Finite Group Databasebeta

← #10948646

Subgroups — #10948646

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
1C3×#2957C211yesmaximal, characteristic, complemented
2#83894C211yesmaximal, characteristic
3#78574C211yesmaximal, characteristic, complemented
4C2×#4566C211yesmaximal, characteristic, complemented
5#70929C214yesmaximal, complemented
6#70928C214yesmaximal, complemented
7#70687C211yesmaximal, characteristic, complemented
8C2×#4494C211yesmaximal, characteristic, complemented
9#69884C211yesmaximal, characteristic, complemented
10#33469—31nomaximal, Sylow
11C2×C3×#135C2211yescharacteristic
12C3×#568—21no—
13C2×#904C2211yescharacteristic
14#5523—21no—
15C3×#444C2214yes—
16C3×#443—22no—
17C2×C3×#108C2211yescharacteristic
18C3×#421—21no—
19C3×#405C2211yescharacteristic
20#5195C2212yes—
21#5194C414yescomplemented
22#5193C2212yes—
23#5179C412yescomplemented
24#5147C412yescomplemented
25#5133—21no—
26C2×#817C2211yescharacteristic
27C2×#812C2211yescharacteristic
28C2×#812C2211yescharacteristic
29#4716C2212yescomplemented
30#4711C2212yescomplemented
31C4×#204—22no—
32#4566C2214yes—
33#4496C2212yescomplemented
34#4495C2212yescomplemented
35#4494C2214yescomplemented
36#4466C2214yescomplemented
37#3414—31no—
38#2957D311yescharacteristic, complemented
39#2952—31no—
40AGL(1,ℤ/16)—34no—
41#2180—34no—
42C2×#365—31no—
43#2148—31no—
44C2×#360—31no—
45#2110—31no—
46C2×C3×#41—21no—
47C2×#235—21no—
48C3×#135—21no—
49C3×#135—21no—
50C3×#135C2×C412yescomplemented
51C3×#135C2312yes—
52C2×C3×Q16C2×C411yescharacteristic, complemented
53C2×C3×SD16—21no—
54C2×C3×D8C2×C411yescharacteristic, complemented
55C3×#131—24no—
56C3×#131—42no—
57C2×C3×M16—21no—
58C2×C3×M16—42no—
59C2×C3×M16D411yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×#117D411yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×Dic12C2311yescharacteristic
64#904—21no—
65#904—21no—
66#904C2×C412yes—
67#904C2×C412yes—
68C2×D24C2311yescharacteristic
69C2×#204—21no—
70#892—41no—
71D3×C4⋊C4—41no—
72#67⋊C4—42no—
73C4×D12—22no—
74C4×Dic6—22no—
75C3×#108—21no—
76C3×#108C2312yes—
77C3×#104—24no—
78C3×#97C2312yes—
79C3×C4×C8D412yes—
80#829C2×C414yescomplemented
81#828C2×C414yescomplemented
82#827C2×C414yescomplemented
83#826C2×C414yescomplemented
84#821—22no—
85#817—21no—
86#817C2312yes—
87#816—22no—
88#812—21no—
89#812C2×C412yescomplemented
90#812C2312yes—
91#812C2312yes—
92C2×C3⋊C16—21no—
93C2×C3⋊C16C2×C412yescomplemented
94C2×#135—31no—
95C2×#135C2×D311yescharacteristic
96#568—21no—
97#520—61no—
98#510—32no—
99#510—32no—
100#509—32no—
101#509—32no—
102#508—34no—
103C2×M32—31no—
104C2×M32—31no—
105#444C2×D314yes—
106#443—22no—
107#442—32no—
108#441—32no—
109C4×SD16—62no—
110C2×#108C2×D311yescharacteristic
111C2×#106—31no—
112#421—21no—
113#417—61no—
114#405C2×D311yescharacteristic
115#365—34no—
116#362—32no—
117#361—32no—
118#360—34no—
119#347—34no—
120C3×#41—22no—
121C3×#41—22no—
122C3×#41—41no—
123C3×#41—41no—
124C2×C3×Q8—21no—
125C2×C3×D4—21no—
126C2×C3×D4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C2×#67—41no—
130C22×Dic3—41no—
131#235—22no—
132#235—22no—
133#235—41no—
134#235—41no—
135C2×D12—21no—
136C2×C4×D3—41no—
137C2×C4×D3—42no—
138C2×Dic6—21no—
139C3×Q16—21no—
140C3×Q16C22×C412yes—
141C3×SD16—22no—
142C3×SD16—22no—
143C3×D8—21no—
144C3×D8C22×C412yes—
145C3×M16—21no—
146C3×M16—21no—
147C3×M16—22no—
148C3×M16—44no—
149C3×M16—41no—
150C2×C3×C8—44no—
151C2×C3×C8—41no—
152C2×C3×C8C22×C411yescharacteristic
153C2×C3×C8C22×C411yescharacteristic
154C2×C3×C8C2×D412yes—
155C2×C3×C8C2×D411yescharacteristic
156C2×C3×C8#4111yescharacteristic
157C2×C3×C8#4111yescharacteristic
158C2×C3×C8C2411yescharacteristic
159C3×C4⋊C4—21no—
160C3×C22⋊C4—21no—
161C3×C42C2×D412yes—
162(C2×D3)⋊C4—42no—
163C4⋊Dic3—22no—
164C4⋊Dic3—41no—
165Dic3⋊C4—42no—
166C4×Dic3—42no—
167Dic12—21no—
168Dic12C22×C412yes—
169D24—21no—
170D24C22×C412yes—
171#204—22no—
172#204—22no—
173C3⋊C16—22no—
174C3⋊C16C22×C414yes—
175C2×#41—21no—
176C2×#41—61no—
177#135—21no—
178#135—21no—
179#135—32no—
180#135—32no—
181#135—61no—
182#135—61no—
183#135C4×D312yescomplemented
184#135C22×D312yes—
185C2×Q16—31no—
186C2×Q16C4×D311yescharacteristic, complemented
187C2×SD16—21no—
188C2×SD16—61no—
189C2×D8—31no—
190C2×D8C4×D311yescharacteristic, complemented
191#131—24no—
192#131—42no—
193C2×M16—21no—
194C2×M16—42no—
195C2×M16#6711yescharacteristic
196C22×C8C22×D311yescharacteristic
197C4×Q8—62no—
198C4×D4—62no—
199C4×D4—122no—
200#117—121no—
201#117#6711yescharacteristic
202C2×C4⋊C4—61no—
203C2×C4⋊C4—121no—
204Q32—34no—
205SD32—34no—
206SD32—34no—
207D16—34no—
208M32—32no—
209M32—32no—
210M32—32no—
211M32—61no—
212C2×C16—32no—
213C2×C16—61no—
214#108—21no—
215#108C22×D312yes—
216#106—32no—
217#106—61no—
218#104—24no—
219#103—62no—
220#102—62no—
221#97C22×D312yes—
222C4×C8#6712yes—
223C23×C3—41no—
224C23×C3AGL(1,ℤ/8)11yescharacteristic
225C22×D3—41no—
226C3×Q8—22no—
227C3×Q8—41no—
228C3×D4—22no—
229C3×D4—42no—
230C3×D4—42no—
231C3×D4—41no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
239C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
240C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
241C2×C3×C4C22×D411yescharacteristic
242#67—42no—
243#67—42no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246C2×Dic3—161no—
247D12—22no—
248D12—41no—
249C4×D3—42no—
250C4×D3—42no—
251C4×D3—162no—
252Dic6—22no—
253Dic6—41no—
254C3×C8—22no—
255C3×C8—42no—
256C3×C8C4×D411yescharacteristic
257C3×C8C4×D411yescharacteristic
258C3×C8C23×C411yescharacteristic
259C3×C8C2×#4111yescharacteristic
260#41—22no—
261#41—22no—
262#41—41no—
263#41—41no—
264#41—62no—
265#41—62no—
266#41—121no—
267#41—121no—
268C2×Q8—21no—
269C2×Q8—61no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—61no—
273C2×D4—121no—
274C22×C4—41no—
275C22×C4—122no—
276C22×C4—121no—
277C22×C4—121no—
278C22×C4C2×#6711yescharacteristic
279Q16—21no—
280Q16—32no—
281Q16—61no—
282Q16C2×C4×D312yes—
283SD16—22no—
284SD16—22no—
285SD16—62no—
286SD16—62no—
287D8—21no—
288D8—32no—
289D8—61no—
290D8C2×C4×D312yes—
291M16—21no—
292M16—21no—
293M16—22no—
294M16—44no—
295M16—41no—
296C2×C8—44no—
297C2×C8—41no—
298C2×C8C2×C4×D311yescharacteristic
299C2×C8C2×C4×D311yescharacteristic
300C2×C8#23511yescharacteristic
301C2×C8#23511yescharacteristic
302C2×C8C2×#6712yes—
303C2×C8C2×#6711yescharacteristic
304C2×C8C23×D311yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—62no—
307C4⋊C4—122no—
308C4⋊C4—121no—
309C22⋊C4—21no—
310C22⋊C4—122no—
311C42—122no—
312C42C2×#6712yes—
313C16—34no—
314C16—62no—
315C22×C3—41no—
316C22×C3—42no—
317C22×C3—161no—
318C22×C3#44211yescharacteristic
319C22×C3#44211yescharacteristic
320C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
321C2×D3—42no—
322C2×D3—81no—
323C2×D3—81no—
324C3×C4—42no—
325C3×C4—42no—
326C3×C4#44211yescharacteristic
327C3×C4#44211yescharacteristic
328C3×C4C2×C4×D411yescharacteristic
329C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
330Dic3—42no—
331Dic3—82no—
332C23—41no—
333C23—121no—
334C23#94911yescharacteristic
335Q8—22no—
336Q8—41no—
337Q8—62no—
338Q8—121no—
339D4—22no—
340D4—42no—
341D4—42no—
342D4—41no—
343D4—62no—
344D4—122no—
345D4—122no—
346D4—121no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—121no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—482no—
356C2×C4—481no—
357C2×C4C4×#6711yescharacteristic
358C2×C4C4×#6711yescharacteristic
359C2×C4#93211yescharacteristic
360C2×C4#93211yescharacteristic
361C2×C4#94911yescharacteristic
362C2×C4C22×#6711yescharacteristic
363C8—22no—
364C8—42no—
365C8C4×#6711yescharacteristic
366C8C4×#6711yescharacteristic
367C8C22×C4×D311yescharacteristic
368C8C2×#23511yescharacteristic
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—82no—
373C2×C3C2×#44211yescharacteristic
374D3—82no—
375C22—41no—
376C22—42no—
377C22—122no—
378C22—161no—
379C22—241no—
380C22—241no—
381C22#501411yescharacteristic
382C22#501411yescharacteristic
383C22C2×#94911yescharacteristic
384C4—42no—
385C4—42no—
386C4—122no—
387C4—242no—
388C4#501411yescharacteristic
389C4#501411yescharacteristic
390C4C2×C4×#6711yescharacteristic
391C4C2×#93211yescharacteristic
392C3#3346911yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—242no—
398C2C2×#501411yescharacteristic
subgroupG/H
centerC2C2×#5014
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC3×#2957C2
socleC2×C3C2×#442
prime radical#83894C2
cosocleC2×C3×C8C24
hypercenter#2957D3
hypocenterC3#33469
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
1. C3×#2957C2semidirect
3. #78574C2semidirect
4. C2×#4566C2semidirect
5. #70929C2semidirect
6. #70928C2semidirect
7. #70687C2semidirect
8. C2×#4494C2semidirect
9. #69884C2semidirect
21. #5194C4semidirect
23. #5179C4semidirect
24. #5147C4semidirect
29. #4716C22semidirect
30. #4711C22semidirect
33. #4496C22semidirect
34. #4495C22semidirect
35. #4494C22semidirect
36. #4466C22semidirect
38. #2957D3semidirect
50. C3×#135C2×C4semidirect
52. C2×C3×Q16C2×C4semidirect
54. C2×C3×D8C2×C4semidirect
80. #829C2×C4semidirect
81. #828C2×C4semidirect
82. #827C2×C4semidirect
83. #826C2×C4semidirect
89. #812C2×C4semidirect
93. C2×C3⋊C16C2×C4semidirect
183. #135C4×D3semidirect
186. C2×Q16C4×D3semidirect
190. C2×D8C4×D3semidirect
392. C3#33469semidirect

Maximal subgroups

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

subgroupindex
C3×#29572
#838942
#785742
C2×#45662
#709292
#709282
#706872
C2×#44942
#698842
#334693