Finite Group Databasebeta

← #10695163

Subgroups — #10695163

Order 768.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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#83281C211yesmaximal, characteristic
2C3×#1492C211yesmaximal, characteristic
3#71802C211yesmaximal, characteristic
4#70516C211yesmaximal, characteristic, complemented
5#70501C211yesmaximal, characteristic, complemented
6#69738C211yesmaximal, characteristic, complemented
7C2×#4454C211yesmaximal, characteristic, complemented
8#9863—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13#5233—22no—
14#5079C411yescharacteristic
15#5079C411yescharacteristic
16#5070C411yescharacteristic
17#5070C411yescharacteristic
18#5056—22no—
19#4924C2211yescharacteristic
20C2×C4×C3⋊C8C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22C3×D4⋊C8—22no—
23#4558—22no—
24D4⋊(C3⋊C8)—22no—
25#4484—22no—
26#4456—22no—
27#4454—22no—
28#3131—31no—
29#1837—31no—
30#1493—31no—
31#1493—31no—
32#1492—31no—
33#1492D311yescharacteristic
34C2×#322—31no—
35C2×C3×#41—21no—
36C2×C3×M16—21no—
37C22×C3×C8—21no—
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×#127C2×C411yescharacteristic
40C3×#124—21no—
41C3×#121—21no—
42C3×#121—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C2×C3×C42C2311yescharacteristic
46#926—21no—
47#924—21no—
48#924—21no—
49C2×#208—21no—
50C22×C3⋊C8—21no—
51C22×C3⋊C8D412yes—
52#870—21no—
53#869C2×C411yescharacteristic
54#869C2×C411yescharacteristic
55C4×Dic6—21no—
56C4×Dic6—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4×C8—21no—
60C3×C4×C8D412yes—
61#835—22no—
62#835—22no—
63#832—22no—
64#832—22no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)—21no—
67C4×C3⋊C8—21no—
68C4×C3⋊C8C2×C411yescharacteristic
69C4×C3⋊C8C2×C411yescharacteristic
70C4×C3⋊C8D412yes—
71C4×C3⋊C8D412yes—
72#533—31no—
73#532C2×D311yescharacteristic
74#488—62no—
75#487—31no—
76#487—31no—
77#486—31no—
78#486—31no—
79#423—31no—
80#423C2×D311yescharacteristic
81#416—62no—
82C2×C4×C8—31no—
83C2×C4×C8—31no—
84C2×C4×C8C2×D311yescharacteristic
85#341—62no—
86Q8⋊C8—62no—
87Q8⋊C8—62no—
88D4⋊C8—22no—
89D4⋊C8—62no—
90#322—62no—
91C3×#41—42no—
92C3×#41—42no—
93C2×C3×Q8—21no—
94C2×C3×D4—21no—
95C2×C3×D4—21no—
96C2×C3×D4—21no—
97C2×C3×D4—41no—
98C22×C3×C4—41no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C2×Dic6—21no—
103C2×Dic6—21no—
104C3×M16—42no—
105C2×C3×C8—22no—
106C2×C3×C8—42no—
107C2×C3×C8—41no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C22⋊C4—41no—
111C3×C22⋊C4—41no—
112C3×C42C22×C411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42C2×D411yescharacteristic
116#217—41no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121Dic3⋊C4—41no—
122Dic3⋊C4—41no—
123C4×Dic3—41no—
124#208—42no—
125C2×C3⋊C8—22no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—22no—
128C2×C3⋊C8—42no—
129C2×C3⋊C8—41no—
130C2×C3⋊C8C22⋊C412yes—
131C2×C3⋊C8C22⋊C412yes—
132C2×#41—21no—
133C2×M16—21no—
134C2×M16—61no—
135C22×C8—21no—
136C22×C8—32no—
137C22×C8—61no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140C4⋊Q8C4×D311yescharacteristic
141#127C4×D311yescharacteristic
142#125—61no—
143#124—21no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146#121—21no—
147#121—21no—
148C4×Q8—61no—
149C4×Q8—61no—
150C4×D4—21no—
151C4×D4—21no—
152#117—61no—
153C2×C42C22×D311yescharacteristic
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—61no—
157C4⋊C8—61no—
158#103—62no—
159#103—62no—
160#102—62no—
161#102—62no—
162C4×C8—21no—
163C4×C8—31no—
164C4×C8—31no—
165C4×C8—32no—
166C4×C8—32no—
167C4×C8—61no—
168C4×C8#6712yes—
169C23×C3—41no—
170C23×C3#12411yescharacteristic
171C3×Q8—42no—
172C3×D4—42no—
173C3×D4—42no—
174C3×D4—42no—
175C3×D4—82no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—41no—
179C2×C3×C4—81no—
180C2×C3×C4—81no—
181C2×C3×C4C2×C22⋊C411yescharacteristic
182C2×C3×C4C4×D411yescharacteristic
183C2×C3×C4C4×D411yescharacteristic
184C2×C3×C4#12111yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4#12711yescharacteristic
187C2×C3×C4#13512yes—
188C2×C3×C4#13512yes—
189C2×Dic3—41no—
190C2×Dic3—41no—
191Dic6—42no—
192Dic6—42no—
193C3×C8—42no—
194C3×C8—81no—
195C3⋊C8—22no—
196C3⋊C8—22no—
197C3⋊C8—42no—
198C3⋊C8—81no—
199#41—42no—
200#41—42no—
201C2×Q8—21no—
202C2×Q8—61no—
203C2×Q8—61no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—41no—
208C22×C4—41no—
209C22×C4#23511yescharacteristic
210C22×C4D3×D411yescharacteristic
211C22×C4#23711yescharacteristic
212M16—42no—
213M16—122no—
214C2×C8—22no—
215C2×C8—32no—
216C2×C8—32no—
217C2×C8—42no—
218C2×C8—41no—
219C2×C8—62no—
220C2×C8—62no—
221C2×C8—62no—
222C2×C8—122no—
223C2×C8—121no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—121no—
231C4⋊C4—121no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—121no—
235C42—121no—
236C42C2×C4×D311yescharacteristic
237C42D3×D411yescharacteristic
238C42D3×D411yescharacteristic
239C42C2×#6711yescharacteristic
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—81no—
243C22×C3—81no—
244C22×C3—81no—
245C22×C3#39011yescharacteristic
246C22×C3#48812yes—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—81no—
250C3×C4#41612yes—
251C3×C4#44412yes—
252C3×C4#44412yes—
253C3×C4#49511yescharacteristic
254C3×C4#49511yescharacteristic
255Dic3—81no—
256Dic3—81no—
257C23—41no—
258C23#88511yescharacteristic
259Q8—42no—
260Q8—122no—
261Q8—122no—
262D4—42no—
263D4—42no—
264D4—42no—
265D4—82no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—121no—
272C2×C4—121no—
273C2×C4D3×C22⋊C411yescharacteristic
274C2×C4#67⋊C411yescharacteristic
275C2×C4#88411yescharacteristic
276C2×C4#91212yes—
277C2×C4#91612yes—
278C2×C4C4×#6711yescharacteristic
279C2×C4#93911yescharacteristic
280C2×C4#94011yescharacteristic
281C8—42no—
282C8—62no—
283C8—62no—
284C8—81no—
285C8—122no—
286C8—241no—
287C2×C3—22no—
288C2×C3—161no—
289C2×C3#195811yescharacteristic
290C2×C3#197511yescharacteristic
291C2×C3#197511yescharacteristic
292C22—21no—
293C22—21no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22#476812yes—
298C22#495711yescharacteristic
299C4—21no—
300C4—21no—
301C4—81no—
302C4—241no—
303C4—241no—
304C4#477212yes—
305C4#482312yes—
306C4#514512yes—
307C4#515511yescharacteristic
308C4#516811yescharacteristic
309C3#986311yescharacteristic, complemented, Sylow
310C2—22no—
311C2—161no—
312C2#7000011yescharacteristic
313C2#7015811yescharacteristic
314C2#7037411yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1492C2
socleC22×C3#390
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1492D3
hypocenterC3#9863
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1492

Complements

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

NH ≅ G/Ntype
4. #70516C2semidirect
5. #70501C2semidirect
6. #69738C2semidirect
7. C2×#4454C2semidirect
309. C3#9863semidirect

Maximal subgroups

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

subgroupindex
#832812
C3×#14922
#718022
#705162
#705012
#697382
C2×#44542
#98633