Finite Group Databasebeta

← #10708142

Subgroups — #10708142

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#77235C211yesmaximal, characteristic
2#69707C211yesmaximal, characteristic
3#68978C211yesmaximal, characteristic
4#68140C211yesmaximal, characteristic
5C3×#1354C211yesmaximal, characteristic
6#67495C211yesmaximal, characteristic
7#67453C211yesmaximal, characteristic
8#12240—31nomaximal, Sylow
9#5500C2211yescharacteristic
10C3×#395C2211yescharacteristic
11#5230—21no—
12#5228—21no—
13#5121—21no—
14#4982—21no—
15#4981C2211yescharacteristic
16#4975C2211yescharacteristic
17#4972—21no—
18#4721—21no—
19#4654—21no—
20#4653—21no—
21#4647—21no—
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4555—21no—
25#4528C2211yescharacteristic
26#4467—21no—
27#4452—21no—
28#2792—31no—
29#2060—31no—
30#1611—31no—
31#1605—31no—
32#1354D311yescharacteristic
33#1346—31no—
34#1325—31no—
35C22×C3×Q8D411yescharacteristic
36C2×C3×M16—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4C2311yescharacteristic
39C2×#216—41no—
40#926—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3D411yescharacteristic
43C2×C4⋊Dic3D411yescharacteristic
44#924—21no—
45#924—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3D411yescharacteristic
53#890—41no—
54#888—41no—
55C3×C22⋊C8—21no—
56C3×#95—21no—
57C3×#95D411yescharacteristic
58#835—41no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95D411yescharacteristic
65#830—21no—
66#818—41no—
67#817—41no—
68#817—41no—
69#816—41no—
70#544—31no—
71#483—61no—
72#451—61no—
73#428—61no—
74#398—31no—
75#398—61no—
76#395—61no—
77#395C2×D311yescharacteristic
78#389—61no—
79#389—61no—
80#387—61no—
81#387—61no—
82#384—31no—
83#343C2×D311yescharacteristic
84#339—61no—
85#328—61no—
86#328—61no—
87#324—31no—
88#324C2×D311yescharacteristic
89C2×C3×Q8—41no—
90C2×C3×Q8—81no—
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—21no—
101C2×Dic6—21no—
102C3×M16—22no—
103C3×M16—41no—
104C2×C3×C8—41no—
105C2×C3×C8—41no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—81no—
108#217—41no—
109#216—82no—
110C4⋊Dic3—24no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3—41no—
116C4⋊Dic3—41no—
117Dic3⋊C4—41no—
118Dic3⋊C4—41no—
119Dic3⋊C4—41no—
120Dic3⋊C4—41no—
121Dic3⋊C4—81no—
122Dic3⋊C4—81no—
123Dic3⋊C4—81no—
124Dic3⋊C4—81no—
125C4×Dic3—21no—
126C4×Dic3—21no—
127C4×Dic3—41no—
128C4×Dic3—81no—
129C2×C3⋊C8—41no—
130C22×Q8D1211yescharacteristic
131C2×Q16—121no—
132C2×M16—21no—
133C4⋊Q8—121no—
134#125—121no—
135C22⋊Q8—61no—
136C22⋊Q8—61no—
137#117—61no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—31no—
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⋊C4—61no—
147C2×C4⋊C4C22×D311yescharacteristic
148C2×C42—31no—
149C2×C42—61no—
150#107—121no—
151#106—121no—
152#106—121no—
153#103—121no—
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#95D1211yescharacteristic
165C23×C3#12011yescharacteristic
166C3×Q8—42no—
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#13511yescharacteristic
181C2×C3×C4#13711yescharacteristic
182C2×Dic3—22no—
183C2×Dic3—22no—
184C2×Dic3—41no—
185C2×Dic3—41no—
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—81no—
197Dic6—81no—
198C3×C8—42no—
199C3×C8—81no—
200C3⋊C8—81no—
201C2×Q8—41no—
202C2×Q8—61no—
203C2×Q8—61no—
204C2×Q8—81no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—61no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4C2×D1211yescharacteristic
212C22×C4D3×D411yescharacteristic
213C22×C4D3×D411yescharacteristic
214Q16—242no—
215M16—22no—
216M16—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—121no—
220C4⋊C4—41no—
221C4⋊C4—64no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—81no—
227C4⋊C4—121no—
228C4⋊C4—121no—
229C4⋊C4—121no—
230C4⋊C4—121no—
231C4⋊C4—121no—
232C4⋊C4—121no—
233C4⋊C4—241no—
234C4⋊C4—241no—
235C4⋊C4—241no—
236C4⋊C4—241no—
237C22⋊C4—121no—
238C42—61no—
239C42—61no—
240C42—121no—
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—
253Dic3—42no—
254Dic3—42no—
255Dic3—81no—
256Dic3—81no—
257Dic3—81no—
258Dic3—81no—
259C23#88211yescharacteristic
260Q8—42no—
261Q8—81no—
262Q8—241no—
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—62no—
277C2×C4—62no—
278C2×C4—121no—
279C2×C4—121no—
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#91611yescharacteristic
291C2×C4#91811yescharacteristic
292C8—42no—
293C8—81no—
294C8—241no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3#160311yescharacteristic
298C2×C3#220011yescharacteristic
299C2×C3#220511yescharacteristic
300C22—21no—
301C22—21no—
302C22#474111yescharacteristic
303C22#480811yescharacteristic
304C22#482611yescharacteristic
305C4—21no—
306C4—21no—
307C4—42no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4—122no—
312C4—122no—
313C4—241no—
314C4—241no—
315C4—241no—
316C4—241no—
317C3#1224011yescharacteristic, complemented, Sylow
318C2—21no—
319C2—21no—
320C2#6832611yescharacteristic
321C2#6913311yescharacteristic
322C2#6916911yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1354C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1354D3
hypocenterC3#12240
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1354

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
#772352
#697072
#689782
#681402
C3×#13542
#674952
#674532
#122403