Finite Group Databasebeta

← #10708136

Subgroups — #10708136

Order 768.

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

normal subgroups only

This group has 350 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#77912C211yesmaximal, characteristic
2C3×#1611C211yesmaximal, characteristic
3#71904C211yesmaximal, characteristic
4#71900C211yesmaximal, characteristic
5#67650C211yesmaximal, characteristic
6#67457C211yesmaximal, characteristic
7#67453C211yesmaximal, characteristic
8#12240—31nomaximal, Sylow
9C3×#544C2211yescharacteristic
10C3×#483—21no—
11C3×#398C2211yescharacteristic
12#5050—21no—
13#4995—21no—
14#4984—21no—
15#4981—21no—
16#4979C2211yescharacteristic
17#4978—21no—
18#4975C2211yescharacteristic
19#4656—21no—
20#4652—21no—
21#4647—21no—
22C3×#328—21no—
23C3×#324C2211yescharacteristic
24#4538—21no—
25#4532—21no—
26#4530C2211yescharacteristic
27#4528C2211yescharacteristic
28#2792—31no—
29#2060—31no—
30#1611D311yescharacteristic
31#1605—31no—
32#1354—31no—
33#1346—31no—
34#1325—31no—
35C22×Dic6D411yescharacteristic
36C3×C4⋊Q8—41no—
37C3×#125—41no—
38C3×C22⋊Q8—21no—
39C3×C22⋊Q8—21no—
40C3×#117—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C4⋊C4C2311yescharacteristic
44C2×#216—41no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3D411yescharacteristic
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3D411yescharacteristic
54C2×#208—21no—
55C3×#106—41no—
56C3×#103—41no—
57C3×C22⋊C8—21no—
58C3×#95—21no—
59C3×#95D411yescharacteristic
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95D411yescharacteristic
66#830—21no—
67#810—41no—
68#808—41no—
69#807—41no—
70#544C2×D311yescharacteristic
71#483—21no—
72#451—61no—
73#428—61no—
74#398—61no—
75#398C2×D311yescharacteristic
76#395—31no—
77#395—61no—
78#389—61no—
79#389—61no—
80#387—61no—
81#387—61no—
82#384—31no—
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—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—21no—
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C2×Dic6—41no—
101C2×Dic6—81no—
102C2×C3×C8—41no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C42—41no—
115#216—82no—
116C4⋊Dic3—41no—
117C4⋊Dic3—81no—
118C4⋊Dic3—81no—
119Dic3⋊C4—24no—
120Dic3⋊C4—81no—
121Dic3⋊C4—81no—
122Dic3⋊C4—81no—
123C4×Dic3—21no—
124C4×Dic3—21no—
125C4×Dic3—81no—
126#208—22no—
127#208—41no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8—41no—
130C22×Q8—31no—
131C2×Q16—121no—
132C2×M16—61no—
133C4⋊Q8—41no—
134#125—41no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137#117—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4#6711yescharacteristic
147C2×C4⋊C4C22×D311yescharacteristic
148C2×C42—31no—
149C2×C42—61no—
150#107—121no—
151#106—41no—
152#106—121no—
153#103—41no—
154#103—121no—
155C22⋊C8—21no—
156C22⋊C8—61no—
157#95—21no—
158#95—31no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95#6711yescharacteristic
165C23×C3#12011yescharacteristic
166C3×Q8—81no—
167C3×Q8—81no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—22no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—41no—
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—41no—
182C2×C3×C4#13511yescharacteristic
183C2×C3×C4#13711yescharacteristic
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—
196Dic6—42no—
197Dic6—81no—
198C3×C8—81no—
199C3⋊C8—42no—
200C3⋊C8—81no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—121no—
204C2×Q8—241no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—61no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4D3×D411yescharacteristic
212C22×C4D3×D411yescharacteristic
213C22×C4C2×#6711yescharacteristic
214Q16—242no—
215M16—62no—
216M16—121no—
217C2×C8—41no—
218C2×C8—121no—
219C2×C8—121no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—64no—
231C4⋊C4—121no—
232C4⋊C4—241no—
233C4⋊C4—241no—
234C4⋊C4—241no—
235C4⋊C4—241no—
236C4⋊C4—241no—
237C22⋊C4—41no—
238C42—41no—
239C42—61no—
240C42—61no—
241C42—241no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3#45211yescharacteristic
245C22×C3#45611yescharacteristic
246C22×C3B(4,𝔽2)11yescharacteristic
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—42no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4—81no—
253C3×C4—81no—
254C3×C4—81no—
255Dic3—42no—
256Dic3—42no—
257Dic3—81no—
258Dic3—81no—
259C23#93711yescharacteristic
260Q8—81no—
261Q8—81no—
262Q8—122no—
263Q8—241no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4#91511yescharacteristic
291C2×C4#91611yescharacteristic
292C8—81no—
293C8—122no—
294C8—241no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3#160311yescharacteristic
298C2×C3#220011yescharacteristic
299C2×C3#220511yescharacteristic
300C22—21no—
301C22—21no—
302C22#503211yescharacteristic
303C22#517311yescharacteristic
304C22#520511yescharacteristic
305C4—21no—
306C4—21no—
307C4—42no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—122no—
314C4—122no—
315C4—241no—
316C4—241no—
317C3#1224011yescharacteristic, complemented, Sylow
318C2—21no—
319C2—21no—
320C2#7097411yescharacteristic
321C2#7099511yescharacteristic
322C2#7100111yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1611C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1611D3
hypocenterC3#12240
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1611

Complements

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

NH ≅ G/Ntype
317. C3#12240semidirect

Maximal subgroups

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

subgroupindex
#779122
C3×#16112
#719042
#719002
#676502
#674572
#674532
#122403