Finite Group Databasebeta

← #10690140

Subgroups — #10690140

Order 768.

376 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 89 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#79317C211yesmaximal, characteristic
2#78479C211yesmaximal, characteristic
3#70803C211yesmaximal, characteristic
4#70777C211yesmaximal, characteristic
5#70771C211yesmaximal, characteristic
6C3×#1302C211yesmaximal, characteristic
7#67662C211yesmaximal, characteristic
8#9006—31nomaximal, Sylow
9#5470C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14#5106—22no—
15#4926C2211yescharacteristic
16#4925C2211yescharacteristic
17#4828—22no—
18#4720—22no—
19#4702—21no—
20#4702—21no—
21#4702C412yes—
22#4702C412yes—
23#4701—22no—
24#4486—21no—
25#4484—22no—
26#4482—21no—
27#4456—22no—
28#3162—31no—
29#2987—31no—
30#1544—31no—
31#1497—31no—
32#1497—31no—
33#1302—31no—
34#1302D311yescharacteristic
35C22×C3×C8—21no—
36C2×C3×C42C2311yescharacteristic
37#926—21no—
38#926D412yes—
39#924—21no—
40#924—21no—
41#924—21no—
42#924—21no—
43#870—21no—
44#870C2×C412yes—
45#869C2×C411yescharacteristic
46#869C2×C411yescharacteristic
47#869D411yescharacteristic
48#869D411yescharacteristic
49C4×Dic6—21no—
50C4×Dic6—21no—
51C4×Dic6—21no—
52C4×Dic6—21no—
53C3×C4⋊C8—21no—
54C3×C4⋊C8D412yes—
55C3×C22⋊C8—21no—
56C3×C22⋊C8—21no—
57C3×#97—21no—
58C3×#97—21no—
59C3×#97C2×C412yes—
60C3×C4×C8—21no—
61#830—21no—
62#830—21no—
63#818—22no—
64#818—22no—
65#817—22no—
66#817—22no—
67#815—22no—
68Dic3⋊C8—24no—
69C8×Dic3—22no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)D412yes—
72#803—21no—
73C4×C3⋊C8—21no—
74#533—31no—
75#533—31no—
76#501—32no—
77#501—32no—
78#501—61no—
79#501—61no—
80#446—62no—
81#446—62no—
82#433—62no—
83#432—31no—
84#432C2×D311yescharacteristic
85#431—31no—
86#431C2×D311yescharacteristic
87#427—62no—
88C2×#97C2×D311yescharacteristic
89#332—61no—
90#332—61no—
91Q8⋊C8—62no—
92Q8⋊C8—62no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4111yescharacteristic
95C22×C3×C4#4111yescharacteristic
96C2×Dic6—21no—
97C2×Dic6—21no—
98C2×Dic6—21no—
99C2×Dic6—21no—
100C2×C3×C8—24no—
101C2×C3×C8—22no—
102C2×C3×C8—22no—
103C2×C3×C8—41no—
104C3×C42C22×C411yescharacteristic
105C3×C42C2×D411yescharacteristic
106C3×C42C2×D411yescharacteristic
107C3×C42C2×D411yescharacteristic
108#217—22no—
109#217—41no—
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3C22⋊C412yes—
117C4⋊Dic3C22⋊C412yes—
118Dic3⋊C4—22no—
119Dic3⋊C4—22no—
120Dic3⋊C4—41no—
121Dic3⋊C4—41no—
122C4×Dic3—22no—
123C4×Dic3—41no—
124C2×C3⋊C8—22no—
125C2×C3⋊C8—41no—
126C22×C8—21no—
127C4⋊Q8—31no—
128C4⋊Q8—31no—
129C4⋊Q8—31no—
130C4⋊Q8—31no—
131#125—32no—
132#125—61no—
133C22⋊Q8—61no—
134C22⋊Q8—61no—
135C22⋊Q8—61no—
136C22⋊Q8—61no—
137C4×Q8—61no—
138C4×Q8—61no—
139C4×Q8—61no—
140C4×Q8—61no—
141#117—32no—
142#117—61no—
143C2×C42C22×D311yescharacteristic
144#107—62no—
145#107—62no—
146#106—62no—
147#106—62no—
148C4⋊C8—21no—
149C4⋊C8—32no—
150C4⋊C8—64no—
151C4⋊C8—61no—
152C4⋊C8D1212yes—
153C22⋊C8—21no—
154C22⋊C8—21no—
155C22⋊C8—61no—
156C22⋊C8—61no—
157#97—21no—
158#97—21no—
159#97—62no—
160#97—61no—
161#97C4×D312yes—
162C4×C8—21no—
163C4×C8—62no—
164C4×C8—61no—
165C23×C3#12311yescharacteristic
166C2×C3×C4—22no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4C2×C22⋊C411yescharacteristic
170C2×C3×C4C4×D411yescharacteristic
171C2×C3×C4C4×D411yescharacteristic
172C2×C3×C4#12011yescharacteristic
173C2×C3×C4#12111yescharacteristic
174C2×C3×C4#12111yescharacteristic
175C2×Dic3—22no—
176C2×Dic3—22no—
177C2×Dic3—41no—
178C2×Dic3—41no—
179Dic6—42no—
180Dic6—42no—
181Dic6—81no—
182Dic6—81no—
183C3×C8—24no—
184C3×C8—42no—
185C3×C8—81no—
186C3⋊C8—42no—
187C3⋊C8—81no—
188C2×Q8—61no—
189C2×Q8—61no—
190C2×Q8—61no—
191C2×Q8—61no—
192C22×C4#23511yescharacteristic
193C22×C4D3×D411yescharacteristic
194C22×C4#23911yescharacteristic
195C2×C8—24no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—41no—
199C2×C8—62no—
200C2×C8—121no—
201C4⋊C4—32no—
202C4⋊C4—32no—
203C4⋊C4—62no—
204C4⋊C4—61no—
205C4⋊C4—62no—
206C4⋊C4—61no—
207C4⋊C4—61no—
208C4⋊C4—61no—
209C4⋊C4—61no—
210C4⋊C4—61no—
211C4⋊C4—121no—
212C4⋊C4—121no—
213C22⋊C4—62no—
214C22⋊C4—121no—
215C42—62no—
216C42—121no—
217C42C2×C4×D311yescharacteristic
218C42C2×D1211yescharacteristic
219C42D3×D411yescharacteristic
220C42D3×D411yescharacteristic
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3#38611yescharacteristic
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—21no—
229C3×C4—21no—
230C3×C4#45612yes—
231Dic3—42no—
232Dic3—42no—
233Dic3—81no—
234Dic3—81no—
235C23#89411yescharacteristic
236Q8—122no—
237Q8—122no—
238Q8—241no—
239Q8—241no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—121no—
246C2×C4—121no—
247C2×C4C4×D1211yescharacteristic
248C2×C4D3×C22⋊C411yescharacteristic
249C2×C4#88211yescharacteristic
250C2×C4#88411yescharacteristic
251C2×C4#89311yescharacteristic
252C2×C4#89511yescharacteristic
253C8—24no—
254C8—42no—
255C8—81no—
256C8—122no—
257C8—241no—
258C2×C3—41no—
259C2×C3#188111yescharacteristic
260C2×C3#188111yescharacteristic
261C2×C3#188611yescharacteristic
262C22—21no—
263C22—21no—
264C22—21no—
265C22#466511yescharacteristic
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—21no—
271C4—122no—
272C4—122no—
273C4—241no—
274C4—241no—
275C4#482612yes—
276C3#900611yescharacteristic, complemented, Sylow
277C2—41no—
278C2#6876511yescharacteristic
279C2#6894711yescharacteristic
280C2#6903511yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1302C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#1302D3
hypocenterC3#9006
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1302

Complements

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

NH ≅ G/Ntype
276. C3#9006semidirect

Maximal subgroups

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

subgroupindex
#793172
#784792
#708032
#707772
#707712
C3×#13022
#676622
#90063