Finite Group Databasebeta

← #10692439

Subgroups — #10692439

Order 768.

368 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#2985C211yesmaximal, characteristic
2#83372C211yesmaximal, characteristic
3#71831C211yesmaximal, characteristic
4#70815C211yesmaximal, characteristic
5#70768C211yesmaximal, characteristic
6#67755C211yesmaximal, characteristic
7#67747C211yesmaximal, characteristic
8#9374—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#446—22no—
12C3×#446—21no—
13C3×C8×Q8—21no—
14C3×#433—22no—
15C3×#432C2211yescharacteristic
16C3×#431C2211yescharacteristic
17#5094—21no—
18#5091—21no—
19#5069C412yes—
20#5064C412yes—
21#4996—22no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4547—21no—
26#4547—21no—
27#4465—22no—
28#4456—21no—
29#4456—21no—
30#3237—31no—
31#2985D311yescharacteristic
32#1552—31no—
33#1537—31no—
34#1481—31no—
35#1290—31no—
36#1287—31no—
37C3×C4⋊Q8D411yescharacteristic
38C3×C4⋊Q8D411yescharacteristic
39C3×#125C2×C412yes—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×#117D412yes—
45C2×C3×C42C2311yescharacteristic
46#926—21no—
47#924—21no—
48#924—21no—
49C22×C3⋊C8—21no—
50#870—21no—
51#869C2×C411yescharacteristic
52#869C2×C411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6—21no—
55C3×C4⋊C8—22no—
56C3×C4⋊C8—22no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×#97—22no—
62C3×#97D412yes—
63C3×C4×C8—22no—
64C3×C4×C8D411yescharacteristic
65C3×C4×C8D411yescharacteristic
66#830—21no—
67#830—21no—
68#810—22no—
69#810—22no—
70#808—22no—
71#807—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76#803—21no—
77#803C2×C412yes—
78#533—31no—
79#533C2×D311yescharacteristic
80#501—32no—
81#491—32no—
82#485—62no—
83#477—61no—
84#475—61no—
85#446—22no—
86#446—21no—
87C8×Q8—21no—
88#433—22no—
89#432—31no—
90#432—31no—
91#432C2×D311yescharacteristic
92#431C2×D311yescharacteristic
93C2×C4⋊C8—31no—
94#332—61no—
95#332—61no—
96#330—62no—
97Q8⋊C8—61no—
98Q8⋊C8—61no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C2×Dic6—21no—
105C2×Dic6—21no—
106C2×C3×C8—21no—
107C2×C3×C8—22no—
108C2×C3×C8—21no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4C22⋊C412yes—
114C3×C4⋊C4C22⋊C412yes—
115C3×C22⋊C4—22no—
116C3×C42—22no—
117C3×C42C22×C411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121#217—41no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128C4×Dic3—41no—
129C2×C3⋊C8—22no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—81no—
133C22×C8—61no—
134C4⋊Q8—31no—
135C4⋊Q8—31no—
136C4⋊Q8D1211yescharacteristic
137C4⋊Q8D1211yescharacteristic
138#125—61no—
139#125C4×D312yes—
140C22⋊Q8—21no—
141C22⋊Q8—21no—
142C22⋊Q8—61no—
143C22⋊Q8—61no—
144C4×Q8—21no—
145C4×Q8—21no—
146C4×Q8—61no—
147C4×Q8—61no—
148#117—61no—
149#117#6712yes—
150C2×C42C22×D311yescharacteristic
151#107—62no—
152#106—62no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—61no—
158C4⋊C8—61no—
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—22no—
168#97—32no—
169#97—61no—
170#97#6712yes—
171C4×C8—22no—
172C4×C8#6711yescharacteristic
173C4×C8#6711yescharacteristic
174C23×C3#12311yescharacteristic
175C3×Q8—41no—
176C3×Q8—41no—
177C3×Q8—81no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4C2×C22⋊C411yescharacteristic
185C2×C3×C4C4×D411yescharacteristic
186C2×C3×C4C4×D411yescharacteristic
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×Dic3—41no—
191C2×Dic3—41no—
192Dic6—42no—
193Dic6—42no—
194C3×C8—41no—
195C3×C8—42no—
196C3×C8—41no—
197C3⋊C8—42no—
198C3⋊C8—81no—
199C3⋊C8—81no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—61no—
203C2×Q8—61no—
204C22×C4#23511yescharacteristic
205C22×C4#23511yescharacteristic
206C22×C4C2×#6711yescharacteristic
207C2×C8—21no—
208C2×C8—22no—
209C2×C8—21no—
210C2×C8—62no—
211C2×C8—121no—
212C2×C8—121no—
213C2×C8—241no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—121no—
223C4⋊C4—121no—
224C4⋊C4(C2×D3)⋊C412yes—
225C4⋊C4(C2×D3)⋊C412yes—
226C22⋊C4—22no—
227C22⋊C4—121no—
228C42—22no—
229C42—121no—
230C42C2×C4×D311yescharacteristic
231C42C2×D1211yescharacteristic
232C42C2×#6711yescharacteristic
233C42C2×#6711yescharacteristic
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—42no—
244C3×C4—42no—
245C3×C4#47111yescharacteristic
246C3×C4#47311yescharacteristic
247Dic3—81no—
248Dic3—81no—
249C23#92911yescharacteristic
250Q8—41no—
251Q8—41no—
252Q8—81no—
253Q8—122no—
254Q8—122no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—121no—
262C2×C4—121no—
263C2×C4C2×(C2×D3)⋊C411yescharacteristic
264C2×C4C4×#6711yescharacteristic
265C2×C4C4×#6711yescharacteristic
266C2×C4#93011yescharacteristic
267C2×C4#93011yescharacteristic
268C2×C4#95311yescharacteristic
269C8—41no—
270C8—42no—
271C8—41no—
272C8—122no—
273C8—241no—
274C8—241no—
275C2×C3—41no—
276C2×C3#188111yescharacteristic
277C2×C3#190811yescharacteristic
278C2×C3#191211yescharacteristic
279C22—21no—
280C22—21no—
281C22—21no—
282C22#521311yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—42no—
289C4—42no—
290C4—241no—
291C4—241no—
292C4#514111yescharacteristic
293C4#514311yescharacteristic
294C3#937411yescharacteristic, complemented, Sylow
295C2—41no—
296C2#7166311yescharacteristic
297C2#7201911yescharacteristic
298C2#7202011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2985C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#2985D3
hypocenterC3#9374
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2985

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
C3×#29852
#833722
#718312
#708152
#707682
#677552
#677472
#93743