
215. C2×C16C4×D3semidirect
392. C3#33468semidirect

Maximal subgroups

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

subgroupindex
#820092
C2×#47172
#784942
C2×C3×#3472
#714502
#714422
#714272
#690912
#690902
#334683