Finite Group Databasebeta

← #10826953

Subgroups — #10826953

Order 768.

1908 conjugacy classes of subgroups; only the 263 normal subgroups are shown.

normal subgroups only

This group has 1908 conjugacy classes of subgroups — too many to draw. Only its 263 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
1C22×#924C212yesmaximal
2C2×#4739C214yesmaximal, complemented
3C3×#1779C211yesmaximal, characteristic
4#69964C214yesmaximal, complemented
5#69930C214yesmaximal, complemented
6#23921—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C23×Dic6C2212yes—
9C22×#217—22no—
10C22×C4⋊Dic3C2212yes—
11C2×#924—22no—
12C2×#924—22no—
13C2×#924—216no—
14C2×#924C2214yes—
15C2×#924C2214yes—
16C22×Dic3⋊C4—22no—
17C22×Dic12C2214yescomplemented
18C2×C3×C22⋊C8C2214yes—
19C2×C3×C22⋊C8C2212yes—
20C2×#816C2218yescomplemented
21#4739—216no—
22#4551C2214yescomplemented
23C22×C22⋊Q8—32no—
24C2×#451—34no—
25#2001—34no—
26#1872—34no—
27#1779D311yescharacteristic
28C25×C3D411yescharacteristic
29C23×C3×C4—28no—
30C23×C3×C4—42no—
31C23×C3×C4D412yes—
32C23×C3×C4D411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C23×C3×C4C2312yes—
35C23×Dic3—22no—
36C23×Dic3—22no—
37C22×Dic6—22no—
38C22×Dic6—28no—
39C22×Dic6—28no—
40C22×Dic6—44no—
41C22×Dic6C2314yes—
42C22×C3×C8C2314yes—
43C2×#217—216no—
44C2×#217—24no—
45C2×#217—24no—
46C2×C4⋊Dic3—22no—
47C2×C4⋊Dic3—28no—
48C2×C4⋊Dic3C2314yes—
49#924—216no—
50#924—216no—
51#924—28no—
52#924—28no—
53#924—28no—
54#924D4116yes—
55C2×Dic3⋊C4—216no—
56C2×Dic3⋊C4—24no—
57C2×Dic3⋊C4—24no—
58C2×Dic12—216no—
59C2×Dic12—44no—
60C3×C22⋊C8—28no—
61C3×C22⋊C8—42no—
62#816—216no—
63C23×Q8—32no—
64C24×C4C2×D311yescharacteristic
65C22×Q16—34no—
66C2×C22⋊Q8—34no—
67C2×C22⋊Q8—34no—
68C2×C22⋊Q8—616no—
69C2×C22⋊Q8—62no—
70C2×C22⋊Q8—62no—
71C22×C4⋊C4—32no—
72C22×C4⋊C4—62no—
73C22×C22⋊C4—62no—
74#451—616no—
75C2×#103—38no—
76C2×C22⋊C8C2×D314yes—
77C2×C22⋊C8C2×D312yes—
78#340—34no—
79C24×C3—24no—
80C24×C3—24no—
81C24×C3—41no—
82C24×C3—41no—
83C24×C3Q1614yescomplemented
84C24×C3C2×D411yescharacteristic
85C24×C3C2×D412yes—
86C22×C3×C4—24no—
87C22×C3×C4—24no—
88C22×C3×C4—24no—
89C22×C3×C4—24no—
90C22×C3×C4—24no—
91C22×C3×C4—24no—
92C22×C3×C4—22no—
93C22×C3×C4—41no—
94C22×C3×C4—42no—
95C22×C3×C4—416no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D412yes—
98C22×C3×C4C2×D418yes—
99C22×C3×C4C2411yescharacteristic
100C22×Dic3—28no—
101C22×Dic3—28no—
102C22×Dic3—24no—
103C22×Dic3—24no—
104C22×Dic3—44no—
105C22×Dic3—44no—
106C22×Dic3—44no—
107C22×Dic3—44no—
108C2×Dic6—28no—
109C2×Dic6—28no—
110C2×Dic6—28no—
111C2×Dic6—24no—
112C2×Dic6—432no—
113C2×Dic6—44no—
114C2×Dic6—44no—
115C2×Dic6C2×D418yes—
116C2×C3×C8—28no—
117C2×C3×C8—24no—
118#217—216no—
119#217—216no—
120C4⋊Dic3—28no—
121C4⋊Dic3—24no—
122C4⋊Dic3C2×D418yes—
123Dic3⋊C4—216no—
124Dic3⋊C4—216no—
125Dic12—416no—
126C25D1211yescharacteristic
127C22×Q8—34no—
128C22×Q8—68no—
129C22×Q8—62no—
130C22×Q8—68no—
131C22×Q8—124no—
132C23×C4—28no—
133C23×C4—42no—
134C23×C4—62no—
135C23×C4—62no—
136C23×C4D1212yes—
137C23×C4D1211yescharacteristic
138C23×C4C22×D311yescharacteristic
139C23×C4C22×D312yes—
140C2×Q16—616no—
141C2×Q16—124no—
142C22×C8C22×D314yes—
143C22⋊Q8—316no—
144C22⋊Q8—68no—
145C22⋊Q8—68no—
146C22⋊Q8—616no—
147C22⋊Q8—616no—
148C22⋊Q8—68no—
149C2×C4⋊C4—34no—
150C2×C4⋊C4—68no—
151C2×C4⋊C4—616no—
152C2×C4⋊C4—62no—
153C2×C4⋊C4—64no—
154C2×C4⋊C4—64no—
155C2×C22⋊C4—616no—
156C2×C22⋊C4—64no—
157C2×C22⋊C4—64no—
158#103—616no—
159C22⋊C8—28no—
160C22⋊C8—42no—
161C23×C3—22no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—22no—
166C23×C3—21no—
167C23×C3—22no—
168C23×C3—21no—
169C23×C3—216no—
170C23×C3—44no—
171C23×C3—44no—
172C23×C3—44no—
173C23×C3—48no—
174C23×C3#12014yes—
175C23×C3C2×Q1612yes—
176C23×C3C2×Q1614yes—
177C23×C3#13714yes—
178C23×C3C22×D411yescharacteristic
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—216no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—44no—
185C2×C3×C4—48no—
186C2×C3×C4—44no—
187C2×C3×C4#12014yes—
188C2×C3×C4C22×D414yes—
189C2×C3×C4H(5,𝔽2)12yes—
190C2×Dic3—28no—
191C2×Dic3—28no—
192C2×Dic3—416no—
193C2×Dic3—416no—
194C2×Dic3—44no—
195C2×Dic3—44no—
196C2×Dic3—44no—
197C2×Dic3—44no—
198Dic6—216no—
199Dic6—416no—
200Dic6—48no—
201C3×C8—44no—
202C24—24no—
203C24—24no—
204C24—41no—
205C24—41no—
206C24Dic1214yescomplemented
207C24C2×D1211yescharacteristic
208C24C2×D1212yes—
209C2×Q8—38no—
210C2×Q8—68no—
211C2×Q8—64no—
212C2×Q8—68no—
213C2×Q8—68no—
214C2×Q8—1232no—
215C2×Q8—124no—
216C2×Q8—124no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—416no—
227C22×C4—68no—
228C22×C4—68no—
229C22×C4—64no—
230C22×C4—64no—
231C22×C4—124no—
232C22×C4—124no—
233C22×C4—124no—
234C22×C4—124no—
235C22×C4C2×D1211yescharacteristic
236C22×C4C2×D1212yes—
237C22×C4D3×D418yes—
238C22×C4C23×D311yescharacteristic
239Q16—1216no—
240C2×C8—28no—
241C2×C8—24no—
242C4⋊C4—38no—
243C4⋊C4—616no—
244C4⋊C4—68no—
245C4⋊C4—616no—
246C4⋊C4—64no—
247C22⋊C4—616no—
248C22⋊C4—616no—
249C22×C3—24no—
250C22×C3—24no—
251C22×C3—24no—
252C22×C3—24no—
253C22×C3—24no—
254C22×C3—24no—
255C22×C3—22no—
256C22×C3—24no—
257C22×C3—416no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—42no—
261C22×C3#45118yescomplemented
262C22×C3C2×#12012yes—
263C22×C3#53411yescharacteristic
264C22×C3C22×Q1611yescharacteristic
265C22×C3C2×#13712yes—
266C22×C3#57511yescharacteristic
267C3×C4—24no—
268C3×C4—41no—
269C3×C4D4214yes—
270Dic3—48no—
271Dic3—48no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—216no—
281C23—44no—
282C23—44no—
283C23—48no—
284C23—44no—
285C23#88214yes—
286C23C2×Dic1212yes—
287C23C2×Dic1214yes—
288C23#90914yes—
289C23C22×D1211yescharacteristic
290Q8—616no—
291Q8—1216no—
292Q8—128no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—216no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—44no—
299C2×C4—48no—
300C2×C4—44no—
301C2×C4—68no—
302C2×C4—68no—
303C2×C4—1216no—
304C2×C4—1216no—
305C2×C4—124no—
306C2×C4—124no—
307C2×C4—124no—
308C2×C4—124no—
309C2×C4#88214yes—
310C2×C4C2×D3×D414yes—
311C2×C4#100412yes—
312C8—44no—
313C2×C3—28no—
314C2×C3—42no—
315C2×C3#240311yescharacteristic
316C2×C3C2×#45112yes—
317C2×C3#300712yes—
318C2×C3#318911yescharacteristic
319C2×C3#319511yescharacteristic
320C22—24no—
321C22—24no—
322C22—24no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22—24no—
327C22—22no—
328C22—416no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22#473918yescomplemented
333C22C2×#88212yes—
334C22#549311yescharacteristic
335C22C22×Dic1211yescharacteristic
336C22C2×#90912yes—
337C22#574811yescharacteristic
338C4—24no—
339C4—41no—
340C4—128no—
341C4—128no—
342C4#559014yes—
343C3#2392111yescharacteristic, complemented, Sylow
344C2—28no—
345C2—42no—
346C2#7385211yescharacteristic
347C2C2×#473912yes—
348C2#7861711yescharacteristic
349C2#7864012yes—
350C2#7864611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1779C2
socleC23×C3C22×D4
prime radicalC25×C3D4
hypercenter#1779D3
hypocenterC3#23921
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#4739C2semidirect
4. #69964C2semidirect
5. #69930C2semidirect
17. C22×Dic12C22semidirect
20. C2×#816C22semidirect
22. #4551C22semidirect
83. C24×C3Q16semidirect
206. C24Dic12semidirect
261. C22×C3#451semidirect
332. C22#4739semidirect
343. C3#23921semidirect

Maximal subgroups

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

subgroupindex
C22×#9242
C2×#47392
C3×#17792
#699642
#699302
#239213