Finite Group Databasebeta

← #10693512

Subgroups — #10693512

Order 768.

466 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#70465C211yesmaximal, characteristic, complemented
2#70457C211yesmaximal, characteristic
3C2×#4485C211yesmaximal, characteristic
4#69580C211yesmaximal, characteristic
5#69567C211yesmaximal, characteristic, complemented
6C3×#1296C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#9563—31nomaximal, Sylow
9C2×#874C2211yescharacteristic
10C2×C3×C4⋊C8C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C3×#384C2211yescharacteristic
13C2×#821—21no—
14C2×#821—21no—
15C2×C8×Dic3—21no—
16C2×#809—21no—
17C2×#809—21no—
18C2×#807—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22#4550—21no—
23#4529—21no—
24#4485C414yescomplemented
25#1982—31no—
26#1957—31no—
27#1947—31no—
28C8×C4⋊C4—31no—
29C2×#331—31no—
30#1296D311yescharacteristic, complemented
31#1277—31no—
32C22×D12—21no—
33C22×D12—21no—
34C22×D12—21no—
35C22×C3×C8—21no—
36C22×C3×C8—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C42C2311yescharacteristic
41C2×C4⋊Dic3D411yescharacteristic
42C2×C4⋊Dic3D411yescharacteristic
43C2×C4×Dic3—21no—
44C22×C3⋊C8—21no—
45C22×C3⋊C8D411yescharacteristic
46C22×C3⋊C8D411yescharacteristic
47#874—21no—
48#874—21no—
49#874—21no—
50#874C2×C412yes—
51C3×C4⋊C8—21no—
52C3×C4⋊C8—21no—
53C3×C4⋊C8—21no—
54C3×C4⋊C8C2×C412yes—
55C3⋊#95—21no—
56#821—42no—
57#821—42no—
58C8×Dic3—42no—
59#809—42no—
60#809—42no—
61#807—42no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)C2×C412yes—
64C4×C3⋊C8—21no—
65C4×C3⋊C8—21no—
66C2×#127—31no—
67C2×#107—61no—
68C2×C4⋊C8—31no—
69C2×C4⋊C8C2×D311yescharacteristic
70C2×C4⋊C8C2×D311yescharacteristic
71C2×#102—61no—
72C2×#102—61no—
73C2×#102—61no—
74C2×#102—61no—
75C2×C4×C8—31no—
76C2×C4×C8—61no—
77#384C2×D311yescharacteristic
78C4×C4⋊C4—31no—
79#340—61no—
80#336—61no—
81#331—34no—
82C23×D3—41no—
83C22×C3×C4—41no—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4#4111yescharacteristic
87C22×Dic3—21no—
88C22×Dic3—21no—
89C2×D12—22no—
90C2×D12—22no—
91C2×D12—22no—
92C2×D12—41no—
93C2×D12—41no—
94C2×D12—41no—
95C2×D12—41no—
96C2×D12—41no—
97C2×D12—41no—
98C2×D12—41no—
99C2×D12—41no—
100C2×D12—41no—
101C2×D12—41no—
102C2×D12—41no—
103C2×D12—81no—
104C2×D12—81no—
105C2×C3×C8—22no—
106C2×C3×C8—41no—
107C2×C3×C8—41no—
108C2×C3×C8—41no—
109C2×C3×C8—41no—
110C2×C3×C8—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—81no—
116C3×C42C22×C411yescharacteristic
117C3×C42#4111yescharacteristic
118C3×C42#4111yescharacteristic
119C3×C42#4111yescharacteristic
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3D812yes—
123C4⋊Dic3SD1612yes—
124C4×Dic3—41no—
125C4×Dic3—41no—
126C2×C3⋊C8—21no—
127C2×C3⋊C8—21no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8—21no—
130C2×C3⋊C8—21no—
131C2×C3⋊C8—21no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8D812yes—
135C22×D4—61no—
136C22×D4—61no—
137C22×D4—61no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—31no—
141C22×C8—31no—
142C22×C8—61no—
143#127—32no—
144#127—61no—
145#127—61no—
146#127—61no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C42—61no—
153C2×C42C22×D311yescharacteristic
154#107—122no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—32no—
159C4⋊C8—61no—
160C4⋊C8C4×D312yes—
161#102—122no—
162#102—122no—
163#102—122no—
164#102—122no—
165C4×C8—61no—
166C4×C8—61no—
167C4×C8—122no—
168#95—61no—
169C23×C3#12111yescharacteristic
170C22×D3—42no—
171C22×D3—81no—
172C22×D3—81no—
173C22×D3—81no—
174C22×D3—81no—
175C22×D3—81no—
176C22×D3—81no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—81no—
180C2×C3×C4—81no—
181C2×C3×C4—81no—
182C2×C3×C4#11711yescharacteristic
183C2×C3×C4C4×D411yescharacteristic
184C2×C3×C4C4×D411yescharacteristic
185C2×C3×C4#12311yescharacteristic
186C2×C3×C4#12411yescharacteristic
187C2×C3×C4#12611yescharacteristic
188C2×C3×C4C2×D811yescharacteristic
189C2×C3×C4C2×D811yescharacteristic
190C2×C3×C4C2×SD1611yescharacteristic
191C2×C3×C4#13511yescharacteristic
192C2×C3×C4#13511yescharacteristic
193C2×C3×C4#13511yescharacteristic
194C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
195C2×C3×C4#13711yescharacteristic
196C2×Dic3—22no—
197C2×Dic3—22no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202D12—42no—
203D12—42no—
204D12—42no—
205D12—42no—
206D12—82no—
207D12—82no—
208D12—81no—
209D12—81no—
210D12—81no—
211D12—81no—
212C3×C8—42no—
213C3×C8—81no—
214C3⋊C8—22no—
215C3⋊C8—42no—
216C3⋊C8—41no—
217C24—121no—
218C2×D4—62no—
219C2×D4—62no—
220C2×D4—62no—
221C2×D4—121no—
222C2×D4—121no—
223C2×D4—121no—
224C2×D4—121no—
225C2×D4—121no—
226C2×D4—121no—
227C2×D4—121no—
228C2×D4—121no—
229C2×D4—121no—
230C2×D4—121no—
231C2×D4—121no—
232C2×D4—241no—
233C2×D4—241no—
234C22×C4—41no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4#23511yescharacteristic
238C22×C4D3×D411yescharacteristic
239C22×C4D3×D411yescharacteristic
240C2×C8—22no—
241C2×C8—32no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—61no—
248C2×C8—61no—
249C2×C8—62no—
250C2×C8—61no—
251C2×C8—61no—
252C2×C8—61no—
253C2×C8—121no—
254C2×C8—121no—
255C4⋊C4—32no—
256C4⋊C4—32no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—81no—
264C42—121no—
265C42—121no—
266C42C2×C4×D311yescharacteristic
267C42#23511yescharacteristic
268C42#23711yescharacteristic
269C42#23911yescharacteristic
270C22×C3#38811yescharacteristic
271C22×C3#45911yescharacteristic
272C22×C3#46411yescharacteristic
273C22×C3#46511yescharacteristic
274C22×C3#46611yescharacteristic
275C22×C3#47111yescharacteristic
276C22×C3#47211yescharacteristic
277C2×D3—82no—
278C2×D3—82no—
279C2×D3—82no—
280C2×D3—161no—
281C2×D3—161no—
282C2×D3—161no—
283C2×D3—161no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—161no—
287C3×C4C4×D811yescharacteristic
288C3×C4C4×D811yescharacteristic
289C3×C4C4×SD1611yescharacteristic
290C3×C4#44011yescharacteristic
291C3×C4#48011yescharacteristic
292C3×C4#48211yescharacteristic
293C3×C4#48611yescharacteristic
294C3×C4#48811yescharacteristic
295Dic3—42no—
296Dic3—42no—
297C23—122no—
298C23—241no—
299C23—241no—
300C23—241no—
301C23—241no—
302C23—241no—
303C23—241no—
304C23#88411yescharacteristic
305D4—122no—
306D4—122no—
307D4—122no—
308D4—122no—
309D4—242no—
310D4—242no—
311D4—241no—
312D4—241no—
313D4—241no—
314D4—241no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—62no—
318C2×C4—62no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4#87211yescharacteristic
327C2×C4#67⋊C411yescharacteristic
328C2×C4#88511yescharacteristic
329C2×C4#89311yescharacteristic
330C2×C4#89411yescharacteristic
331C2×C4#89811yescharacteristic
332C2×C4D3×D811yescharacteristic
333C2×C4D3×D811yescharacteristic
334C2×C4D3×SD1611yescharacteristic
335C2×C4#91411yescharacteristic
336C2×C4#91611yescharacteristic
337C2×C4#91611yescharacteristic
338C2×C4#91811yescharacteristic
339C2×C4#91911yescharacteristic
340C8—42no—
341C8—62no—
342C8—81no—
343C8—121no—
344C8—122no—
345C2×C3#166511yescharacteristic
346C2×C3#166711yescharacteristic
347C2×C3#167811yescharacteristic
348C2×C3#168211yescharacteristic
349C2×C3#192511yescharacteristic
350C2×C3#193111yescharacteristic
351C2×C3#193411yescharacteristic
352D3—162no—
353C22—242no—
354C22—242no—
355C22—242no—
356C22—481no—
357C22—481no—
358C22—481no—
359C22—481no—
360C22#466911yescharacteristic
361C22#474911yescharacteristic
362C22#475411yescharacteristic
363C22#477511yescharacteristic
364C22#478611yescharacteristic
365C22#479811yescharacteristic
366C22#480711yescharacteristic
367C4—21no—
368C4—21no—
369C4—122no—
370C4—122no—
371C4—161no—
372C4#475611yescharacteristic
373C4#476011yescharacteristic
374C4#480011yescharacteristic
375C4#481711yescharacteristic
376C4#485211yescharacteristic
377C4#486611yescharacteristic
378C4#487211yescharacteristic
379C4#488211yescharacteristic
380C3#956311yescharacteristic, complemented, Sylow
381C2—482no—
382C2#6839511yescharacteristic
383C2#6844011yescharacteristic
384C2#6847011yescharacteristic
385C2#6851011yescharacteristic
386C2#6879111yescharacteristic
387C2#6885211yescharacteristic
388C2#6898311yescharacteristic
subgroupG/H
centerC23#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socleC23×C3#121
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1296D3
hypocenterC3#9563
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
1. #70465C2semidirect
5. #69567C2semidirect
6. C3×#1296C2semidirect
7. #67703C2semidirect
24. #4485C4semidirect
30. #1296D3semidirect
380. C3#9563semidirect

Maximal subgroups

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

subgroupindex
#704652
#704572
C2×#44852
#695802
#695672
C3×#12962
#677032
#95633