Finite Group Databasebeta

← #10692440

Subgroups — #10692440

Order 768.

368 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 89 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#79321C211yesmaximal, characteristic
2#79316C211yesmaximal, characteristic
3#70815C211yesmaximal, characteristic
4#70439C211yesmaximal, characteristic
5#70429C211yesmaximal, characteristic
6C3×#1290C211yesmaximal, characteristic
7#67756C211yesmaximal, characteristic
8#9374—31nomaximal, Sylow
9#5470C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#4926C2211yescharacteristic
15#4925C2211yescharacteristic
16#4851—21no—
17#4850—21no—
18#4831—21no—
19#4830—21no—
20#4723—22no—
21#4720—22no—
22#4703C412yes—
23#4702C412yes—
24#4701—22no—
25#4486—21no—
26#4484—21no—
27#4484—21no—
28#4482—21no—
29#4465—22no—
30#3237—31no—
31#2985—31no—
32#1552—31no—
33#1537—31no—
34#1481—31no—
35#1290D311yescharacteristic
36#1287—31no—
37C22×C3×C8—21no—
38C2×C3×C42C2311yescharacteristic
39#926—21no—
40#926D412yes—
41#924—21no—
42#924—21no—
43#924—21no—
44#924—21no—
45#870—21no—
46#870C2×C412yes—
47#869C2×C411yescharacteristic
48#869C2×C411yescharacteristic
49#869D411yescharacteristic
50#869D411yescharacteristic
51C4×Dic6—21no—
52C4×Dic6—21no—
53C4×Dic6—21no—
54C4×Dic6—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×#97C2×C412yes—
62C3×#97D412yes—
63#830—21no—
64#830—21no—
65#818—22no—
66#817—22no—
67#816—22no—
68#816—22no—
69#815—22no—
70Dic3⋊C8—22no—
71Dic3⋊C8—22no—
72C8×Dic3—22no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75#803—21no—
76C4×C3⋊C8D411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78#533—31no—
79#533—31no—
80#501—32no—
81#491—32no—
82#485—62no—
83#477—61no—
84#475—61no—
85#446—61no—
86#446—62no—
87C8×Q8—61no—
88#433—62no—
89#432—31no—
90#432C2×D311yescharacteristic
91#432C2×D311yescharacteristic
92#431—31no—
93C2×C4⋊C8C2×D311yescharacteristic
94#332—61no—
95#332—61no—
96#330—62no—
97Q8⋊C8—61no—
98Q8⋊C8—61no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C2×Dic6—21no—
103C2×Dic6—21no—
104C2×Dic6—21no—
105C2×Dic6—21no—
106C2×C3×C8—22no—
107C2×C3×C8—22no—
108C2×C3×C8—41no—
109C2×C3×C8—81no—
110C3×C42C22×C411yescharacteristic
111C3×C42C2×D411yescharacteristic
112C3×C42C2×D411yescharacteristic
113C3×C42C2×D411yescharacteristic
114#217—22no—
115#217—41no—
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3C22⋊C412yes—
123C4⋊Dic3C22⋊C412yes—
124Dic3⋊C4—22no—
125Dic3⋊C4—22no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128C4×Dic3—22no—
129C4×Dic3—41no—
130C2×C3⋊C8—21no—
131C2×C3⋊C8—21no—
132C2×C3⋊C8—41no—
133C22×C8—21no—
134C4⋊Q8—31no—
135C4⋊Q8—31no—
136C4⋊Q8—31no—
137C4⋊Q8—31no—
138#125—32no—
139#125—61no—
140C22⋊Q8—61no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143C22⋊Q8—61no—
144C4×Q8—61no—
145C4×Q8—61no—
146C4×Q8—61no—
147C4×Q8—61no—
148#117—32no—
149#117—61no—
150C2×C42C22×D311yescharacteristic
151#107—62no—
152#106—62no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—62no—
158C4⋊C8—62no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161#103—62no—
162#103—62no—
163C22⋊C8—21no—
164C22⋊C8—21no—
165C22⋊C8—61no—
166C22⋊C8—61no—
167#97—62no—
168#97—61no—
169#97C4×D312yes—
170#97D1212yes—
171C4×C8—31no—
172C4×C8—31no—
173C4×C8—62no—
174C23×C3#12311yescharacteristic
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4C2×C22⋊C411yescharacteristic
180C2×C3×C4C4×D411yescharacteristic
181C2×C3×C4C4×D411yescharacteristic
182C2×C3×C4#12011yescharacteristic
183C2×C3×C4#12111yescharacteristic
184C2×C3×C4#12111yescharacteristic
185C2×Dic3—22no—
186C2×Dic3—22no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189Dic6—41no—
190Dic6—41no—
191Dic6—42no—
192Dic6—42no—
193Dic6—81no—
194C3×C8—42no—
195C3×C8—42no—
196C3×C8—81no—
197C3⋊C8—41no—
198C3⋊C8—41no—
199C3⋊C8—81no—
200C2×Q8—61no—
201C2×Q8—61no—
202C2×Q8—61no—
203C2×Q8—61no—
204C22×C4#23511yescharacteristic
205C22×C4D3×D411yescharacteristic
206C22×C4#23911yescharacteristic
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—61no—
211C2×C8—61no—
212C2×C8—81no—
213C2×C8—121no—
214C4⋊C4—32no—
215C4⋊C4—32no—
216C4⋊C4—62no—
217C4⋊C4—61no—
218C4⋊C4—62no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—121no—
225C4⋊C4—121no—
226C22⋊C4—62no—
227C22⋊C4—121no—
228C42—62no—
229C42—121no—
230C42C2×C4×D311yescharacteristic
231C42C2×D1211yescharacteristic
232C42D3×D411yescharacteristic
233C42D3×D411yescharacteristic
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3#38611yescharacteristic
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4#47111yescharacteristic
244C3×C4#47311yescharacteristic
245Dic3—42no—
246Dic3—42no—
247Dic3—81no—
248Dic3—81no—
249C23#89411yescharacteristic
250Q8—122no—
251Q8—122no—
252Q8—121no—
253Q8—121no—
254Q8—241no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—121no—
262C2×C4—121no—
263C2×C4C4×D1211yescharacteristic
264C2×C4D3×C22⋊C411yescharacteristic
265C2×C4#88211yescharacteristic
266C2×C4#88411yescharacteristic
267C2×C4#89311yescharacteristic
268C2×C4#89511yescharacteristic
269C8—42no—
270C8—42no—
271C8—81no—
272C8—121no—
273C8—121no—
274C8—241no—
275C2×C3—41no—
276C2×C3#188111yescharacteristic
277C2×C3#190811yescharacteristic
278C2×C3#191211yescharacteristic
279C22—21no—
280C22—21no—
281C22—21no—
282C22#466511yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—122no—
289C4—122no—
290C4—241no—
291C4—241no—
292C4#474811yescharacteristic
293C4#475511yescharacteristic
294C3#937411yescharacteristic, complemented, Sylow
295C2—41no—
296C2#6870811yescharacteristic
297C2#6894711yescharacteristic
298C2#6900711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1290C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#1290D3
hypocenterC3#9374
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1290

Complements

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

NH ≅ G/Ntype
294. C3#9374semidirect

Maximal subgroups

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

subgroupindex
#793212
#793162
#708152
#704392
#704292
C3×#12902
#677562
#93743