Finite Group Databasebeta

← #10692706

Subgroups — #10692706

Order 768.

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

normal subgroups only

This group has 384 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#83198C211yesmaximal, characteristic
2#79317C211yesmaximal, characteristic
3C3×#1556C211yesmaximal, characteristic
4#71776C211yesmaximal, characteristic
5#70815C211yesmaximal, characteristic
6#67754C211yesmaximal, characteristic
7#67753C211yesmaximal, characteristic
8#9422—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13#5225—22no—
14#5082—22no—
15#5079C412yes—
16#5079C412yes—
17#4926C2211yescharacteristic
18#4926C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4828—22no—
21#4720—22no—
22#4701—22no—
23C3×#332—22no—
24#4547—22no—
25#4465—22no—
26#3139—31no—
27#2987—31no—
28#1556—31no—
29#1556D311yescharacteristic
30#1537—31no—
31#1291—31no—
32#1291—31no—
33C22×C3×Q8—21no—
34C3×C4⋊Q8—21no—
35C3×C4⋊Q8—21no—
36C3×C4⋊Q8—21no—
37C3×C4⋊Q8C2×C412yes—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C42C2311yescharacteristic
41#926D412yes—
42#924—21no—
43#924—21no—
44C2×#208—21no—
45#870D412yes—
46#869C2×C411yescharacteristic
47#869C2×C411yescharacteristic
48C4×Dic6—21no—
49C4×Dic6—21no—
50C3×C4⋊C8—21no—
51C3×C4⋊C8D412yes—
52C3×C22⋊C8—21no—
53C3×C22⋊C8—21no—
54C3×#97—21no—
55C3×#97—21no—
56#835—24no—
57#835—24no—
58#830—21no—
59#830—21no—
60#815—22no—
61Dic3⋊C8—24no—
62C8×Dic3—22no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)D412yes—
65#803—21no—
66#803—21no—
67#803—21no—
68C4×C3⋊C8C2×C412yes—
69#533—31no—
70C2×C4⋊Q8C2×D311yescharacteristic
71#491—62no—
72#487—32no—
73#487—32no—
74#446—62no—
75#446—62no—
76#433—62no—
77#432—31no—
78#432—31no—
79#432C2×D311yescharacteristic
80#432C2×D311yescharacteristic
81#419—62no—
82C4×M16—31no—
83#332—22no—
84#332—62no—
85#330—62no—
86C2×C3×Q8—22no—
87C2×C3×Q8—44no—
88C2×C3×Q8—41no—
89C22×C3×C4—41no—
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4#4111yescharacteristic
92C22×C3×C4#4111yescharacteristic
93C2×Dic6—21no—
94C2×Dic6—21no—
95C2×C3×C8—22no—
96C2×C3×C8—41no—
97C3×C4⋊C4—22no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—41no—
101C3×C42C22×C411yescharacteristic
102C3×C42C2×D411yescharacteristic
103C3×C42C2×D411yescharacteristic
104C3×C42C2×D411yescharacteristic
105#217—22no—
106C4⋊Dic3—21no—
107C4⋊Dic3—21no—
108C4⋊Dic3C22⋊C412yes—
109C4⋊Dic3C22⋊C412yes—
110Dic3⋊C4—22no—
111Dic3⋊C4—22no—
112C4×Dic3—22no—
113#208—42no—
114C2×C3⋊C8—22no—
115C2×C3⋊C8—22no—
116C2×C3⋊C8—41no—
117C22×Q8—21no—
118C2×M16—61no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—31no—
123C4⋊Q8—31no—
124C4⋊Q8C4×D312yes—
125#125—32no—
126C22⋊Q8—61no—
127C22⋊Q8—61no—
128C4×Q8—61no—
129C4×Q8—61no—
130#117—32no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C42C22×D311yescharacteristic
134C4⋊C8—21no—
135C4⋊C8—32no—
136C4⋊C8—64no—
137C4⋊C8—61no—
138C4⋊C8#6712yes—
139#103—64no—
140#103—64no—
141C22⋊C8—21no—
142C22⋊C8—21no—
143C22⋊C8—61no—
144C22⋊C8—61no—
145#97—21no—
146#97—21no—
147#97—62no—
148#97—61no—
149#97—61no—
150#97—61no—
151C4×C8—32no—
152C4×C8—62no—
153C23×C3#12311yescharacteristic
154C3×Q8—44no—
155C3×Q8—82no—
156C2×C3×C4—22no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—42no—
160C2×C3×C4—82no—
161C2×C3×C4C2×C22⋊C411yescharacteristic
162C2×C3×C4C4×D411yescharacteristic
163C2×C3×C4C4×D411yescharacteristic
164C2×C3×C4#12011yescharacteristic
165C2×C3×C4#12111yescharacteristic
166C2×C3×C4#12111yescharacteristic
167C2×Dic3—22no—
168C2×Dic3—22no—
169Dic6—81no—
170Dic6—81no—
171C3×C8—42no—
172C3×C8—81no—
173C3⋊C8—42no—
174C3⋊C8—42no—
175C3⋊C8—81no—
176C2×Q8—22no—
177C2×Q8—44no—
178C2×Q8—41no—
179C2×Q8—61no—
180C2×Q8—61no—
181C22×C4—41no—
182C22×C4#23511yescharacteristic
183C22×C4D3×D411yescharacteristic
184C22×C4#23711yescharacteristic
185M16—122no—
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—62no—
189C2×C8—62no—
190C2×C8—121no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—32no—
194C4⋊C4—32no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—61no—
198C4⋊C4—62no—
199C4⋊C4—61no—
200C4⋊C4—62no—
201C22⋊C4—62no—
202C42—62no—
203C42C2×C4×D311yescharacteristic
204C42D3×D411yescharacteristic
205C42D3×D411yescharacteristic
206C42C2×#6711yescharacteristic
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3#38611yescharacteristic
211C3×C4—21no—
212C3×C4—21no—
213C3×C4—21no—
214C3×C4—21no—
215C3×C4—21no—
216C3×C4—82no—
217C3×C4#45412yes—
218Dic3—42no—
219Dic3—42no—
220C23#88311yescharacteristic
221Q8—44no—
222Q8—82no—
223Q8—241no—
224Q8—241no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—82no—
232C2×C4D3×C22⋊C411yescharacteristic
233C2×C4#67⋊C411yescharacteristic
234C2×C4#88411yescharacteristic
235C2×C4C4×#6711yescharacteristic
236C2×C4#93711yescharacteristic
237C2×C4#93811yescharacteristic
238C8—42no—
239C8—81no—
240C8—122no—
241C8—122no—
242C8—241no—
243C2×C3—41no—
244C2×C3#188811yescharacteristic
245C2×C3#190211yescharacteristic
246C2×C3#190211yescharacteristic
247C22—21no—
248C22—21no—
249C22—21no—
250C22#495611yescharacteristic
251C4—21no—
252C4—21no—
253C4—21no—
254C4—21no—
255C4—21no—
256C4—82no—
257C4—122no—
258C4—122no—
259C4#520112yes—
260C3#942211yescharacteristic, complemented, Sylow
261C2—41no—
262C2#7012211yescharacteristic
263C2#7032511yescharacteristic
264C2#7034911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#1556D3
hypocenterC3#9422
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
260. C3#9422semidirect

Maximal subgroups

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

subgroupindex
#831982
#793172
C3×#15562
#717762
#708152
#677542
#677532
#94223