Finite Group Databasebeta

← #10692164

Subgroups — #10692164

Order 768.

400 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 91 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#83279C211yesmaximal, characteristic
2#83273C211yesmaximal, characteristic
3C3×#1553C211yesmaximal, characteristic
4#70814C211yesmaximal, characteristic, complemented
5#70787C211yesmaximal, characteristic, complemented
6C3⋊#1308C211yesmaximal, characteristic, complemented
7#67747C211yesmaximal, characteristic, complemented
8#9335—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#431C2211yescharacteristic
13#5234—22no—
14#5081—21no—
15#5071—21no—
16#5034C412yes—
17#5033C412yes—
18#5011—22no—
19#5011—22no—
20#5009—22no—
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24C3×#332—21no—
25C3×#331—21no—
26#4469—22no—
27#4465—22no—
28#3234—31no—
29#2980—31no—
30#1553D311yescharacteristic
31#1534—31no—
32#1470—31no—
33#1308—31no—
34#1287—31no—
35C2×C3×#41—21no—
36C3×C4⋊Q8D411yescharacteristic
37C3×#127D411yescharacteristic
38C3×#124—21no—
39C3×#121—21no—
40C3×#121C2×C412yes—
41C3×C4×D4—21no—
42C3×C4×D4D412yes—
43C2×C3×C42C2311yescharacteristic
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×#208—22no—
49C22×C3⋊C8—22no—
50#870—21no—
51#870—21no—
52#870—21no—
53#870—21no—
54C3×C4⋊C8—21no—
55C3×C4⋊C8—21no—
56C3×C22⋊C8—21no—
57C3×C22⋊C8—21no—
58C3×#97—21no—
59C3×C4×C8—21no—
60#832—22no—
61#832—22no—
62#830—24no—
63#830—21no—
64#830C2×C412yes—
65#830D412yes—
66#808—22no—
67#807—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71#803—21no—
72#803—21no—
73#803D412yes—
74#532C2×D311yescharacteristic
75C2×#125—31no—
76#501—61no—
77#490—61no—
78#481—32no—
79#480—32no—
80#478—62no—
81#435—62no—
82#435—62no—
83#432—31no—
84#432—31no—
85#432—31no—
86#432C2×D311yescharacteristic
87#431C2×D311yescharacteristic
88#408—62no—
89#332—21no—
90#331—21no—
91#330—62no—
92#324—62no—
93C3×#41—81no—
94C3×#41—81no—
95C2×C3×Q8—21no—
96C2×C3×D4—22no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4C22⋊C412yes—
100C22×C3×C4—22no—
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×Dic3—21no—
105C22×Dic3—21no—
106C2×C3×C8—41no—
107C2×C3×C8—41no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4C22⋊C412yes—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C42C2×D411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42#4111yescharacteristic
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120Dic3⋊C4—41no—
121Dic3⋊C4—41no—
122Dic3⋊C4—41no—
123Dic3⋊C4—41no—
124#208—82no—
125C2×C3⋊C8—22no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—22no—
128C2×C3⋊C8—82no—
129C2×#41—21no—
130C2×M16—62no—
131C22×C8—62no—
132C4⋊Q8#6711yescharacteristic
133#127#6711yescharacteristic
134#125—61no—
135#125—61no—
136#125—61no—
137#125—61no—
138#124—21no—
139#121—21no—
140#121C4×D312yes—
141C4×D4—21no—
142C4×D4D1212yes—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C42C22×D311yescharacteristic
148#107—62no—
149#106—62no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8—61no—
153C4⋊C8—61no—
154C4⋊C8—61no—
155#102—62no—
156#102—62no—
157C22⋊C8—21no—
158C22⋊C8—21no—
159C22⋊C8—32no—
160C22⋊C8—32no—
161C22⋊C8—64no—
162C22⋊C8—61no—
163#97—21no—
164#97—32no—
165#97—61no—
166#97—61no—
167C4×C8—21no—
168C23×C3—22no—
169C23×C3C4×D411yescharacteristic
170C3×Q8—81no—
171C3×D4—42no—
172C3×D4—81no—
173C3×D4—81no—
174C3×D4—82no—
175C2×C3×C4—22no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4C2×C22⋊C411yescharacteristic
182C2×C3×C4C4×D411yescharacteristic
183C2×C3×C4#12011yescharacteristic
184C2×C3×C4#12111yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4#12311yescharacteristic
187C2×Dic3—42no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—81no—
191C3×C8—81no—
192C3×C8—81no—
193C3⋊C8—42no—
194C3⋊C8—42no—
195C3⋊C8—42no—
196#41—81no—
197#41—81no—
198C2×Q8—21no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4(C2×D3)⋊C412yes—
203C22×C4—22no—
204C22×C4—61no—
205C22×C4—61no—
206C22×C4C2×C4×D311yescharacteristic
207C22×C4D3×D411yescharacteristic
208C22×C4#23711yescharacteristic
209M16—242no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—62no—
213C2×C8—62no—
214C2×C8—62no—
215C2×C8—242no—
216C4⋊C4—21no—
217C4⋊C4—61no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—121no—
222C4⋊C4—121no—
223C4⋊C4—121no—
224C4⋊C4—121no—
225C4⋊C4(C2×D3)⋊C412yes—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C42C2×D1211yescharacteristic
229C42D3×D411yescharacteristic
230C42#23711yescharacteristic
231C42C2×#6711yescharacteristic
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—42no—
236C22×C3—42no—
237C22×C3—42no—
238C22×C3#38611yescharacteristic
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—42no—
246Dic3—81no—
247Dic3—81no—
248C23—22no—
249C23#67⋊C411yescharacteristic
250Q8—81no—
251D4—42no—
252D4—81no—
253D4—81no—
254D4—82no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—122no—
262C2×C4—121no—
263C2×C4—121no—
264C2×C4—241no—
265C2×C4#67⋊C411yescharacteristic
266C2×C4#88211yescharacteristic
267C2×C4#88611yescharacteristic
268C2×C4C2×(C2×D3)⋊C411yescharacteristic
269C2×C4#93911yescharacteristic
270C2×C4#93911yescharacteristic
271C8—81no—
272C8—81no—
273C8—122no—
274C8—122no—
275C8—122no—
276C2×C3—41no—
277C2×C3—82no—
278C2×C3#188111yescharacteristic
279C2×C3#190411yescharacteristic
280C2×C3#191511yescharacteristic
281C22—21no—
282C22—21no—
283C22—21no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22#495811yescharacteristic
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—42no—
295C4—241no—
296C4—241no—
297C3#933511yescharacteristic, complemented, Sylow
298C2—41no—
299C2—82no—
300C2#7013811yescharacteristic
301C2#7029311yescharacteristic
302C2#7031411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1553C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1553D3
hypocenterC3#9335
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1553

Complements

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

NH ≅ G/Ntype
4. #70814C2semidirect
5. #70787C2semidirect
6. C3⋊#1308C2semidirect
7. #67747C2semidirect
297. C3#9335semidirect

Maximal subgroups

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

subgroupindex
#832792
#832732
C3×#15532
#708142
#707872
C3⋊#13082
#677472
#93353