Finite Group Databasebeta

← #10708356

Subgroups — #10708356

Order 768.

364 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 55 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#76549C211yesmaximal, characteristic
2#69673C211yesmaximal, characteristic
3#68140C211yesmaximal, characteristic
4#68139C211yesmaximal, characteristic
5C3×#1319C211yesmaximal, characteristic
6#67458C211yesmaximal, characteristic
7#67457C211yesmaximal, characteristic
8#12273—31nomaximal, Sylow
9#5500C2211yescharacteristic
10C3×#391C2211yescharacteristic
11#5230—21no—
12#5225—21no—
13#4981C2211yescharacteristic
14#4979C2211yescharacteristic
15C4×Dic3⋊C4—21no—
16#4739—21no—
17#4721—21no—
18#4652—21no—
19#4652—21no—
20#4647—21no—
21#4646—21no—
22C3×#324C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4530C2211yescharacteristic
25#4523—21no—
26#4467—21no—
27#4467—21no—
28#2586—31no—
29#2028—31no—
30#1611—31no—
31#1605—31no—
32#1354—31no—
33#1346—31no—
34#1319D311yescharacteristic
35C22×C3×Q8D411yescharacteristic
36C22×Dic6D411yescharacteristic
37C2×C3×C4⋊C4C2311yescharacteristic
38C2×C3×C42D411yescharacteristic
39#926—21no—
40C2×C4⋊Dic3D411yescharacteristic
41#924—21no—
42#924—21no—
43C2×Dic3⋊C4—21no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C2×C4×Dic3—21no—
50C2×#208—21no—
51C2×Dic12—41no—
52#890—41no—
53#888—41no—
54C3×C22⋊C8—21no—
55C3×C22⋊C8—21no—
56C3×#95—21no—
57C3×#95—21no—
58#835—41no—
59#835—41no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95D411yescharacteristic
66C3⋊#95D411yescharacteristic
67#817—41no—
68#816—41no—
69#816—41no—
70#544—31no—
71#483—61no—
72#451—61no—
73#419—61no—
74#398—31no—
75#395—31no—
76#395—61no—
77#391—61no—
78#391C2×D311yescharacteristic
79#389—61no—
80#387—61no—
81#387—61no—
82C4×C4⋊C4—61no—
83#343—31no—
84#339—61no—
85#328—61no—
86#328—61no—
87#324C2×D311yescharacteristic
88#324C2×D311yescharacteristic
89C2×C3×Q8—21no—
90C2×C3×Q8—21no—
91C2×C3×Q8—41no—
92C2×C3×Q8—41no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C2×Dic6—41no—
105C2×Dic6—42no—
106C2×C3×C8—41no—
107C2×C3×C8—41no—
108C3×C4⋊C4—41no—
109C3×C42—41no—
110#217—41no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3—41no—
116Dic3⋊C4—24no—
117Dic3⋊C4—42no—
118Dic3⋊C4—41no—
119Dic3⋊C4—41no—
120Dic3⋊C4—41no—
121Dic3⋊C4—41no—
122Dic3⋊C4—81no—
123Dic3⋊C4—81no—
124C4×Dic3—42no—
125C4×Dic3—41no—
126C4×Dic3—81no—
127#208—41no—
128#208—41no—
129C2×C3⋊C8—41no—
130Dic12—82no—
131C22×Q8—31no—
132C22×Q8D1211yescharacteristic
133C2×Q16—121no—
134C2×M16—61no—
135C4⋊Q8—121no—
136#125—121no—
137C22⋊Q8—61no—
138C22⋊Q8—61no—
139#117—61no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4C22×D311yescharacteristic
147C2×C42—61no—
148C2×C42—61no—
149C2×C42D1211yescharacteristic
150#106—121no—
151#103—121no—
152#103—121no—
153#103—121no—
154#103—121no—
155C22⋊C8—21no—
156C22⋊C8—21no—
157#95—21no—
158#95—21no—
159#95—31no—
160#95—31no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166C23×C3#12011yescharacteristic
167C3×Q8—42no—
168C3×Q8—81no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—22no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4#13511yescharacteristic
182C2×C3×C4#13711yescharacteristic
183C2×Dic3—22no—
184C2×Dic3—22no—
185C2×Dic3—22no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197Dic6—42no—
198Dic6—81no—
199Dic6—81no—
200Dic6—81no—
201C3×C8—81no—
202C3×C8—81no—
203C3⋊C8—81no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8—121no—
211C2×Q8—122no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4C2×D1211yescharacteristic
219C22×C4D3×D411yescharacteristic
220C22×C4D3×D411yescharacteristic
221Q16—242no—
222M16—121no—
223M16—121no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—121no—
227C4⋊C4—41no—
228C4⋊C4—64no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—121no—
234C4⋊C4—121no—
235C4⋊C4—121no—
236C4⋊C4—122no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—241no—
240C4⋊C4—241no—
241C22⋊C4—121no—
242C42—41no—
243C42—121no—
244C42—122no—
245C42—241no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3#45211yescharacteristic
249C22×C3#45511yescharacteristic
250C22×C3B(4,𝔽2)11yescharacteristic
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—42no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—81no—
257Dic3—42no—
258Dic3—42no—
259Dic3—42no—
260Dic3—81no—
261Dic3—81no—
262Dic3—81no—
263C23#88211yescharacteristic
264Q8—42no—
265Q8—81no—
266Q8—122no—
267Q8—241no—
268Q8—241no—
269Q8—241no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4#90411yescharacteristic
297C2×C4#90911yescharacteristic
298C8—81no—
299C8—81no—
300C8—241no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3#160611yescharacteristic
304C2×C3#219711yescharacteristic
305C2×C3#220511yescharacteristic
306C22—21no—
307C22—21no—
308C22#473311yescharacteristic
309C22#474111yescharacteristic
310C22#474611yescharacteristic
311C4—21no—
312C4—21no—
313C4—42no—
314C4—41no—
315C4—41no—
316C4—81no—
317C4—122no—
318C4—122no—
319C4—122no—
320C4—241no—
321C4—241no—
322C4—241no—
323C3#1227311yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2#6810911yescharacteristic
327C2#6915311yescharacteristic
328C2#6916911yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1319C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1319D3
hypocenterC3#12273
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
323. C3#12273semidirect

Maximal subgroups

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

subgroupindex
#765492
#696732
#681402
#681392
C3×#13192
#674582
#674572
#122733