Finite Group Databasebeta

← #10708263

Subgroups — #10708263

Order 768.

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

normal subgroups only

This group has 332 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#78338C211yesmaximal, characteristic
2#69719C211yesmaximal, characteristic
3#68978C211yesmaximal, characteristic
4#68139C211yesmaximal, characteristic
5C3×#1353C211yesmaximal, characteristic
6#67494C211yesmaximal, characteristic
7#67453C211yesmaximal, characteristic
8#12259—31nomaximal, Sylow
9#5500C2211yescharacteristic
10C3×#400C2211yescharacteristic
11#5121—21no—
12#4981—21no—
13#4979C2211yescharacteristic
14#4975C2211yescharacteristic
15#4970—21no—
16#4739—21no—
17#4661—21no—
18#4658—21no—
19#4657—21no—
20#4656—21no—
21#4651—21no—
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4555—21no—
25#4528C2211yescharacteristic
26#4467—21no—
27#4452—21no—
28#2843—31no—
29#2060—31no—
30#1617—31no—
31#1605—31no—
32#1354—31no—
33#1353D311yescharacteristic
34#1325—31no—
35C22×Dic6D411yescharacteristic
36C2×C3×M16—21no—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C4⋊C4C2311yescharacteristic
39#926—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3D411yescharacteristic
43#924—21no—
44#924—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×C4×Dic3D411yescharacteristic
50C2×Dic12—41no—
51#890—41no—
52#888—41no—
53C3×C22⋊C8—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95D411yescharacteristic
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95D411yescharacteristic
64#830—21no—
65#818—41no—
66#817—41no—
67#816—41no—
68#810—41no—
69#807—41no—
70#544—31no—
71#484—61no—
72#451—61no—
73#428—61no—
74#400—61no—
75#400C2×D311yescharacteristic
76#398—61no—
77#398—61no—
78#398—61no—
79#395—31no—
80#395—61no—
81#384—31no—
82#383—61no—
83#343C2×D311yescharacteristic
84#339—61no—
85#328—61no—
86#328—61no—
87#324—31no—
88#324C2×D311yescharacteristic
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×Dic3—21no—
95C22×Dic3—21no—
96C22×Dic3—21no—
97C22×Dic3—21no—
98C2×Dic6—21no—
99C2×Dic6—21no—
100C2×Dic6—41no—
101C2×Dic6—81no—
102C3×M16—22no—
103C3×M16—41no—
104C2×C3×C8—41no—
105C2×C3×C8—41no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108#217—41no—
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3—41no—
114C4⋊Dic3—81no—
115C4⋊Dic3—81no—
116Dic3⋊C4—41no—
117Dic3⋊C4—41no—
118Dic3⋊C4—81no—
119Dic3⋊C4—81no—
120Dic3⋊C4—81no—
121Dic3⋊C4—81no—
122C4×Dic3—21no—
123C4×Dic3—41no—
124C4×Dic3Q1612yescomplemented
125C2×C3⋊C8—41no—
126Dic12—82no—
127C22×Q8—31no—
128C2×Q16—121no—
129C2×M16—21no—
130C4⋊Q8—121no—
131#125—121no—
132C22⋊Q8—61no—
133C22⋊Q8—61no—
134#117—61no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4D1211yescharacteristic
143C2×C4⋊C4C22×D311yescharacteristic
144C2×C42—31no—
145#107—121no—
146#107—121no—
147#106—121no—
148#103—121no—
149#103—121no—
150C22⋊C8—21no—
151C22⋊C8—61no—
152#95—21no—
153#95—21no—
154#95—31no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95—61no—
160#95—61no—
161#95D1211yescharacteristic
162C23×C3#12011yescharacteristic
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—42no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4C2×Q1611yescharacteristic
175C2×C3×C4#13711yescharacteristic
176C2×Dic3—22no—
177C2×Dic3—41no—
178C2×Dic3—41no—
179C2×Dic3—41no—
180C2×Dic3—41no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190Dic6—42no—
191Dic6—81no—
192Dic6—81no—
193Dic6—81no—
194C3×C8—42no—
195C3×C8—81no—
196C3⋊C8—81no—
197C2×Q8—61no—
198C2×Q8—61no—
199C2×Q8—121no—
200C2×Q8—241no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—61no—
204C22×C4—61no—
205C22×C4—61no—
206C22×C4—61no—
207C22×C4C2×D1211yescharacteristic
208C22×C4D3×D411yescharacteristic
209C22×C4D3×D411yescharacteristic
210Q16—242no—
211M16—22no—
212M16—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—121no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—121no—
223C4⋊C4—121no—
224C4⋊C4—121no—
225C4⋊C4—241no—
226C4⋊C4—241no—
227C4⋊C4—241no—
228C4⋊C4—241no—
229C4⋊C4—241no—
230C4⋊C4—241no—
231C22⋊C4—121no—
232C42—32no—
233C42—61no—
234C42—121no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3#45111yescharacteristic
238C22×C3#45611yescharacteristic
239C22×C3B(4,𝔽2)11yescharacteristic
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—41no—
243C3×C4—41no—
244C3×C4—81no—
245C3×C4—81no—
246Dic3—42no—
247Dic3—81no—
248Dic3—81no—
249Dic3—81no—
250Dic3—81no—
251Dic3—81no—
252C23#88211yescharacteristic
253Q8—122no—
254Q8—241no—
255Q8—241no—
256Q8—241no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—62no—
269C2×C4—121no—
270C2×C4—121no—
271C2×C4—121no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4#91511yescharacteristic
283C2×C4D3×Q1611yescharacteristic
284C8—42no—
285C8—81no—
286C8—241no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3#160511yescharacteristic
290C2×C3#220011yescharacteristic
291C2×C3#220311yescharacteristic
292C22—21no—
293C22—21no—
294C22#474111yescharacteristic
295C22#480911yescharacteristic
296C22#482611yescharacteristic
297C4—21no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4—122no—
304C4—241no—
305C4—241no—
306C4—241no—
307C4—241no—
308C4—241no—
309C3#1225911yescharacteristic, complemented, Sylow
310C2—21no—
311C2—21no—
312C2#6832711yescharacteristic
313C2#6913911yescharacteristic
314C2#6916211yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1353C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1353D3
hypocenterC3#12259
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1353

Complements

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

NH ≅ G/Ntype
124. C4×Dic3Q16semidirect
309. C3#12259semidirect

Maximal subgroups

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

subgroupindex
#783382
#697192
#689782
#681392
C3×#13532
#674942
#674532
#122593