Finite Group Databasebeta

← #10690618

Subgroups — #10690618

Order 768.

512 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79263C211yesmaximal, characteristic, complemented
2C2×#4710C211yesmaximal, characteristic
3#70783C211yesmaximal, characteristic
4#70767C211yesmaximal, characteristic, complemented
5#70756C211yesmaximal, characteristic
6C3×#1281C211yesmaximal, characteristic, complemented
7#67660C211yesmaximal, characteristic, complemented
8#9090—31nomaximal, Sylow
9C22×#204C411yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×#97C2211yescharacteristic
15C2×#821C411yescharacteristic
16C2×#818C411yescharacteristic
17C2×#816C411yescharacteristic
18#4925C2211yescharacteristic
19#4925C2211yescharacteristic
20#4830—22no—
21#4718—22no—
22#4710—22no—
23#4710—22no—
24C8×Dic6—22no—
25#4483—22no—
26#4468—22no—
27#4456—22no—
28#2982—31no—
29C2×#440—31no—
30#1479—31no—
31#1478—31no—
32#1460—31no—
33#1281D311yescharacteristic, complemented
34#1275—31no—
35C22×D12C2×C411yescharacteristic
36C22×C4×D3—21no—
37C22×Dic6C2×C411yescharacteristic
38C22×C3×C8C2×C411yescharacteristic
39C22×C3×C8C2×C411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41C2×(C2×D3)⋊C4—21no—
42C2×C4⋊Dic3C2×C411yescharacteristic
43C2×C4⋊Dic3C2×C411yescharacteristic
44C2×Dic3⋊C4—21no—
45C2×C4×Dic3—21no—
46C2×#204—22no—
47C2×#204—22no—
48C2×#204C814yescomplemented
49C4×D12—21no—
50C4×D12—21no—
51C4×D12—21no—
52C4×D12—21no—
53C4×Dic6—21no—
54C4×Dic6—21no—
55C4×Dic6D412yes—
56C4×Dic6D412yes—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8D412yes—
61C3×#97—21no—
62C3×#97—21no—
63C3×C4×C8—21no—
64C3×C4×C8—21no—
65#830—21no—
66#830—21no—
67#821—22no—
68#818—22no—
69#816—22no—
70Dic3⋊C8—24no—
71C8×Dic3—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D412yes—
76C22×SD16—31no—
77C2×C4×Q8—31no—
78C2×C4×D4—31no—
79C8×Q8—62no—
80C8×Q8—62no—
81#440—62no—
82#440—62no—
83#431—31no—
84#431—31no—
85#431—62no—
86#431C2×D311yescharacteristic
87#431C2×D311yescharacteristic
88C2×#107—31no—
89C2×#103—31no—
90C2×#102—31no—
91C2×#97C2×D311yescharacteristic
92Q8⋊C8—62no—
93D4⋊C8—62no—
94#323—62no—
95C23×D3—21no—
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4#4111yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C2×D12—21no—
103C2×D12—21no—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—21no—
107C2×D12C2×C812yes—
108C2×C4×D3—41no—
109C2×C4×D3—41no—
110C2×C4×D3—82no—
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×Dic6C22⋊C412yes—
115C2×Dic6C2×C812yes—
116C2×Dic6M1612yes—
117C2×C3×C8—21no—
118C2×C3×C8—21no—
119C2×C3×C8—21no—
120C2×C3×C8—22no—
121C2×C3×C8—21no—
122C2×C3×C8—21no—
123C2×C3×C8—41no—
124C2×C3×C8C2×C812yes—
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129(C2×D3)⋊C4—41no—
130(C2×D3)⋊C4—41no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3C22⋊C412yes—
135Dic3⋊C4—22no—
136Dic3⋊C4—22no—
137C4×Dic3—22no—
138C4×Dic3—22no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—41no—
141#204—28no—
142C22×Q8—31no—
143C22×D4—31no—
144C23×C4—61no—
145C2×SD16—34no—
146C2×SD16—62no—
147C2×SD16—62no—
148C22×C8C4×D311yescharacteristic
149C22×C8C4×D311yescharacteristic
150C4×Q8—32no—
151C4×Q8—32no—
152C4×Q8—61no—
153C4×Q8—61no—
154C4×D4—61no—
155C4×D4—61no—
156C4×D4—61no—
157C4×D4—61no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C42—61no—
163C2×C42C22×D311yescharacteristic
164#107—62no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—64no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170#103—62no—
171#102—62no—
172C22⋊C8—21no—
173C22⋊C8—61no—
174C22⋊C8—61no—
175C22⋊C8D1212yes—
176#97—21no—
177#97—21no—
178C4×C8—21no—
179C4×C8—21no—
180C4×C8—32no—
181C4×C8—64no—
182C4×C8—61no—
183C23×C3C4×D411yescharacteristic
184C22×D3—22no—
185C22×D3—41no—
186C22×D3—41no—
187C22×D3—81no—
188C22×D3—81no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4C2×C22⋊C411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12311yescharacteristic
197C2×C3×C4C22×C811yescharacteristic
198C2×C3×C4C2×M1611yescharacteristic
199C2×C3×C4#13111yescharacteristic
200C2×C3×C4#13111yescharacteristic
201C2×Dic3—22no—
202C2×Dic3—22no—
203C2×Dic3—22no—
204C2×Dic3—22no—
205C2×Dic3—41no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208D12—22no—
209D12—22no—
210D12—41no—
211D12—41no—
212C4×D3—82no—
213C4×D3—82no—
214Dic6—22no—
215Dic6—22no—
216Dic6—22no—
217Dic6C22⋊C814yes—
218C3×C8—22no—
219C3×C8—42no—
220C3×C8—41no—
221C3×C8—81no—
222C3⋊C8—42no—
223C3⋊C8—81no—
224C24—61no—
225C2×Q8—32no—
226C2×Q8—32no—
227C2×Q8—32no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×D4—32no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—121no—
241C22×C4—121no—
242C22×C4—242no—
243C22×C4C2×C4×D311yescharacteristic
244C22×C4C2×D1211yescharacteristic
245C22×C4#23511yescharacteristic
246SD16—68no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—22no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—41no—
254C2×C8—62no—
255C2×C8—121no—
256C2×C8D3×C812yes—
257C4⋊C4—32no—
258C4⋊C4—62no—
259C4⋊C4—61no—
260C4⋊C4—62no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C42—62no—
266C42—62no—
267C42D3×D411yescharacteristic
268C42D3×D411yescharacteristic
269C42D3×D411yescharacteristic
270C42#23911yescharacteristic
271C22×C3—21no—
272C22×C3—21no—
273C22×C3#38611yescharacteristic
274C22×C3C8×D411yescharacteristic
275C22×C3#43511yescharacteristic
276C2×D3—42no—
277C2×D3—82no—
278C2×D3—81no—
279C2×D3—81no—
280C2×D3—82no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4C2×C22⋊C811yescharacteristic
286C3×C4#40811yescharacteristic
287C3×C4C8×D411yescharacteristic
288C3×C4#43511yescharacteristic
289Dic3—24no—
290Dic3—42no—
291Dic3—81no—
292C23—62no—
293C23—121no—
294C23—121no—
295C23—241no—
296C23—241no—
297C23C4×D1211yescharacteristic
298Q8—34no—
299Q8—62no—
300Q8—62no—
301Q8—62no—
302D4—62no—
303D4—62no—
304D4—121no—
305D4—121no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—121no—
313C2×C4—242no—
314C2×C4—242no—
315C2×C4—241no—
316C2×C4—241no—
317C2×C4D3×C22⋊C411yescharacteristic
318C2×C4#88211yescharacteristic
319C2×C4#88411yescharacteristic
320C2×C4#89311yescharacteristic
321C2×C4#89411yescharacteristic
322C2×C4#89511yescharacteristic
323C2×C4C2×D3×C811yescharacteristic
324C2×C4#90111yescharacteristic
325C2×C4D3×M1611yescharacteristic
326C2×C4#90711yescharacteristic
327C8—22no—
328C8—42no—
329C8—41no—
330C8—81no—
331C8—122no—
332C8—241no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3#186911yescharacteristic
336C2×C3#190311yescharacteristic
337C2×C3#190511yescharacteristic
338D3—82no—
339C22—21no—
340C22—21no—
341C22—122no—
342C22—242no—
343C22—242no—
344C22—241no—
345C22—241no—
346C22#466511yescharacteristic
347C22C8×D1211yescharacteristic
348C22#470611yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—64no—
354C4—122no—
355C4—241no—
356C4D3×C22⋊C811yescharacteristic
357C4#472711yescharacteristic
358C4#483411yescharacteristic
359C4#483611yescharacteristic
360C3#909011yescharacteristic, complemented, Sylow
361C2—21no—
362C2—21no—
363C2—242no—
364C2#6866811yescharacteristic
365C2#6903011yescharacteristic
366C2#6903811yescharacteristic
subgroupG/H
centerC22#4665
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1281C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1281D3
hypocenterC3#9090
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
1. #79263C2semidirect
4. #70767C2semidirect
6. C3×#1281C2semidirect
7. #67660C2semidirect
33. #1281D3semidirect
48. C2×#204C8semidirect
360. C3#9090semidirect

Maximal subgroups

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

subgroupindex
#792632
C2×#47102
#707832
#707672
#707562
C3×#12812
#676602
#90903