Finite Group Databasebeta

← #10692163

Subgroups — #10692163

Order 768.

400 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83019C211yesmaximal, characteristic
2#79337C211yesmaximal, characteristic
3C3×#1534C211yesmaximal, characteristic, complemented
4#71737C211yesmaximal, characteristic, complemented
5#70817C211yesmaximal, characteristic
6C3⋊#1308C211yesmaximal, characteristic, complemented
7#67756C211yesmaximal, characteristic, complemented
8#9335—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13#5067—21no—
14#5066—22no—
15#5064—21no—
16#4968C412yes—
17#4967C412yes—
18#4926C2211yescharacteristic
19#4926C2211yescharacteristic
20#4925C2211yescharacteristic
21#4836—22no—
22#4727—22no—
23#4706—22no—
24C3×#330—22no—
25#4528—22no—
26#4486—21no—
27#4485—21no—
28#3234—31no—
29#2980—31no—
30#1553—31no—
31#1534D311yescharacteristic, complemented
32#1470—31no—
33#1308—31no—
34#1287—31no—
35C2×#235—21no—
36C3×#125—21no—
37C3×#125—21no—
38C3×#125—21no—
39C3×#125—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C42C2311yescharacteristic
45#930—21no—
46#930C2×C412yes—
47C2×ΓL(1,𝔽25)—22no—
48C2×D3×C8—22no—
49#875—21no—
50#874D411yescharacteristic
51C4×D12—21no—
52C4×D12D412yes—
53#869D411yescharacteristic
54C3×C4⋊C8—21no—
55C3×C4⋊C8—21no—
56C3×C22⋊C8—21no—
57C3×C22⋊C8—21no—
58C3×#97—21no—
59C3×#97D412yes—
60#830—21no—
61#830C2×C412yes—
62#830D412yes—
63(C2×D3)⋊C8—24no—
64#809—22no—
65#809—22no—
66#808—22no—
67#807—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71#803—21no—
72#803—21no—
73C4×C3⋊C8—21no—
74#532—31no—
75C2×#125C2×D311yescharacteristic
76#501—61no—
77#490—61no—
78#481—32no—
79#480—32no—
80#478—62no—
81#435—62no—
82#435—62no—
83#432—31no—
84#432—31no—
85#432C2×D311yescharacteristic
86#432C2×D311yescharacteristic
87#431—31no—
88#408—62no—
89#332—61no—
90#331—61no—
91#330—22no—
92#324—62no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4#4111yescharacteristic
98C2×#67—22no—
99#235—81no—
100#235—81no—
101C2×D12—21no—
102C2×D12—21no—
103C2×D12C22⋊C412yes—
104C2×C4×D3—22no—
105C2×Dic6—21no—
106C2×C3×C8—22no—
107C2×C3×C8—41no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C42C2×D411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42#4111yescharacteristic
120(C2×D3)⋊C4—22no—
121(C2×D3)⋊C4—22no—
122C4⋊Dic3—21no—
123C4⋊Dic3C22⋊C412yes—
124C2×C3⋊C8—22no—
125C2×C3⋊C8—22no—
126C2×C3⋊C8—41no—
127ΓL(1,𝔽25)—82no—
128D3×C8—82no—
129C2×#41—61no—
130C2×M16—62no—
131C22×C8—62no—
132C4⋊Q8—31no—
133#127—31no—
134#125—21no—
135#125—21no—
136#125—21no—
137#125—21no—
138#124—61no—
139#121—32no—
140#121—61no—
141C4×D4—32no—
142C4×D4—61no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4C4×D311yescharacteristic
146C2×C4⋊C4C4×D311yescharacteristic
147C2×C42C22×D311yescharacteristic
148#107—62no—
149#106—62no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8—61no—
153C4⋊C8—61no—
154C4⋊C8—61no—
155#102—62no—
156#102—62no—
157C22⋊C8—21no—
158C22⋊C8—21no—
159C22⋊C8—32no—
160C22⋊C8—32no—
161C22⋊C8—64no—
162C22⋊C8—61no—
163#97—21no—
164#97—61no—
165#97—61no—
166#97#6712yes—
167C4×C8—61no—
168C23×C3C4×D411yescharacteristic
169C22×D3—22no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—42no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—81no—
177C2×C3×C4C2×C22⋊C411yescharacteristic
178C2×C3×C4C4×D411yescharacteristic
179C2×C3×C4#12011yescharacteristic
180C2×C3×C4#12111yescharacteristic
181C2×C3×C4#12111yescharacteristic
182C2×C3×C4#12311yescharacteristic
183#67—82no—
184C2×Dic3—22no—
185D12—42no—
186D12—81no—
187D12—81no—
188C4×D3—42no—
189C4×D3—42no—
190Dic6—81no—
191C3×C8—42no—
192C3×C8—81no—
193C3⋊C8—42no—
194C3⋊C8—42no—
195C3⋊C8—81no—
196#41—241no—
197#41—241no—
198C2×Q8—61no—
199C2×D4—32no—
200C2×D4—62no—
201C2×D4—61no—
202C2×D4—61no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—62no—
206C22×C4C2×C4×D311yescharacteristic
207C22×C4D3×D411yescharacteristic
208C22×C4#23711yescharacteristic
209M16—242no—
210C2×C8—22no—
211C2×C8—41no—
212C2×C8—62no—
213C2×C8—62no—
214C2×C8—121no—
215C2×C8—242no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—32no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—61no—
226C22⋊C4—62no—
227C22⋊C4—62no—
228C42#23511yescharacteristic
229C42D3×D411yescharacteristic
230C42D3×D411yescharacteristic
231C42C2×#6711yescharacteristic
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3#38611yescharacteristic
236C2×D3—42no—
237C2×D3—42no—
238C2×D3—42no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—81no—
246C3×C4—81no—
247Dic3—42no—
248C23—62no—
249C23#67⋊C411yescharacteristic
250Q8—241no—
251D4—122no—
252D4—241no—
253D4—241no—
254D4—242no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—62no—
262C2×C4—81no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4D3×C22⋊C411yescharacteristic
266C2×C4#88311yescharacteristic
267C2×C4#88411yescharacteristic
268C2×C4C4×#6711yescharacteristic
269C2×C4#93711yescharacteristic
270C2×C4#93811yescharacteristic
271C8—42no—
272C8—81no—
273C8—122no—
274C8—122no—
275C8—241no—
276C2×C3—41no—
277C2×C3#188111yescharacteristic
278C2×C3#190411yescharacteristic
279C2×C3#191511yescharacteristic
280D3—82no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—122no—
285C22—122no—
286C22—122no—
287C22#495611yescharacteristic
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—81no—
295C4—81no—
296C4—122no—
297C3#933511yescharacteristic, complemented, Sylow
298C2—41no—
299C2—242no—
300C2#7012411yescharacteristic
301C2#7040711yescharacteristic
302C2#7041611yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1534C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1534D3
hypocenterC3#9335
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1534

Complements

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

NH ≅ G/Ntype
3. C3×#1534C2semidirect
4. #71737C2semidirect
6. C3⋊#1308C2semidirect
7. #67756C2semidirect
31. #1534D3semidirect
297. C3#9335semidirect

Maximal subgroups

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

subgroupindex
#830192
#793372
C3×#15342
#717372
#708172
C3⋊#13082
#677562
#93353