Finite Group Databasebeta

← #10715549

Subgroups — #10715549

Order 768.

312 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 71 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
2C2×C3×#356C211yesmaximal, characteristic
3C2×#4548C211yesmaximal, characteristic
4#67551C214yesmaximal, complemented
5#13599—31nomaximal, Sylow
6C2×C3×C4⋊Q8C2211yescharacteristic
7#5470C411yescharacteristic
8C2×C3×#101—21no—
9C2×#836—21no—
10#5225—21no—
11#5225—21no—
12#5082C412yes—
13#5082C2212yes—
14#5079—21no—
15#5079—21no—
16C4×#208C411yescharacteristic
17C3×#356—21no—
18C3×#356C2212yes—
19#4548—21no—
20#4548C2212yes—
21#3139—31no—
22C2×#356—31no—
23C2×#356D311yescharacteristic
24#1413—34no—
25C22×C3×Q8D411yescharacteristic
26C22×C3×Q8D411yescharacteristic
27C2×C3×M16—41no—
28C3×C4⋊Q8—41no—
29C3×C4⋊Q8C2×C411yescharacteristic
30C3×C4⋊Q8D411yescharacteristic
31C3×C4⋊Q8D411yescharacteristic
32C3×C4⋊Q8C2311yescharacteristic
33C2×C3×C4⋊C4—41no—
34C2×C3×C42C2×C411yescharacteristic
35#926—21no—
36#924—41no—
37C2×#208—21no—
38C2×#208—41no—
39#870C2×C412yes—
40#869—21no—
41C4×Dic6—41no—
42C3×#101—22no—
43C3×#101—41no—
44#836—22no—
45#836—41no—
46#835—22no—
47#835—22no—
48#835—22no—
49#835—22no—
50#803C2×C412yes—
51C4×C3⋊C8—21no—
52#533—31no—
53C2×C4⋊Q8C2×D311yescharacteristic
54#491—32no—
55#491—32no—
56#487—61no—
57#487—61no—
58#419—61no—
59#419—61no—
60C2×#101—21no—
61C2×#101—61no—
62C4×M16—31no—
63#356—21no—
64#356—32no—
65#356—61no—
66#356C2×D312yes—
67C2×C3×Q8—21no—
68C2×C3×Q8—21no—
69C2×C3×Q8—41no—
70C2×C3×Q8—41no—
71C2×C3×Q8—41no—
72C2×C3×Q8—41no—
73C2×C3×Q8C22⋊C411yescharacteristic
74C2×C3×Q8C22⋊C411yescharacteristic
75C2×C3×Q8C2×D411yescharacteristic
76C2×C3×Q8C2×D411yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4C22⋊C411yescharacteristic
81C2×Dic6—41no—
82C3×M16—42no—
83C3×M16—81no—
84C2×C3×C8—81no—
85C3×C4⋊C4—41no—
86C3×C4⋊C4—41no—
87C3×C4⋊C4—41no—
88C3×C4⋊C4—41no—
89C3×C42—21no—
90C3×C42C22⋊C411yescharacteristic
91C3×C42C22×C411yescharacteristic
92#217—41no—
93C4⋊Dic3—22no—
94C4⋊Dic3—41no—
95Dic3⋊C4—42no—
96C4×Dic3—41no—
97#208—41no—
98#208—41no—
99#208—42no—
100#208—81no—
101C2×C3⋊C8—22no—
102C2×C3⋊C8—81no—
103C22×Q8D1211yescharacteristic
104C22×Q8#6711yescharacteristic
105C2×M16—41no—
106C2×M16—61no—
107C2×M16—121no—
108C4⋊Q8—41no—
109C4⋊Q8—61no—
110C4⋊Q8C4×D311yescharacteristic
111C4⋊Q8D1211yescharacteristic
112C4⋊Q8#6711yescharacteristic
113C4⋊Q8C22×D311yescharacteristic
114#125—32no—
115C22⋊Q8—121no—
116C4×Q8—121no—
117#117—61no—
118C2×C4⋊C4—41no—
119C2×C42C4×D311yescharacteristic
120#103—62no—
121#103—62no—
122#103—62no—
123#103—62no—
124#101—22no—
125#101—41no—
126#101—62no—
127#101—121no—
128#97—32no—
129C4×C8—61no—
130C23×C3#9911yescharacteristic
131C3×Q8—41no—
132C3×Q8—41no—
133C3×Q8—41no—
134C3×Q8—41no—
135C3×Q8—81no—
136C3×Q8—81no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—42no—
146C2×C3×C4—41no—
147C2×C3×C4—41no—
148C2×C3×C4—41no—
149C2×C3×C4—41no—
150C2×C3×C4#9911yescharacteristic
151C2×C3×C4C2×C22⋊C411yescharacteristic
152C2×Dic3—42no—
153Dic6—82no—
154C3×C8—82no—
155C3⋊C8—42no—
156C3⋊C8—82no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—121no—
164C2×Q8(C2×D3)⋊C411yescharacteristic
165C2×Q8(C2×D3)⋊C411yescharacteristic
166C2×Q8C2×D1211yescharacteristic
167C2×Q8C2×#6711yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4(C2×D3)⋊C411yescharacteristic
172M16—42no—
173M16—81no—
174M16—121no—
175M16—121no—
176M16—122no—
177M16—241no—
178C2×C8—62no—
179C2×C8—81no—
180C2×C8—241no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—62no—
186C4⋊C4—122no—
187C4⋊C4—121no—
188C22⋊C4—121no—
189C42—21no—
190C42—121no—
191C42(C2×D3)⋊C411yescharacteristic
192C42C2×C4×D311yescharacteristic
193C22×C3—41no—
194C22×C3#35411yescharacteristic
195C22×C3#35411yescharacteristic
196C22×C3C2×#9911yescharacteristic
197C3×C4—22no—
198C3×C4—21no—
199C3×C4—21no—
200C3×C4—41no—
201C3×C4—41no—
202C3×C4—41no—
203C3×C4—41no—
204C3×C4—41no—
205Dic3—82no—
206C23#80611yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—41no—
210Q8—41no—
211Q8—81no—
212Q8—81no—
213Q8—242no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—122no—
228C2×C4#80611yescharacteristic
229C2×C4C2×(C2×D3)⋊C411yescharacteristic
230C8—82no—
231C8—122no—
232C8—242no—
233C2×C3—21no—
234C2×C3—21no—
235C2×C3—21no—
236C2×C3C2×#35411yescharacteristic
237C22—41no—
238C22#447311yescharacteristic
239C22#447311yescharacteristic
240C22C2×#80611yescharacteristic
241C4—22no—
242C4—21no—
243C4—21no—
244C4—41no—
245C4—41no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—242no—
250C3#1359911yescharacteristic, complemented, Sylow
251C2—21no—
252C2—21no—
253C2—21no—
254C2C2×#447311yescharacteristic
subgroupG/H
centerC2C2×#4473
commutatorC3×C42C22×C4
FrattiniC4⋊Q8C22×D3
FittingC2×C3×#356C2
socleC2×C3C2×#354
prime radicalC23×C3#99
cosocleC3×C4⋊Q8C23
hypercenterC2×#356D3
hypocenterC3#13599
LCSG ▷ C3×C42 ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
4. #67551C2semidirect
250. C3#13599semidirect

Maximal subgroups

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

subgroupindex
#831982
C2×C3×#3562
C2×#45482
#675512
#135993