Finite Group Databasebeta

← #10691824

Subgroups — #10691824

Order 768.

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

normal subgroups only

This group has 374 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#79316C211yesmaximal, characteristic
2#79273C211yesmaximal, characteristic
3#70813C211yesmaximal, characteristic
4#70813C211yesmaximal, characteristic
5#70803C211yesmaximal, characteristic
6C3×#1287C211yesmaximal, characteristic
7#67740C211yesmaximal, characteristic
8#9286—31nomaximal, Sylow
9#5470C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#4925C2211yescharacteristic
15C2×#803C2211yescharacteristic
16#4831—21no—
17#4830—21no—
18#4829—22no—
19#4723—22no—
20#4720—22no—
21#4701—22no—
22#4682C411yescharacteristic
23#4681C411yescharacteristic
24#4681C411yescharacteristic
25#4680C411yescharacteristic
26#4484—21no—
27#4484—21no—
28#4465—22no—
29#4465—22no—
30#3231—31no—
31#2985—31no—
32#1531—31no—
33#1531—31no—
34#1497—31no—
35#1287D311yescharacteristic
36#1275—31no—
37C2×C3×C42C2311yescharacteristic
38#926—21no—
39#926D412yes—
40#924—21no—
41#924—21no—
42#924—21no—
43#924—21no—
44C22×C3⋊C8—21no—
45#870—21no—
46#870D412yes—
47#869C2×C411yescharacteristic
48#869C2×C411yescharacteristic
49#869C2×C411yescharacteristic
50#869C2×C411yescharacteristic
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×C22⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×#97—21no—
62C3×#97D412yes—
63C3×C4×C8C2×C411yescharacteristic
64C3×C4×C8C2×C411yescharacteristic
65#830—21no—
66#818—22no—
67#817—22no—
68#816—22no—
69#816—22no—
70#815—22no—
71Dic3⋊C8—22no—
72Dic3⋊C8—22no—
73C8×Dic3—22no—
74C4⋊(C3⋊C8)—21no—
75#803—21no—
76#803—21no—
77C4×C3⋊C8D411yescharacteristic
78C4×C3⋊C8D411yescharacteristic
79#533—31no—
80#533—31no—
81#500—31no—
82#498—31no—
83#487—31no—
84#487—31no—
85#485—62no—
86#479—62no—
87#446—61no—
88#446—62no—
89C8×Q8—61no—
90#433—62no—
91#432C2×D311yescharacteristic
92#432C2×D311yescharacteristic
93#431—31no—
94#431C2×D311yescharacteristic
95C2×#97—31no—
96#330—62no—
97#330—62no—
98Q8⋊C8—61no—
99Q8⋊C8—61no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C2×Dic6—21no—
104C2×Dic6—21no—
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×C3×C8—21no—
108C2×C3×C8—21no—
109C2×C3×C8—22no—
110C2×C3×C8—41no—
111C3×C42C22×C411yescharacteristic
112C3×C42C2×D411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42C2×D411yescharacteristic
115#217—22no—
116#217—41no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3C22⋊C412yes—
124C4⋊Dic3C22⋊C412yes—
125Dic3⋊C4—22no—
126Dic3⋊C4—22no—
127Dic3⋊C4—41no—
128Dic3⋊C4—41no—
129C4×Dic3—22no—
130C4×Dic3—41no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C22×C8—61no—
137C4⋊Q8—31no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140C4⋊Q8—31no—
141#125—32no—
142#125—61no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146C22⋊Q8—61no—
147C4×Q8—61no—
148C4×Q8—61no—
149C4×Q8—61no—
150C4×Q8—61no—
151#117—32no—
152#117—61no—
153C2×C42C22×D311yescharacteristic
154#107—62no—
155#106—62no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—62no—
160C4⋊C8—62no—
161C4⋊C8—61no—
162#103—62no—
163#103—62no—
164C22⋊C8—21no—
165C22⋊C8—21no—
166C22⋊C8—21no—
167C22⋊C8—61no—
168#97—21no—
169#97—62no—
170#97—61no—
171#97—61no—
172#97D1212yes—
173C4×C8—31no—
174C4×C8—31no—
175C4×C8—62no—
176C4×C8C4×D311yescharacteristic
177C4×C8C4×D311yescharacteristic
178C23×C3#12311yescharacteristic
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4C2×C22⋊C411yescharacteristic
184C2×C3×C4C4×D411yescharacteristic
185C2×C3×C4C4×D411yescharacteristic
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12111yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×Dic3—22no—
190C2×Dic3—22no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193Dic6—41no—
194Dic6—41no—
195Dic6—42no—
196Dic6—41no—
197Dic6—41no—
198Dic6—81no—
199C3×C8—42no—
200C3×C8—41no—
201C3×C8—41no—
202C3×C8—81no—
203C3⋊C8—41no—
204C3⋊C8—41no—
205C3⋊C8—42no—
206C2×Q8—61no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×Q8—61no—
210C22×C4#23511yescharacteristic
211C22×C4D3×D411yescharacteristic
212C22×C4#23911yescharacteristic
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—22no—
216C2×C8—41no—
217C2×C8—62no—
218C2×C8—61no—
219C2×C8—61no—
220C2×C8—121no—
221C2×C8—121no—
222C4⋊C4—32no—
223C4⋊C4—32no—
224C4⋊C4—61no—
225C4⋊C4—62no—
226C4⋊C4—61no—
227C4⋊C4—62no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C22⋊C4—62no—
235C22⋊C4—121no—
236C42—62no—
237C42—121no—
238C42C2×C4×D311yescharacteristic
239C42C2×D1211yescharacteristic
240C42D3×D411yescharacteristic
241C42D3×D411yescharacteristic
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#44311yescharacteristic
252C3×C4#44411yescharacteristic
253Dic3—42no—
254Dic3—42no—
255Dic3—81no—
256Dic3—81no—
257C23#89411yescharacteristic
258Q8—122no—
259Q8—121no—
260Q8—121no—
261Q8—121no—
262Q8—121no—
263Q8—241no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—121no—
271C2×C4—121no—
272C2×C4C4×D1211yescharacteristic
273C2×C4D3×C22⋊C411yescharacteristic
274C2×C4#88211yescharacteristic
275C2×C4#88411yescharacteristic
276C2×C4#89311yescharacteristic
277C2×C4#89511yescharacteristic
278C8—42no—
279C8—41no—
280C8—41no—
281C8—81no—
282C8—122no—
283C8—121no—
284C8—121no—
285C2×C3—41no—
286C2×C3#187611yescharacteristic
287C2×C3#190211yescharacteristic
288C2×C3#191211yescharacteristic
289C22—21no—
290C22—21no—
291C22—21no—
292C22#466511yescharacteristic
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—122no—
299C4—122no—
300C4—241no—
301C4—241no—
302C4#489411yescharacteristic
303C4#489511yescharacteristic
304C3#928611yescharacteristic, complemented, Sylow
305C2—41no—
306C2#6870811yescharacteristic
307C2#6894611yescharacteristic
308C2#6903711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1287C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#1287D3
hypocenterC3#9286
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1287

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
#793162
#792732
#708132
#708032
C3×#12872
#677402
#92863