Finite Group Databasebeta

← #10765739

Subgroups — #10765739

Order 768.

666 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78468C211yesmaximal, characteristic
2#78458C211yesmaximal, characteristic
3#78445C211yesmaximal, characteristic, complemented
4C2×C3×#322C211yesmaximal, characteristic, complemented
5#69738C211yesmaximal, characteristic, complemented
6#69735C211yesmaximal, characteristic
7#69731C211yesmaximal, characteristic, complemented
8#68082C214yesmaximal, complemented
9#68076C214yesmaximal, complemented
10#17960—31nomaximal, Sylow
11C2×#901—21no—
12#5477C2211yescharacteristic
13#5470C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C2×C3×C4×C8C2211yescharacteristic
17#5114C414yes—
18#5110—21no—
19#5102—21no—
20#5101—21no—
21#4924C2211yescharacteristic
22#4924C2211yescharacteristic
23#4694C411yescharacteristic
24#4693C411yescharacteristic
25C4×#204C2214yes—
26#4688—21no—
27#4688C2212yes—
28C8×D12—21no—
29C8×D12C2212yes—
30#4681C411yescharacteristic
31#4680C411yescharacteristic
32#4679—21no—
33#4679C2212yes—
34C8×Dic6—21no—
35C8×Dic6C2212yes—
36C3×#322C2212yes—
37C3×#322C2212yes—
38#4459C2214yes—
39#4458C2214yes—
40#4456C2214yes—
41#3167—31no—
42#2960—31no—
43#2931—31no—
44#1592—34no—
45SD16⋊C8—34no—
46#1567—31no—
47#1493—31no—
48#1492—31no—
49C2×#322D311yescharacteristic, complemented
50C2×#235—21no—
51C22×C3×C8—21no—
52C22×C3×C8C2×C412yes—
53C22×C3×C8D411yescharacteristic
54C22×C3×C8D411yescharacteristic
55C2×C3×C42C2311yescharacteristic
56#930—21no—
57#930C2×C412yes—
58#926—21no—
59#924—21no—
60#924C2×C412yes—
61C2×#208—21no—
62C2×#208—21no—
63C2×#204C2×C414yes—
64#901—44no—
65C2×ΓL(1,𝔽25)—22no—
66C2×D3×C8—22no—
67#875—21no—
68#874C2×C411yescharacteristic
69C4×D12—21no—
70C4×D12C2312yes—
71#870—21no—
72#869C2×C411yescharacteristic
73#869C2×C411yescharacteristic
74#869C2×C411yescharacteristic
75C4×Dic6—21no—
76C4×Dic6C2312yes—
77C3×C4×C8—21no—
78C3×C4×C8C2×C411yescharacteristic
79C3×C4×C8C2×C411yescharacteristic
80C3×C4×C8D411yescharacteristic
81C3×C4×C8D411yescharacteristic
82C3×C4×C8C2312yes—
83C3×C4×C8C2312yes—
84C3×C4×C8C2311yescharacteristic
85C3×C4×C8C2311yescharacteristic
86#821C2×C414yes—
87(C2×D3)⋊C8—22no—
88(C2×D3)⋊C8—22no—
89#817C2×C414yes—
90#816C2×C414yes—
91#815—22no—
92Dic3⋊C8—22no—
93Dic3⋊C8—22no—
94C8×Dic3—22no—
95C4⋊(C3⋊C8)—21no—
96C4⋊(C3⋊C8)—21no—
97C4⋊(C3⋊C8)C2312yes—
98C4⋊(C3⋊C8)C2312yes—
99C2×#131—61no—
100#533—31no—
101#532—31no—
102#498—31no—
103#492—31no—
104#487—31no—
105#486—31no—
106#465—34no—
107#446—32no—
108#446—61no—
109C8×Q8—32no—
110C8×Q8—61no—
111C4×SD16—34no—
112#436—32no—
113#436—61no—
114C8×D4—32no—
115C8×D4—61no—
116#424—61no—
117#423—31no—
118#423—31no—
119#408—61no—
120#405—61no—
121C2×C4×C8C2×D311yescharacteristic
122C2×C4×C8C2×D311yescharacteristic
123C2×C4×C8C2×D311yescharacteristic
124#335—34no—
125Q8⋊C8—34no—
126D4⋊C8—34no—
127#322C2×D312yes—
128#322C2×D312yes—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C2×#67—22no—
133#235—42no—
134#235—42no—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12C22×C412yes—
138C2×C4×D3—22no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6C22×C412yes—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C2×C3×C8—24no—
145C2×C3×C8—22no—
146C2×C3×C8—42no—
147C2×C3×C8C22×C412yes—
148C2×C3×C8C22×C412yes—
149C2×C3×C8C2×D411yescharacteristic
150C2×C3×C8C2×D411yescharacteristic
151C2×C3×C8C2×D411yescharacteristic
152C2×C3×C8C2×D411yescharacteristic
153C3×C42C22×C411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156C3×C42C2411yescharacteristic
157#217—22no—
158(C2×D3)⋊C4—22no—
159(C2×D3)⋊C4—22no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3C22×C412yes—
165C4⋊Dic3C22×C412yes—
166Dic3⋊C4—22no—
167Dic3⋊C4—22no—
168C4×Dic3—22no—
169#208—42no—
170#208—42no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173#204—28no—
174ΓL(1,𝔽25)—44no—
175D3×C8—44no—
176C2×#41—61no—
177C2×SD16—34no—
178#131—124no—
179C2×M16—61no—
180C2×M16—61no—
181C2×M16—62no—
182C22×C8—21no—
183C22×C8—62no—
184C22×C8C4×D312yes—
185C22×C8D1211yescharacteristic
186C22×C8D1211yescharacteristic
187C4⋊Q8—31no—
188C4⋊Q8—31no—
189C4⋊Q8—31no—
190#127—31no—
191#125—61no—
192#124—61no—
193C22⋊Q8—32no—
194C22⋊Q8—61no—
195#121—32no—
196#121—61no—
197C4×Q8—32no—
198C4×Q8—61no—
199C4×D4—32no—
200C4×D4—61no—
201#117—61no—
202C2×C42C22×D311yescharacteristic
203#106—34no—
204C4⋊C8—32no—
205C4⋊C8—32no—
206C4⋊C8—62no—
207C4⋊C8—62no—
208C4⋊C8—61no—
209C4⋊C8—61no—
210#103—34no—
211#102—34no—
212C22⋊C8—62no—
213C22⋊C8—62no—
214#97—62no—
215C4×C8—21no—
216C4×C8—62no—
217C4×C8C4×D311yescharacteristic
218C4×C8C4×D311yescharacteristic
219C4×C8D1211yescharacteristic
220C4×C8D1211yescharacteristic
221C4×C8C22×D312yes—
222C4×C8C22×D312yes—
223C4×C8C22×D311yescharacteristic
224C4×C8C22×D311yescharacteristic
225C23×C3C4×D411yescharacteristic
226C22×D3—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4C4×D411yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)12yes—
233C2×C3×C4#13712yes—
234C2×C3×C4C23×C411yescharacteristic
235C2×C3×C4C22×D411yescharacteristic
236C2×C3×C4C2×#4111yescharacteristic
237#67—44no—
238C2×Dic3—22no—
239C2×Dic3—22no—
240C2×Dic3—22no—
241D12—24no—
242D12—42no—
243D12—42no—
244C4×D3—42no—
245C4×D3—42no—
246Dic6—24no—
247Dic6—42no—
248Dic6—42no—
249C3×C8—22no—
250C3×C8—21no—
251C3×C8—21no—
252C3×C8—21no—
253C3×C8—21no—
254C3×C8—42no—
255C3×C8#13114yescomplemented
256C3⋊C8—42no—
257C3⋊C8—42no—
258#41—122no—
259#41—122no—
260C2×Q8—32no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×D4—32no—
264C2×D4—62no—
265C2×D4—61no—
266C2×D4—61no—
267C22×C4—62no—
268C22×C4C2×C4×D311yescharacteristic
269C22×C4C2×D1211yescharacteristic
270C22×C4#23511yescharacteristic
271SD16—68no—
272M16—124no—
273M16—122no—
274M16—122no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—42no—
280C2×C8—62no—
281C2×C8—62no—
282C2×C8—124no—
283C2×C8C2×C4×D312yes—
284C2×C8C2×C4×D312yes—
285C2×C8C2×D1211yescharacteristic
286C2×C8C2×D1211yescharacteristic
287C2×C8C2×D1211yescharacteristic
288C2×C8C2×D1211yescharacteristic
289C4⋊C4—32no—
290C4⋊C4—32no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—62no—
295C4⋊C4—61no—
296C4⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C42—62no—
301C42C2×C4×D311yescharacteristic
302C42C2×D1211yescharacteristic
303C42#23511yescharacteristic
304C42C23×D311yescharacteristic
305C22×C3—21no—
306C22×C3—21no—
307C22×C3#44012yes—
308C22×C3C2×C4×D411yescharacteristic
309C2×D3—42no—
310C2×D3—42no—
311C2×D3—42no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4#44012yes—
315C3×C4#44412yes—
316C3×C4C2×#13112yes—
317C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
318C3×C4C2×#13711yescharacteristic
319Dic3—42no—
320Dic3—42no—
321Dic3—42no—
322C23—62no—
323C23C4×D1211yescharacteristic
324Q8—64no—
325Q8—122no—
326Q8—122no—
327D4—64no—
328D4—124no—
329D4—122no—
330D4—122no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—62no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4C4×D1211yescharacteristic
339C2×C4C4×D1211yescharacteristic
340C2×C4C4×D1211yescharacteristic
341C2×C4#90812yes—
342C2×C4#90912yes—
343C2×C4C22×C4×D311yescharacteristic
344C2×C4C22×D1211yescharacteristic
345C2×C4C2×#23511yescharacteristic
346C8—22no—
347C8—21no—
348C8—21no—
349C8—21no—
350C8—21no—
351C8—42no—
352C8—122no—
353C8—122no—
354C8#90114yescomplemented
355C2×C3—22no—
356C2×C3#293111yescharacteristic
357C2×C3C2×#44011yescharacteristic
358C2×C3C2×#44411yescharacteristic
359D3—82no—
360C22—21no—
361C22—21no—
362C22—122no—
363C22—122no—
364C22—122no—
365C22#471012yes—
366C22C2×C4×D1211yescharacteristic
367C4—21no—
368C4—21no—
369C4—122no—
370C4—122no—
371C4—122no—
372C4#471012yes—
373C4#471712yes—
374C4C2×#90112yes—
375C4C2×#90811yescharacteristic
376C4C2×#90911yescharacteristic
377C3#1796011yescharacteristic, complemented, Sylow
378C2—22no—
379C2—242no—
380C2#7845811yescharacteristic
381C2C2×#471011yescharacteristic
382C2C2×#471711yescharacteristic
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC2×C3×#322C2
socleC22×C3C2×C4×D4
prime radical#5477C22
cosocleC3×C42C24
hypercenterC2×#322D3
hypocenterC3#17960
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×#322

Complements

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

NH ≅ G/Ntype
3. #78445C2semidirect
4. C2×C3×#322C2semidirect
5. #69738C2semidirect
7. #69731C2semidirect
8. #68082C2semidirect
9. #68076C2semidirect
49. C2×#322D3semidirect
255. C3×C8#131semidirect
354. C8#901semidirect
377. C3#17960semidirect

Maximal subgroups

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

subgroupindex
#784682
#784582
#784452
C2×C3×#3222
#697382
#697352
#697312
#680822
#680762
#179603