Finite Group Databasebeta

← #10681985

Subgroups — #10681985

Order 768.

412 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 83 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#69420C211yesmaximal, characteristic
2C3×#1280C211yesmaximal, characteristic
3#67613C211yesmaximal, characteristic
4#7123—31nomaximal, Sylow
5C22×C4⋊Dic3C411yescharacteristic
6C22×Dic3⋊C4—21no—
7C2×C3×C22⋊C8—21no—
8C2×C3×#95C2211yescharacteristic
9C2×C3⋊#95—21no—
10C2×C3⋊#95C411yescharacteristic
11C2×#830—21no—
12#1444—31no—
13#1280—31no—
14#1280D311yescharacteristic
15C23×C3×C4C2×C411yescharacteristic
16C23×C3×C4D411yescharacteristic
17C23×C3×C4D411yescharacteristic
18C23×Dic3—21no—
19C23×Dic3—21no—
20C22×C3×C8—41no—
21C2×C4⋊Dic3—41no—
22C2×C4⋊Dic3—42no—
23C2×Dic3⋊C4—42no—
24C2×Dic3⋊C4—42no—
25C2×Dic3⋊C4—41no—
26C2×Dic3⋊C4—41no—
27C22×C3⋊C8—41no—
28C3×C22⋊C8—22no—
29C3×C22⋊C8—41no—
30C3×#95—41no—
31C3⋊#95—21no—
32C3⋊#95—42no—
33C3⋊#95C812yescomplemented
34#830—22no—
35#830—41no—
36C22×C4⋊C4—31no—
37C22×C4⋊C4—61no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—61no—
40C2×#95—31no—
41C2×#95—61no—
42C2×#95C2×D311yescharacteristic
43C24×C3C22⋊C411yescharacteristic
44C22×C3×C4—21no—
45C22×C3×C4—21no—
46C22×C3×C4—21no—
47C22×C3×C4—21no—
48C22×C3×C4—21no—
49C22×C3×C4—21no—
50C22×C3×C4—41no—
51C22×C3×C4—41no—
52C22×C3×C4—41no—
53C22×C3×C4—41no—
54C22×C3×C4—41no—
55C22×C3×C4—41no—
56C22×C3×C4C2×C811yescharacteristic
57C22×C3×C4M1611yescharacteristic
58C22×C3×C4D811yescharacteristic
59C22×C3×C4SD1611yescharacteristic
60C22×C3×C4SD1611yescharacteristic
61C22×C3×C4Q1611yescharacteristic
62C22×Dic3—22no—
63C22×Dic3—22no—
64C22×Dic3—22no—
65C22×Dic3—22no—
66C22×Dic3—41no—
67C22×Dic3—41no—
68C22×Dic3—41no—
69C22×Dic3—41no—
70C22×Dic3—41no—
71C22×Dic3—41no—
72C22×Dic3—41no—
73C22×Dic3—41no—
74C22×Dic3—41no—
75C22×Dic3—41no—
76C2×C3×C8—42no—
77C2×C3×C8—81no—
78C2×C3×C8—81no—
79C4⋊Dic3—42no—
80C4⋊Dic3—42no—
81Dic3⋊C4—44no—
82Dic3⋊C4—44no—
83C2×C3⋊C8—42no—
84C2×C3⋊C8—81no—
85C2×C3⋊C8—81no—
86C23×C4—61no—
87C23×C4—61no—
88C23×C4C4×D311yescharacteristic
89C23×C4D1211yescharacteristic
90C23×C4#6711yescharacteristic
91C22×C8—41no—
92C22×C8—121no—
93C2×C4⋊C4—121no—
94C2×C4⋊C4—122no—
95C2×C4⋊C4—122no—
96C2×C4⋊C4—121no—
97C2×C4⋊C4—121no—
98C2×C4⋊C4—122no—
99C22⋊C8—22no—
100C22⋊C8—41no—
101C22⋊C8—62no—
102C22⋊C8—121no—
103#95—32no—
104#95—41no—
105#95—61no—
106#95—122no—
107C23×C3—21no—
108C23×C3—21no—
109C23×C3—21no—
110C23×C3—21no—
111C23×C3C22⋊C811yescharacteristic
112C23×C3#9911yescharacteristic
113C23×C3#10111yescharacteristic
114C23×C3#10211yescharacteristic
115C23×C3#10311yescharacteristic
116C23×C3#10411yescharacteristic
117C23×C3#10411yescharacteristic
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—42no—
125C2×C3×C4—41no—
126C2×C3×C4—41no—
127C2×C3×C4—41no—
128C2×C3×C4—41no—
129C2×C3×C4—41no—
130C2×C3×C4—41no—
131C2×C3×C4—41no—
132C2×C3×C4—41no—
133C2×C3×C4—41no—
134C2×C3×C4—41no—
135C2×C3×C4—41no—
136C2×C3×C4—41no—
137C2×C3×C4—81no—
138C2×C3×C4—81no—
139C2×Dic3—24no—
140C2×Dic3—42no—
141C2×Dic3—42no—
142C2×Dic3—42no—
143C2×Dic3—42no—
144C2×Dic3—42no—
145C2×Dic3—42no—
146C2×Dic3—42no—
147C2×Dic3—42no—
148C2×Dic3—42no—
149C2×Dic3—81no—
150C2×Dic3—81no—
151C2×Dic3—81no—
152C2×Dic3—81no—
153C3×C8—82no—
154C3⋊C8—82no—
155C24(C2×D3)⋊C411yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—62no—
169C22×C4—62no—
170C22×C4—62no—
171C22×C4—62no—
172C22×C4—121no—
173C22×C4—121no—
174C22×C4—121no—
175C22×C4—121no—
176C22×C4—121no—
177C22×C4—121no—
178C22×C4—121no—
179C22×C4—121no—
180C22×C4—121no—
181C22×C4—121no—
182C22×C4D3×C811yescharacteristic
183C22×C4ΓL(1,𝔽25)11yescharacteristic
184C22×C4#20411yescharacteristic
185C22×C4Dic1211yescharacteristic
186C22×C4#21311yescharacteristic
187C22×C4#21511yescharacteristic
188C2×C8—42no—
189C2×C8—81no—
190C2×C8—81no—
191C2×C8—122no—
192C2×C8—241no—
193C2×C8—241no—
194C4⋊C4—122no—
195C4⋊C4—124no—
196C4⋊C4—124no—
197C4⋊C4—122no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3#32411yescharacteristic
211C22×C3D4⋊C811yescharacteristic
212C22×C3Q8⋊C811yescharacteristic
213C22×C3#32711yescharacteristic
214C22×C3#32811yescharacteristic
215C22×C3#33011yescharacteristic
216C22×C3#33211yescharacteristic
217C22×C3AΣL(1,𝔽16)11yescharacteristic
218C22×C3#35211yescharacteristic
219C22×C3#35311yescharacteristic
220C22×C3#35611yescharacteristic
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4—41no—
225C3×C4—41no—
226C3×C4—41no—
227Dic3—44no—
228Dic3—82no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23#80511yescharacteristic
234C23#80611yescharacteristic
235C23#80911yescharacteristic
236C23#81611yescharacteristic
237C23(C2×D3)⋊C811yescharacteristic
238C23#82411yescharacteristic
239C23#82511yescharacteristic
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—64no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4—122no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4—122no—
269C2×C4—122no—
270C2×C4—122no—
271C2×C4—241no—
272C2×C4—241no—
273C2×C4—241no—
274C2×C4—241no—
275C8—82no—
276C8—242no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3#127111yescharacteristic
282C2×C3#132511yescharacteristic
283C2×C3#132611yescharacteristic
284C2×C3#134311yescharacteristic
285C2×C3#134611yescharacteristic
286C2×C3#135211yescharacteristic
287C2×C3#135411yescharacteristic
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22#445111yescharacteristic
301C22#445611yescharacteristic
302C22#446511yescharacteristic
303C22#446711yescharacteristic
304C22#446911yescharacteristic
305C22#447111yescharacteristic
306C22#447411yescharacteristic
307C22#447611yescharacteristic
308C22#447811yescharacteristic
309C22#448211yescharacteristic
310C22#448311yescharacteristic
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—124no—
318C4—242no—
319C3#712311yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2#6744611yescharacteristic
325C2#6745111yescharacteristic
326C2#6745311yescharacteristic
327C2#6745711yescharacteristic
328C2#6747811yescharacteristic
329C2#6748911yescharacteristic
330C2#6749511yescharacteristic
subgroupG/H
centerC23#806
commutatorC22×C3×C4C2×C8
FrattiniC2×#95C2×D3
FittingC3×#1280C2
socleC23×C3#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×#95C22
hypercenter#1280D3
hypocenterC3#7123
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ #1280

Complements

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

NH ≅ G/Ntype
33. C3⋊#95C8semidirect
319. C3#7123semidirect

Maximal subgroups

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

subgroupindex
#694202
C3×#12802
#676132
#71233