Finite Group Databasebeta

← #10708028

Subgroups — #10708028

Order 768.

380 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 57 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#77240C211yesmaximal, characteristic
2C3×#1605C211yesmaximal, characteristic
3#71905C211yesmaximal, characteristic
4#71904C211yesmaximal, characteristic
5#67648C211yesmaximal, characteristic
6#67457C211yesmaximal, characteristic
7#67455C211yesmaximal, characteristic
8#12224—31nomaximal, Sylow
9C3×#544C2211yescharacteristic
10C3×#451—21no—
11C3×#395C2211yescharacteristic
12#5230—21no—
13#5050—21no—
14#5004—21no—
15#4979—21no—
16#4979C2211yescharacteristic
17#4973—21no—
18#4973C2211yescharacteristic
19#4646—21no—
20#4646—21no—
21#4646—21no—
22C3×#328—21no—
23C3×#324C2211yescharacteristic
24#4538—21no—
25#4532—21no—
26#4530C2211yescharacteristic
27#4528C2211yescharacteristic
28#2811—31no—
29#2028—31no—
30#1605—31no—
31#1605D311yescharacteristic
32#1354—31no—
33#1354—31no—
34#1319—31no—
35C22×C3×Q8D411yescharacteristic
36C22×Dic6—21no—
37C22×Dic6D411yescharacteristic
38C22×Dic6D411yescharacteristic
39C2×C3×Q16—41no—
40C3×C4⋊Q8—41no—
41C3×#125—41no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×#117—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4C2311yescharacteristic
47C2×#216—41no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3D411yescharacteristic
53C2×#208—21no—
54C3×#103—41no—
55C3×C22⋊C8—21no—
56C3×#95—21no—
57C3×#95D411yescharacteristic
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95D411yescharacteristic
66#830—21no—
67#810—41no—
68#810—41no—
69#810—41no—
70#544C2×D311yescharacteristic
71#451—21no—
72#451—61no—
73#419—61no—
74#395—31no—
75#395—61no—
76#395—61no—
77#395C2×D311yescharacteristic
78#391—31no—
79#391—61no—
80#391—61no—
81#391—61no—
82#391—61no—
83#343—31no—
84#339—61no—
85#328—21no—
86#328—61no—
87#324—31no—
88#324C2×D311yescharacteristic
89C2×C3×Q8—21no—
90C2×C3×Q8—21no—
91C2×C3×Q8—41no—
92C2×C3×Q8—81no—
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—24no—
103C2×Dic6—21no—
104C2×Dic6—41no—
105C2×Dic6—41no—
106C2×Dic6—41no—
107C2×Dic6—81no—
108C2×Dic6—81no—
109C2×Dic6—81no—
110C2×Dic6Q1612yescomplemented
111C3×Q16—82no—
112C2×C3×C8—41no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—81no—
121C3×C22⋊C4—41no—
122C3×C42—41no—
123#216—82no—
124Dic3⋊C4—81no—
125Dic3⋊C4—81no—
126Dic3⋊C4—81no—
127C4×Dic3—41no—
128C4×Dic3—81no—
129#208—41no—
130#208—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C22×Q8—31no—
134C22×Q8—31no—
135C22×Q8—61no—
136C22×Q8#6711yescharacteristic
137C2×Q16—41no—
138C2×Q16—121no—
139C2×M16—61no—
140C4⋊Q8—41no—
141#125—41no—
142C22⋊Q8—21no—
143C22⋊Q8—21no—
144#117—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4C22×D311yescharacteristic
150C2×C42—31no—
151C2×C42—61no—
152#103—41no—
153#103—121no—
154#103—121no—
155#103—121no—
156C22⋊C8—21no—
157C22⋊C8—61no—
158#95—21no—
159#95—31no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95#6711yescharacteristic
168C23×C3#12011yescharacteristic
169C3×Q8—42no—
170C3×Q8—81no—
171C3×Q8—81no—
172C3×Q8—81no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—22no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4C2×Q1611yescharacteristic
188C2×C3×C4#13711yescharacteristic
189C2×Dic3—22no—
190C2×Dic3—22no—
191C2×Dic3—22no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201Dic6—42no—
202Dic6—42no—
203Dic6—81no—
204Dic6—84no—
205Dic6—81no—
206C3×C8—81no—
207C3⋊C8—81no—
208C3⋊C8—81no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—32no—
212C2×Q8—41no—
213C2×Q8—64no—
214C2×Q8—61no—
215C2×Q8—81no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×Q8—121no—
219C2×Q8—241no—
220C2×Q8—241no—
221C2×Q8—241no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4D3×D411yescharacteristic
229C22×C4D3×D411yescharacteristic
230C22×C4C2×#6711yescharacteristic
231Q16—82no—
232Q16—242no—
233M16—121no—
234M16—121no—
235C2×C8—41no—
236C2×C8—121no—
237C2×C8—121no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—81no—
246C4⋊C4—241no—
247C4⋊C4—241no—
248C4⋊C4—241no—
249C22⋊C4—41no—
250C42—41no—
251C42—121no—
252C42—241no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3#45111yescharacteristic
256C22×C3#45511yescharacteristic
257C22×C3B(4,𝔽2)11yescharacteristic
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—42no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—81no—
264C3×C4—81no—
265C3×C4—81no—
266Dic3—42no—
267Dic3—42no—
268Dic3—42no—
269Dic3—81no—
270C23#93711yescharacteristic
271Q8—42no—
272Q8—81no—
273Q8—81no—
274Q8—81no—
275Q8—122no—
276Q8—122no—
277Q8—241no—
278Q8—244no—
279Q8—241no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—62no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4D3×Q1611yescharacteristic
307C2×C4#91811yescharacteristic
308C8—81no—
309C8—241no—
310C8—241no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3#160211yescharacteristic
314C2×C3#219911yescharacteristic
315C2×C3#220511yescharacteristic
316C22—21no—
317C22—21no—
318C22#503211yescharacteristic
319C22#518611yescharacteristic
320C22#520211yescharacteristic
321C4—21no—
322C4—21no—
323C4—42no—
324C4—41no—
325C4—41no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—122no—
330C4—122no—
331C4—122no—
332C4—241no—
333C3#1222411yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#7098011yescharacteristic
337C2#7099511yescharacteristic
338C2#7099711yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1605C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1605D3
hypocenterC3#12224
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1605

Complements

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

NH ≅ G/Ntype
110. C2×Dic6Q16semidirect
333. C3#12224semidirect

Maximal subgroups

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

subgroupindex
#772402
C3×#16052
#719052
#719042
#676482
#674572
#674552
#122243