Finite Group Databasebeta

← #10681981

Subgroups — #10681981

Order 768.

368 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C3×#1442C211yesmaximal, characteristic
2#67613C211yesmaximal, characteristic
3#67613C211yesmaximal, characteristic
4#7122—31nomaximal, Sylow
5C22×C3×C4⋊C4C411yescharacteristic
6C2×C3×#95—21no—
7C2×C3×#95—21no—
8C2×C3×#95C411yescharacteristic
9C2×C3×#95C2211yescharacteristic
10C2×#830—21no—
11C2×#830—21no—
12#1442D311yescharacteristic
13#1280—31no—
14#1280—31no—
15C23×C3×C4—21no—
16C23×C3×C4—21no—
17C23×C3×C4C2×C411yescharacteristic
18C23×C3×C4D411yescharacteristic
19C23×C3×C4D411yescharacteristic
20C2×C3×C4⋊C4—41no—
21C2×C3×C4⋊C4—42no—
22C22×C3⋊C8—41no—
23C22×C3⋊C8—41no—
24C3×#95—21no—
25C3×#95—41no—
26C3×#95—42no—
27C3×#95—41no—
28C3×#95—41no—
29C3×#95C812yescomplemented
30#830—22no—
31#830—22no—
32#830—41no—
33#830—41no—
34C22×C4⋊C4Dic311yescharacteristic
35C2×C22⋊C8—61no—
36C2×C22⋊C8—61no—
37C2×#95—21no—
38C2×#95—21no—
39C2×#95Dic311yescharacteristic
40C2×#95C2×D311yescharacteristic
41C24×C3C22⋊C411yescharacteristic
42C22×C3×C4—21no—
43C22×C3×C4—21no—
44C22×C3×C4—21no—
45C22×C3×C4—21no—
46C22×C3×C4—21no—
47C22×C3×C4—21no—
48C22×C3×C4—22no—
49C22×C3×C4—22no—
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×C4—41no—
57C22×C3×C4—41no—
58C22×C3×C4—41no—
59C22×C3×C4—41no—
60C22×C3×C4—41no—
61C22×C3×C4—41no—
62C22×C3×C4—41no—
63C22×C3×C4—41no—
64C22×C3×C4—41no—
65C22×C3×C4—41no—
66C22×C3×C4—41no—
67C22×C3×C4—41no—
68C22×C3×C4C2×C811yescharacteristic
69C22×C3×C4M1611yescharacteristic
70C22×C3×C4D811yescharacteristic
71C22×C3×C4SD1611yescharacteristic
72C22×C3×C4SD1611yescharacteristic
73C22×C3×C4Q1611yescharacteristic
74C3×C4⋊C4—42no—
75C3×C4⋊C4—42no—
76C2×C3⋊C8—42no—
77C2×C3⋊C8—42no—
78C2×C3⋊C8—81no—
79C2×C3⋊C8—81no—
80C2×C3⋊C8—81no—
81C2×C3⋊C8—81no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4C2×Dic311yescharacteristic
85C23×C4#6711yescharacteristic
86C23×C4#6711yescharacteristic
87C22×C8—121no—
88C22×C8—121no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C22⋊C8—62no—
92C22⋊C8—62no—
93C22⋊C8—121no—
94C22⋊C8—121no—
95#95—21no—
96#95—41no—
97#95—42no—
98#95—41no—
99#95—41no—
100#95C3⋊C812yescomplemented
101C23×C3—21no—
102C23×C3—21no—
103C23×C3—21no—
104C23×C3—21no—
105C23×C3C22⋊C811yescharacteristic
106C23×C3#9911yescharacteristic
107C23×C3#10111yescharacteristic
108C23×C3#10211yescharacteristic
109C23×C3#10311yescharacteristic
110C23×C3#10411yescharacteristic
111C23×C3#10411yescharacteristic
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—42no—
119C2×C3×C4—41no—
120C2×C3×C4—41no—
121C2×C3×C4—41no—
122C2×C3×C4—41no—
123C2×C3×C4—41no—
124C2×C3×C4—41no—
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—42no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—42no—
137C2×C3×C4—81no—
138C2×C3×C4—81no—
139C2×C3×C4—81no—
140C2×C3×C4—81no—
141C2×C3×C4—81no—
142C2×C3×C4—81no—
143C2×C3×C4—81no—
144C2×C3×C4—81no—
145C2×C3×C4—81no—
146C2×C3×C4—81no—
147C3⋊C8—82no—
148C3⋊C8—82no—
149C24#21711yescharacteristic
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4C2×C3⋊C811yescharacteristic
177C22×C4#20811yescharacteristic
178C22×C4#21311yescharacteristic
179C22×C4#21411yescharacteristic
180C22×C4#21511yescharacteristic
181C22×C4#21611yescharacteristic
182C2×C8—122no—
183C2×C8—122no—
184C2×C8—241no—
185C2×C8—241no—
186C2×C8—241no—
187C2×C8—241no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C22×C3—21no—
191C22×C3—21no—
192C22×C3—21no—
193C22×C3—21no—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—21no—
202C22×C3#32411yescharacteristic
203C22×C3D4⋊C811yescharacteristic
204C22×C3Q8⋊C811yescharacteristic
205C22×C3#32711yescharacteristic
206C22×C3#32811yescharacteristic
207C22×C3#33011yescharacteristic
208C22×C3#33211yescharacteristic
209C22×C3AΣL(1,𝔽16)11yescharacteristic
210C22×C3#35211yescharacteristic
211C22×C3#35411yescharacteristic
212C22×C3#35511yescharacteristic
213C3×C4—41no—
214C3×C4—41no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4—41no—
218C3×C4—41no—
219C3×C4—82no—
220C3×C4—82no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23#83011yescharacteristic
226C23#83211yescharacteristic
227C23#83411yescharacteristic
228C23#83511yescharacteristic
229C23#83611yescharacteristic
230C23#83711yescharacteristic
231C23#83711yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C8—242no—
268C8—242no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3#127111yescharacteristic
274C2×C3#132411yescharacteristic
275C2×C3#132611yescharacteristic
276C2×C3#134411yescharacteristic
277C2×C3#134511yescharacteristic
278C2×C3#135111yescharacteristic
279C2×C3#135311yescharacteristic
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22#452811yescharacteristic
293C22D4⋊(C3⋊C8)11yescharacteristic
294C22#453511yescharacteristic
295C22#453611yescharacteristic
296C22#453711yescharacteristic
297C22#453811yescharacteristic
298C22#453911yescharacteristic
299C22#454111yescharacteristic
300C22#454211yescharacteristic
301C22#454311yescharacteristic
302C22#454711yescharacteristic
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4—82no—
311C3#712211yescharacteristic, complemented, Sylow
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#6763211yescharacteristic
317C2#6763611yescharacteristic
318C2#6763811yescharacteristic
319C2#6764611yescharacteristic
320C2#6764711yescharacteristic
321C2#6764911yescharacteristic
322C2#6765211yescharacteristic
subgroupG/H
centerC23#834
commutatorC22×C3×C4C2×C8
FrattiniC2×#95C2×D3
FittingC3×#1442C2
socleC23×C3#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×#95C22
hypercenter#1442D3
hypocenterC3#7122
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ #1442

Complements

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

NH ≅ G/Ntype
29. C3×#95C8semidirect
100. #95C3⋊C8semidirect
311. C3#7122semidirect

Maximal subgroups

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

subgroupindex
C3×#14422
#676132
#71223