Finite Group Databasebeta

← #10691823

Subgroups — #10691823

Order 768.

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

normal subgroups only

This group has 374 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#83371C211yesmaximal, characteristic
3#71822C211yesmaximal, characteristic
4#70813C211yesmaximal, characteristic
5#70771C211yesmaximal, characteristic
6#67756C211yesmaximal, characteristic
7#67740C211yesmaximal, characteristic
8#9286—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#446—22no—
12C3×#446—21no—
13C3×C8×Q8—21no—
14C3×#433—22no—
15C3×#432C2211yescharacteristic
16C3×#431C2211yescharacteristic
17#5093C411yescharacteristic
18#5093C411yescharacteristic
19#5080C411yescharacteristic
20#5078C411yescharacteristic
21#5063—22no—
22#4996—22no—
23#4926C2211yescharacteristic
24#4925C2211yescharacteristic
25C2×#803C2211yescharacteristic
26#4543—22no—
27#4465—22no—
28#4456—21no—
29#4456—21no—
30#3231—31no—
31#2985D311yescharacteristic
32#1531—31no—
33#1531—31no—
34#1497—31no—
35#1287—31no—
36#1275—31no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×#125D412yes—
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—
74#803—21no—
75#803—21no—
76#803—21no—
77C4×C3⋊C8C2×C411yescharacteristic
78C4×C3⋊C8C2×C411yescharacteristic
79#533—31no—
80#533C2×D311yescharacteristic
81#500—31no—
82#498—31no—
83#487—31no—
84#487—31no—
85#485—62no—
86#479—62no—
87#446—22no—
88#446—21no—
89C8×Q8—21no—
90#433—22no—
91#432—31no—
92#432C2×D311yescharacteristic
93#431—31no—
94#431C2×D311yescharacteristic
95C2×#97—31no—
96#330—62no—
97#330—62no—
98Q8⋊C8—61no—
99Q8⋊C8—61no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×C3×C8—21no—
108C2×C3×C8—22no—
109C2×C3×C8—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4C22⋊C412yes—
115C3×C4⋊C4C22⋊C412yes—
116C3×C22⋊C4—22no—
117C3×C42—22no—
118C3×C42C22×C411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122#217—41no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127Dic3⋊C4—41no—
128Dic3⋊C4—41no—
129C4×Dic3—41no—
130C2×C3⋊C8—22no—
131C2×C3⋊C8—21no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C22×C8—61no—
137C4⋊Q8—31no—
138C4⋊Q8—31no—
139C4⋊Q8C4×D311yescharacteristic
140C4⋊Q8C4×D311yescharacteristic
141#125—61no—
142#125D1212yes—
143C22⋊Q8—21no—
144C22⋊Q8—21no—
145C22⋊Q8—61no—
146C22⋊Q8—61no—
147C4×Q8—21no—
148C4×Q8—21no—
149C4×Q8—61no—
150C4×Q8—61no—
151#117—61no—
152#117#6712yes—
153C2×C42C22×D311yescharacteristic
154#107—62no—
155#106—62no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162#103—62no—
163#103—62no—
164C22⋊C8—21no—
165C22⋊C8—21no—
166C22⋊C8—61no—
167C22⋊C8—61no—
168#97—22no—
169#97—61no—
170#97—61no—
171#97—61no—
172#97#6712yes—
173C4×C8—22no—
174C4×C8—31no—
175C4×C8—31no—
176C4×C8#6711yescharacteristic
177C4×C8#6711yescharacteristic
178C23×C3#12311yescharacteristic
179C3×Q8—41no—
180C3×Q8—41no—
181C3×Q8—81no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4C2×C22⋊C411yescharacteristic
189C2×C3×C4C4×D411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×Dic3—41no—
195C2×Dic3—41no—
196Dic6—42no—
197Dic6—41no—
198Dic6—41no—
199C3×C8—41no—
200C3×C8—42no—
201C3×C8—41no—
202C3⋊C8—41no—
203C3⋊C8—41no—
204C3⋊C8—42no—
205C3⋊C8—81no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—61no—
209C2×Q8—61no—
210C22×C4#23511yescharacteristic
211C22×C4#23511yescharacteristic
212C22×C4C2×#6711yescharacteristic
213C2×C8—21no—
214C2×C8—22no—
215C2×C8—21no—
216C2×C8—62no—
217C2×C8—61no—
218C2×C8—61no—
219C2×C8—121no—
220C2×C8—121no—
221C2×C8—121no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—121no—
231C4⋊C4—121no—
232C4⋊C4(C2×D3)⋊C412yes—
233C4⋊C4(C2×D3)⋊C412yes—
234C22⋊C4—22no—
235C22⋊C4—121no—
236C42—22no—
237C42—121no—
238C42C2×C4×D311yescharacteristic
239C42C2×D1211yescharacteristic
240C42C2×#6711yescharacteristic
241C42C2×#6711yescharacteristic
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3#38611yescharacteristic
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—42no—
252C3×C4—42no—
253C3×C4#44311yescharacteristic
254C3×C4#44411yescharacteristic
255Dic3—81no—
256Dic3—81no—
257C23#92911yescharacteristic
258Q8—41no—
259Q8—41no—
260Q8—81no—
261Q8—122no—
262Q8—121no—
263Q8—121no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—121no—
271C2×C4—121no—
272C2×C4C2×(C2×D3)⋊C411yescharacteristic
273C2×C4C4×#6711yescharacteristic
274C2×C4C4×#6711yescharacteristic
275C2×C4#93011yescharacteristic
276C2×C4#93011yescharacteristic
277C2×C4#95311yescharacteristic
278C8—41no—
279C8—42no—
280C8—41no—
281C8—122no—
282C8—121no—
283C8—121no—
284C8—241no—
285C2×C3—41no—
286C2×C3#187611yescharacteristic
287C2×C3#190211yescharacteristic
288C2×C3#191211yescharacteristic
289C22—21no—
290C22—21no—
291C22—21no—
292C22#521311yescharacteristic
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—42no—
299C4—42no—
300C4—241no—
301C4—241no—
302C4#514411yescharacteristic
303C4#514511yescharacteristic
304C3#928611yescharacteristic, complemented, Sylow
305C2—41no—
306C2#7166311yescharacteristic
307C2#7201011yescharacteristic
308C2#7202211yescharacteristic
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#9286
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
304. C3#9286semidirect

Maximal subgroups

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

subgroupindex
C3×#29852
#833712
#718222
#708132
#707712
#677562
#677402
#92863