Finite Group Databasebeta

← #10692173

Subgroups — #10692173

Order 768.

360 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C3×#2987C211yesmaximal, characteristic
2#83326C211yesmaximal, characteristic
3#71832C211yesmaximal, characteristic
4#70831C211yesmaximal, characteristic
5#70814C211yesmaximal, characteristic
6C3⋊#1308C211yesmaximal, characteristic
7#67756C211yesmaximal, characteristic
8#9336—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#446—22no—
12C3×#446—22no—
13C3×#433—22no—
14C3×#432C2211yescharacteristic
15C3×#432C2211yescharacteristic
16#5082—21no—
17#5081—21no—
18#5044C412yes—
19#5043C412yes—
20#5022—22no—
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23#4925C2211yescharacteristic
24#4547—21no—
25#4546—21no—
26#4469—22no—
27#4465—22no—
28#3236—31no—
29#2987D311yescharacteristic
30#1554—31no—
31#1534—31no—
32#1470—31no—
33#1308—31no—
34#1287—31no—
35C3×C4⋊Q8D411yescharacteristic
36C3×C4⋊Q8D411yescharacteristic
37C3×#125—21no—
38C3×C22⋊Q8—21no—
39C3×C22⋊Q8C2×C412yes—
40C3×C4×Q8—21no—
41C3×C4×Q8D412yes—
42C3×#117—21no—
43C2×C3×C42C2311yescharacteristic
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48#870—21no—
49#870—21no—
50#870—21no—
51#870—21no—
52C3×C4⋊C8—24no—
53C3×C4⋊C8—21no—
54C3×C4⋊C8—21no—
55C3×C22⋊C8—21no—
56C3×C22⋊C8D412yes—
57C3×#97—22no—
58C3×#97—21no—
59C3×#97D412yes—
60C3×C4×C8—22no—
61#835—22no—
62#835—22no—
63#830—21no—
64#830—21no—
65#830C2×C412yes—
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#533C2×D311yescharacteristic
75C2×#125—31no—
76#501—61no—
77#491—61no—
78#484—32no—
79#483—32no—
80#479—62no—
81#446—22no—
82#446—22no—
83#433—22no—
84#432—31no—
85#432—31no—
86#432C2×D311yescharacteristic
87#432C2×D311yescharacteristic
88#431—31no—
89#333—61no—
90#332—61no—
91#330—62no—
92#324—62no—
93C2×C3×Q8—21no—
94C2×C3×Q8C22⋊C412yes—
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4#4111yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C2×C3×C8—22no—
101C2×C3×C8—22no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4C22⋊C412yes—
108C3×C22⋊C4—22no—
109C3×C42—22no—
110C3×C42C2×D411yescharacteristic
111C3×C42C2×D411yescharacteristic
112C3×C42C2×D411yescharacteristic
113C3×C42#4111yescharacteristic
114C4⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118Dic3⋊C4—41no—
119Dic3⋊C4—41no—
120Dic3⋊C4—41no—
121Dic3⋊C4—41no—
122C2×C3⋊C8—22no—
123C2×C3⋊C8—41no—
124C2×C3⋊C8—41no—
125C4⋊Q8D1211yescharacteristic
126C4⋊Q8D1211yescharacteristic
127#125—21no—
128#125—61no—
129#125—61no—
130#125—61no—
131#125—61no—
132C22⋊Q8—21no—
133C22⋊Q8C4×D312yes—
134C4×Q8—21no—
135C4×Q8#6712yes—
136#117—21no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C42C22×D311yescharacteristic
142#107—62no—
143#106—62no—
144C4⋊C8—24no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—61no—
148C4⋊C8—61no—
149C4⋊C8—61no—
150#103—62no—
151#103—62no—
152C22⋊C8—21no—
153C22⋊C8—32no—
154C22⋊C8—61no—
155C22⋊C8—61no—
156C22⋊C8#6712yes—
157#97—22no—
158#97—21no—
159#97—61no—
160#97—61no—
161#97#6712yes—
162C4×C8—22no—
163C4×C8—61no—
164C23×C3C4×D411yescharacteristic
165C3×Q8—42no—
166C3×Q8—81no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4C2×C22⋊C411yescharacteristic
173C2×C3×C4C4×D411yescharacteristic
174C2×C3×C4#12011yescharacteristic
175C2×C3×C4#12111yescharacteristic
176C2×C3×C4#12111yescharacteristic
177C2×C3×C4#12311yescharacteristic
178C2×Dic3—42no—
179C2×Dic3—41no—
180C2×Dic3—41no—
181C2×Dic3—81no—
182C3×C8—42no—
183C3×C8—42no—
184C3⋊C8—42no—
185C3⋊C8—81no—
186C3⋊C8—81no—
187C2×Q8—21no—
188C2×Q8(C2×D3)⋊C412yes—
189C22×C4—61no—
190C22×C4—61no—
191C22×C4C2×C4×D311yescharacteristic
192C22×C4#23511yescharacteristic
193C22×C4C2×#6711yescharacteristic
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—62no—
197C2×C8—121no—
198C2×C8—121no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—61no—
205C4⋊C4—61no—
206C4⋊C4—61no—
207C4⋊C4—61no—
208C4⋊C4—121no—
209C4⋊C4—121no—
210C4⋊C4—121no—
211C4⋊C4—121no—
212C4⋊C4(C2×D3)⋊C412yes—
213C22⋊C4—22no—
214C42—22no—
215C42C2×D1211yescharacteristic
216C42#23511yescharacteristic
217C42C2×#6711yescharacteristic
218C42C2×#6711yescharacteristic
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3#38611yescharacteristic
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—21no—
229C3×C4—42no—
230C3×C4—42no—
231Dic3—81no—
232Dic3—81no—
233C23C4×#6711yescharacteristic
234Q8—42no—
235Q8—81no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—122no—
242C2×C4—121no—
243C2×C4—121no—
244C2×C4—241no—
245C2×C4C2×(C2×D3)⋊C411yescharacteristic
246C2×C4C4×#6711yescharacteristic
247C2×C4#92911yescharacteristic
248C2×C4#93011yescharacteristic
249C2×C4#93011yescharacteristic
250C2×C4#95311yescharacteristic
251C8—42no—
252C8—42no—
253C8—122no—
254C8—241no—
255C8—241no—
256C2×C3—41no—
257C2×C3#188111yescharacteristic
258C2×C3#190511yescharacteristic
259C2×C3#191511yescharacteristic
260C22—21no—
261C22—21no—
262C22—21no—
263C22#521311yescharacteristic
264C4—21no—
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—42no—
272C4—241no—
273C4—241no—
274C3#933611yescharacteristic, complemented, Sylow
275C2—41no—
276C2#7166211yescharacteristic
277C2#7202011yescharacteristic
278C2#7203111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2987C2
socleC22×C3#386
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#2987D3
hypocenterC3#9336
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
274. C3#9336semidirect

Maximal subgroups

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

subgroupindex
C3×#29872
#833262
#718322
#708312
#708142
C3⋊#13082
#677562
#93363