Finite Group Databasebeta

← #10896378

Subgroups — #10896378

Order 768.

686 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73391C211yesmaximal, characteristic
2#73388C211yesmaximal, characteristic
3C3×#1832C211yesmaximal, characteristic
4#70456C212yesmaximal
5#70455C212yesmaximal
6#68693C212yesmaximal
7#68689C212yesmaximal
8#68684C212yesmaximal
9#68683C212yesmaximal
10#29264—31nomaximal, Sylow
11C2×#869C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×C4×Dic6C2212yes—
14C2×C3×C4⋊C8C2212yes—
15C3×#376C2211yescharacteristic
16C2×#818C2212yes—
17C2×#817C2212yes—
18C2×#816C2212yes—
19C2×#816C2212yes—
20C2×#816C2212yes—
21C2×#816C2212yes—
22#4936C2211yescharacteristic
23#4936C2211yescharacteristic
24C4×C4⋊Dic3C2212yes—
25#4929C2212yes—
26#4928C2212yes—
27#4654—22no—
28#4653—22no—
29#4653—22no—
30#4646—22no—
31C3×#336C2211yescharacteristic
32C3×#336C2211yescharacteristic
33C3×#336C2212yes—
34#4551C2212yes—
35#4551C2212yes—
36#2331—31no—
37#2327—31no—
38#2097—32no—
39#2024—32no—
40#1981—32no—
41#1949—32no—
42#1865—32no—
43#1863—32no—
44#1832D311yescharacteristic
45C22×Dic6—21no—
46C22×Dic6C2312yes—
47C22×C3×C8C2312yes—
48C22×C3×C8C2312yes—
49C2×C3×C42C2311yescharacteristic
50C2×C3×C42C2311yescharacteristic
51C2×C3×C42C2311yescharacteristic
52C2×C4⋊Dic3—22no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3C2312yes—
55C2×C4⋊Dic3C2312yes—
56C2×C4⋊Dic3C2312yes—
57C2×Dic3⋊C4—22no—
58C2×Dic3⋊C4—22no—
59C2×C4×Dic3—22no—
60C2×C4×Dic3—22no—
61#869—22no—
62#869—24no—
63#869Q814yescomplemented
64C4×Dic6—42no—
65C4×Dic6—42no—
66C3×C4⋊C8—24no—
67C3×#95D412yes—
68C3×#95D412yes—
69C3⋊#95—21no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95—21no—
76#818—42no—
77#817—42no—
78#816—24no—
79#816—24no—
80#816—42no—
81#816—42no—
82C2×C4⋊Q8—31no—
83C2×C4⋊Q8—31no—
84C2×C4×Q8—32no—
85C2×#107—32no—
86C2×#106—32no—
87C2×C4⋊C8C2×D312yes—
88C2×#103—32no—
89C2×#103—32no—
90C2×#103—32no—
91C2×#103—32no—
92#391—32no—
93#391—62no—
94#389—32no—
95#387—62no—
96#387—62no—
97#387—62no—
98#382—31no—
99#382—31no—
100C4×C4⋊C4—32no—
101#376C2×D311yescharacteristic
102#340—32no—
103#340—32no—
104#336C2×D312yes—
105#336C2×D311yescharacteristic
106#336C2×D311yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D412yes—
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2411yescharacteristic
112C22×Dic3—22no—
113C22×Dic3—22no—
114C22×Dic3—22no—
115C22×Dic3—22no—
116C2×Dic6—22no—
117C2×Dic6—22no—
118C2×Dic6—22no—
119C2×Dic6—22no—
120C2×Dic6—22no—
121C2×Dic6—24no—
122C2×Dic6—82no—
123C2×Dic6C2×Q814yes—
124C2×C3×C8—22no—
125C2×C3×C8—22no—
126C2×C3×C8—22no—
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C2×C3×C8—22no—
130C3×C42—22no—
131C3×C42—22no—
132C3×C42C2×Q812yes—
133C3×C42#4112yes—
134C4⋊Dic3—24no—
135C4⋊Dic3—28no—
136C4⋊Dic3—22no—
137C4⋊Dic3—22no—
138C4⋊Dic3—22no—
139C4⋊Dic3—22no—
140C4⋊Dic3—22no—
141C4⋊Dic3#4114yes—
142Dic3⋊C4—44no—
143Dic3⋊C4—82no—
144C4×Dic3—44no—
145C4×Dic3—82no—
146C22×Q8—32no—
147C22×Q8—61no—
148C22×C8C22×D312yes—
149C22×C8C22×D312yes—
150C4⋊Q8—34no—
151C4⋊Q8—64no—
152C4⋊Q8—62no—
153C4×Q8—122no—
154C4×Q8—122no—
155C2×C4⋊C4—32no—
156C2×C4⋊C4—32no—
157C2×C4⋊C4—32no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—62no—
160C2×C4⋊C4—62no—
161C2×C4⋊C4—62no—
162C2×C42—62no—
163C2×C42—62no—
164C2×C42C22×D311yescharacteristic
165C2×C42C22×D311yescharacteristic
166C2×C42C22×D311yescharacteristic
167#107—122no—
168#106—122no—
169C4⋊C8—24no—
170#103—64no—
171#103—64no—
172#103—122no—
173#103—122no—
174#95—62no—
175#95—61no—
176#95—62no—
177#95—61no—
178#95—62no—
179#95—62no—
180#95—62no—
181#95D1212yes—
182#95D1212yes—
183C23×C3C22×D411yescharacteristic
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4C22⋊Q814yes—
194C2×C3×C4C22×Q811yescharacteristic
195C2×C3×C4C2×#4111yescharacteristic
196C2×C3×C4C2×#4112yes—
197C2×C3×C4H(5,𝔽2)12yes—
198C2×Dic3—24no—
199C2×Dic3—24no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—42no—
203C2×Dic3—42no—
204C2×Dic3—42no—
205C2×Dic3—42no—
206C2×Dic3—42no—
207C2×Dic3—42no—
208C2×Dic3—42no—
209C2×Dic3—42no—
210Dic6—24no—
211Dic6—24no—
212Dic6—42no—
213Dic6—42no—
214Dic6—84no—
215C3×C8—42no—
216C3×C8—42no—
217C2×Q8—34no—
218C2×Q8—64no—
219C2×Q8—62no—
220C2×Q8—62no—
221C2×Q8—62no—
222C2×Q8—62no—
223C2×Q8—62no—
224C2×Q8—242no—
225C22×C4—62no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4C2×D1211yescharacteristic
230C22×C4C2×D1211yescharacteristic
231C22×C4C2×D1212yes—
232C22×C4C2×D1212yes—
233C22×C4C23×D311yescharacteristic
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C4⋊C4—34no—
241C4⋊C4—64no—
242C4⋊C4—68no—
243C4⋊C4—62no—
244C4⋊C4—62no—
245C4⋊C4—62no—
246C4⋊C4—62no—
247C4⋊C4—62no—
248C4⋊C4—124no—
249C4⋊C4—242no—
250C42—22no—
251C42—22no—
252C42—124no—
253C42—242no—
254C42#23712yes—
255C42D3×Q812yes—
256C22×C3C2×C22⋊Q811yescharacteristic
257C22×C3#52511yescharacteristic
258C22×C3#53511yescharacteristic
259C22×C3#57712yes—
260C22×C3#57812yes—
261C3×C4—24no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—42no—
265C3×C4#55412yes—
266C3×C4#55912yes—
267Dic3—44no—
268Dic3—44no—
269Dic3—82no—
270Dic3—82no—
271C23C22×D1211yescharacteristic
272Q8—64no—
273Q8—64no—
274Q8—122no—
275Q8—122no—
276Q8—244no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—64no—
287C2×C4—64no—
288C2×C4—122no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4#89714yes—
299C2×C4C2×#23711yescharacteristic
300C2×C4C2×D3×Q811yescharacteristic
301C2×C4D3×#4112yes—
302C2×C4#100912yes—
303C8—42no—
304C8—42no—
305C2×C3#258011yescharacteristic
306C2×C3#308212yes—
307C2×C3#312311yescharacteristic
308C2×C3#312711yescharacteristic
309C2×C3#322512yes—
310C22C2×#89711yescharacteristic
311C22#552411yescharacteristic
312C22#552511yescharacteristic
313C22#575112yes—
314C22#575312yes—
315C4—24no—
316C4—41no—
317C4—42no—
318C4—41no—
319C4—124no—
320C4—124no—
321C4—242no—
322C4—242no—
323C4#568112yes—
324C4#569012yes—
325C3#2926411yescharacteristic, complemented, Sylow
326C2#7419011yescharacteristic
327C2#7910112yes—
328C2#7938711yescharacteristic
329C2#7939312yes—
330C2#7939611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1832C2
socle, prime radicalC23×C3C22×D4
hypercenter#1832D3
hypocenterC3#29264
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #1832

Complements

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

NH ≅ G/Ntype
63. #869Q8semidirect
325. C3#29264semidirect

Maximal subgroups

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

subgroupindex
#733912
#733882
C3×#18322
#704562
#704552
#686932
#686892
#686842
#686832
#292643