Finite Group Databasebeta

← #10950879

Subgroups — #10950879

Order 768.

592 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3169C211yesmaximal, characteristic, complemented
2#78470C211yesmaximal, characteristic, complemented
3#71385C211yesmaximal, characteristic, complemented
4#71381C211yesmaximal, characteristic, complemented
5#71290C211yesmaximal, characteristic, complemented
6#71288C211yesmaximal, characteristic, complemented
7#71073C214yesmaximal, complemented
8#70885C214yesmaximal, complemented
9C4×#812C211yesmaximal, characteristic, complemented
10#33748—31nomaximal, Sylow
11C3×#532—21no—
12#5477—21no—
13C3×#500C2211yescharacteristic
14C3×#493C2211yescharacteristic
15C3×#486—21no—
16C3×#466—21no—
17C3×#466C2212yes—
18C3×C4×D8—21no—
19C3×C4×D8C2212yes—
20C2×C3×C4×C8C2211yescharacteristic
21C2×#826C2214yes—
22#5115—21no—
23#5115C2212yes—
24C2×#812C2212yes—
25#4695C2211yescharacteristic
26#4694—21no—
27C4×D24—21no—
28C4×D24C2212yes—
29#4682C2211yescharacteristic
30#4562C2214yes—
31#4491C2214yes—
32#4490C2214yes—
33#4461C2212yes—
34C4×C3⋊C16C2211yescharacteristic
35C4×C3⋊C16C2211yescharacteristic
36#3169—31no—
37#3169D311yescharacteristic, complemented
38#2254—31no—
39#2246—31no—
40#2240—31no—
41#2240—31no—
42#2218—34no—
43#2177—34no—
44C4×M32—31no—
45C2×C3×#41—41no—
46C2×#235—41no—
47C2×C3×D8—21no—
48C2×C3×D8C2312yes—
49C22×C3×C8D411yescharacteristic
50C22×C3×C8C2311yescharacteristic
51C3×C4⋊Q8—21no—
52C3×#127—21no—
53C3×#124—41no—
54C3×#121—22no—
55C3×#121—41no—
56C3×C4×D4—22no—
57C3×C4×D4—41no—
58C2×C3×C42D411yescharacteristic
59#930—22no—
60#930—41no—
61C2×D24—21no—
62C2×D24C2312yes—
63#875—41no—
64#874—21no—
65C4×D12—22no—
66C4×D12—41no—
67#869—21no—
68C3×#107—21no—
69C3×#107C2312yes—
70C3×#102—22no—
71C3×#102—22no—
72C3×C4×C8D411yescharacteristic
73C3×C4×C8D411yescharacteristic
74C3×C4×C8C2311yescharacteristic
75C3×C4×C8C2311yescharacteristic
76#826—28no—
77#821—22no—
78#821—22no—
79#818—21no—
80#818C2312yes—
81#812—24no—
82C2×C3⋊C16C2312yes—
83C2×C3⋊C16C2312yes—
84#532—21no—
85#532—61no—
86C2×D16—34no—
87C2×M32—32no—
88#500—31no—
89#500C2×D311yescharacteristic
90#493—31no—
91#493C2×D311yescharacteristic
92#486—21no—
93#486—61no—
94#466—21no—
95#466—32no—
96#466—61no—
97#466C2×D312yes—
98C4×D8—21no—
99C4×D8—32no—
100C4×D8—61no—
101C4×D8C2×D312yes—
102C2×C4×C8C2×D311yescharacteristic
103#367—34no—
104#357—34no—
105#357—34no—
106#346—32no—
107C4×C16—31no—
108C4×C16—31no—
109C3×#41—84no—
110C2×C3×Q8—41no—
111C2×C3×D4—22no—
112C2×C3×D4—42no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C22×C3×C4—42no—
116C22×C3×C4D811yescharacteristic
117C22×C3×C4D811yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C2×#67—42no—
120#235—84no—
121C2×D12—22no—
122C2×D12—41no—
123C2×D12—41no—
124C2×C4×D3—42no—
125C2×Dic6—41no—
126C3×D8—24no—
127C3×D8—42no—
128C2×C3×C8—22no—
129C2×C3×C8—22no—
130C2×C3×C8C2×D411yescharacteristic
131C2×C3×C8C2×D411yescharacteristic
132C2×C3×C8C2×D411yescharacteristic
133C2×C3×C8C2411yescharacteristic
134C3×C4⋊C4—22no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—42no—
137C3×C22⋊C4—42no—
138C3×C42D811yescharacteristic
139C3×C42D811yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142(C2×D3)⋊C4—42no—
143(C2×D3)⋊C4—42no—
144C4⋊Dic3—22no—
145C4⋊Dic3—41no—
146D24—24no—
147D24—42no—
148C3⋊C16—22no—
149C3⋊C16#4114yescomplemented
150C2×#41—41no—
151C2×#41—121no—
152C2×D8—21no—
153C2×D8—32no—
154C2×D8—61no—
155C2×D8C22×D312yes—
156C22×C8#6711yescharacteristic
157C22×C8C22×D311yescharacteristic
158C4⋊Q8—21no—
159C4⋊Q8—61no—
160#127—21no—
161#127—61no—
162#124—41no—
163#124—121no—
164#121—22no—
165#121—41no—
166#121—62no—
167#121—121no—
168C4×D4—22no—
169C4×D4—41no—
170C4×D4—62no—
171C4×D4—121no—
172C2×C42#6711yescharacteristic
173D16—68no—
174M32—64no—
175C2×C16—32no—
176C2×C16—32no—
177#107—21no—
178#107—32no—
179#107—61no—
180#107C22×D312yes—
181#102—22no—
182#102—22no—
183#102—62no—
184#102—62no—
185C4×C8#6711yescharacteristic
186C4×C8#6711yescharacteristic
187C4×C8C22×D311yescharacteristic
188C4×C8C22×D311yescharacteristic
189C23×C3—42no—
190C23×C3C2×D811yescharacteristic
191C22×D3—42no—
192C3×Q8—82no—
193C3×D4—42no—
194C3×D4—42no—
195C3×D4—84no—
196C3×D4—82no—
197C3×D4—82no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—42no—
203C2×C3×C4—84no—
204C2×C3×C4C2×D811yescharacteristic
205C2×C3×C4C2×D811yescharacteristic
206C2×C3×C4C2×D811yescharacteristic
207C2×C3×C4C2×D811yescharacteristic
208C2×C3×C4C2×D811yescharacteristic
209C2×C3×C4C22×D411yescharacteristic
210#67—84no—
211C2×Dic3—42no—
212D12—42no—
213D12—42no—
214D12—82no—
215D12—82no—
216C4×D3—84no—
217Dic6—82no—
218C3×C8—21no—
219C3×C8—21no—
220C3×C8#13512yes—
221C3×C8C2×#4112yes—
222#41—84no—
223#41—244no—
224C2×Q8—41no—
225C2×Q8—121no—
226C2×D4—22no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—62no—
231C2×D4—122no—
232C2×D4—121no—
233C2×D4—121no—
234C22×C4—42no—
235C22×C4—122no—
236C22×C4#21311yescharacteristic
237C22×C4#21311yescharacteristic
238C22×C4C2×#6711yescharacteristic
239D8—24no—
240D8—42no—
241D8—64no—
242D8—122no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8C2×#6711yescharacteristic
246C2×C8C2×#6711yescharacteristic
247C2×C8C2×#6711yescharacteristic
248C2×C8C23×D311yescharacteristic
249C4⋊C4—22no—
250C4⋊C4—41no—
251C4⋊C4—62no—
252C4⋊C4—121no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—122no—
256C22⋊C4—122no—
257C42#21311yescharacteristic
258C42#21311yescharacteristic
259C42C2×#6711yescharacteristic
260C42C2×#6711yescharacteristic
261C16—34no—
262C16—62no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—82no—
267C22×C3—82no—
268C22×C3—82no—
269C22×C3C22×D811yescharacteristic
270C2×D3—82no—
271C2×D3—82no—
272C2×D3—82no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—82no—
277C3×C4#50912yes—
278C3×C4#50912yes—
279C3×C4#53211yescharacteristic
280C3×C4C2×#13511yescharacteristic
281Dic3—82no—
282C23—42no—
283C23—122no—
284C23C2×#21311yescharacteristic
285Q8—82no—
286Q8—242no—
287D4—42no—
288D4—42no—
289D4—84no—
290D4—82no—
291D4—82no—
292D4—122no—
293D4—122no—
294D4—244no—
295D4—242no—
296D4—242no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—84no—
303C2×C4—122no—
304C2×C4—244no—
305C2×C4C2×#21311yescharacteristic
306C2×C4C2×#21311yescharacteristic
307C2×C4C2×#21311yescharacteristic
308C2×C4C2×#21311yescharacteristic
309C2×C4C2×#21311yescharacteristic
310C2×C4C22×#6711yescharacteristic
311C8—21no—
312C8—21no—
313C8#95012yes—
314C8D3×#4112yes—
315C2×C3—41no—
316C2×C3—162no—
317C2×C3#316911yescharacteristic
318C2×C3C2×#50911yescharacteristic
319C2×C3C2×#50911yescharacteristic
320D3—162no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—242no—
328C22—242no—
329C22—242no—
330C22C22×#21311yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—82no—
335C4—242no—
336C4#514712yes—
337C4#519312yes—
338C4C2×#95011yescharacteristic
339C4#583211yescharacteristic
340C3#3374811yescharacteristic, complemented, Sylow
341C2—41no—
342C2—162no—
343C2—482no—
344C2#8329111yescharacteristic
345C2C2×#514711yescharacteristic
346C2C2×#519311yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3169C2
socleC22×C3C22×D8
hypercenter#3169D3
hypocenterC3#33748
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. C3×#3169C2semidirect
2. #78470C2semidirect
3. #71385C2semidirect
4. #71381C2semidirect
5. #71290C2semidirect
6. #71288C2semidirect
7. #71073C2semidirect
8. #70885C2semidirect
9. C4×#812C2semidirect
37. #3169D3semidirect
149. C3⋊C16#41semidirect
340. C3#33748semidirect

Maximal subgroups

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

subgroupindex
C3×#31692
#784702
#713852
#713812
#712902
#712882
#710732
#708852
C4×#8122
#337483