Finite Group Databasebeta

← #10696853

Subgroups — #10696853

Order 768.

456 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 83 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#83273C211yesmaximal, characteristic
2#83271C211yesmaximal, characteristic
3C2×#4703C211yesmaximal, characteristic, complemented
4C3×#1530C211yesmaximal, characteristic
5#70831C211yesmaximal, characteristic, complemented
6#70822C211yesmaximal, characteristic, complemented
7#67662C211yesmaximal, characteristic, complemented
8#10134—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14C2×#816—21no—
15C2×#816—21no—
16#5081—21no—
17#5080—21no—
18#5071—21no—
19#5070—21no—
20#5034—21no—
21#5034—22no—
22#5033—21no—
23#5009—22no—
24#5008—22no—
25#4926C2211yescharacteristic
26#4925C2211yescharacteristic
27#4703—22no—
28#4703—22no—
29C3×#329—22no—
30#4486—22no—
31#4469—22no—
32#3234—31no—
33#3227—31no—
34C2×#491—31no—
35#1556—31no—
36#1530D311yescharacteristic
37#1470—31no—
38#1302—31no—
39C2×C3×#41—21no—
40C22×Dic6—21no—
41C22×C3×C8—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×#127D411yescharacteristic
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3D411yescharacteristic
53C2×C4⋊Dic3D411yescharacteristic
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56#870—21no—
57#870—21no—
58#870—21no—
59#870—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D412yes—
64C3×C4⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×#97—21no—
67C3×#97—21no—
68C3×#97D412yes—
69#832—24no—
70#832—22no—
71#832—22no—
72#830—21no—
73#830D412yes—
74#816—24no—
75#816—24no—
76#808—24no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82C4×C3⋊C8—21no—
83#532C2×D311yescharacteristic
84C2×C4⋊Q8—31no—
85C2×#125—31no—
86#501—61no—
87#498—61no—
88#491—62no—
89#491—62no—
90#490—61no—
91#486—61no—
92#481—62no—
93#481—61no—
94#480—61no—
95#478—62no—
96#476—62no—
97#432—31no—
98#432C2×D311yescharacteristic
99#431—31no—
100C2×#103—61no—
101C2×#103—61no—
102C2×#97C2×D311yescharacteristic
103#332—62no—
104#329—22no—
105#324—62no—
106C3×#41—82no—
107C2×C3×Q8—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—41no—
119C2×Dic6—22no—
120C2×Dic6—44no—
121C2×Dic6—41no—
122C2×C3×C8—22no—
123C2×C3×C8—42no—
124C2×C3×C8—41no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C4⋊Dic3—21no—
134C4⋊Dic3—22no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—22no—
147C2×#41—21no—
148C22×Q8—61no—
149C22×C8—21no—
150C4⋊Q8—32no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8D1211yescharacteristic
155#127D1211yescharacteristic
156#125—61no—
157#125—61no—
158#125—61no—
159#125—61no—
160#124—21no—
161#121—21no—
162#121—21no—
163C4×D4—21no—
164C4×D4—21no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C42C22×D311yescharacteristic
172#107—62no—
173#106—64no—
174#106—62no—
175C4⋊C8—21no—
176C4⋊C8—32no—
177C4⋊C8—61no—
178#103—64no—
179#103—64no—
180#102—64no—
181#102—62no—
182#102—62no—
183C22⋊C8—21no—
184C22⋊C8—32no—
185C22⋊C8—61no—
186#97—21no—
187#97—21no—
188#97—61no—
189#97#6712yes—
190C4×C8—61no—
191C23×C3—41no—
192C23×C3#12111yescharacteristic
193C3×Q8—81no—
194C3×D4—42no—
195C3×D4—42no—
196C3×D4—81no—
197C3×D4—82no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—82no—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12711yescharacteristic
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—82no—
214C2×Dic3—81no—
215Dic6—44no—
216Dic6—82no—
217C3×C8—42no—
218C3×C8—81no—
219C3⋊C8—42no—
220C3⋊C8—42no—
221#41—82no—
222C2×Q8—21no—
223C2×Q8—62no—
224C2×Q8—124no—
225C2×Q8—121no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—41no—
230C22×C4—41no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—121no—
234C22×C4#23511yescharacteristic
235C22×C4D3×D411yescharacteristic
236C22×C4D3×D411yescharacteristic
237C2×C8—22no—
238C2×C8—42no—
239C2×C8—41no—
240C2×C8—62no—
241C2×C8—62no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—61no—
245C4⋊C4—62no—
246C4⋊C4—61no—
247C4⋊C4—62no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42C2×D1211yescharacteristic
259C42D3×D411yescharacteristic
260C42D3×D411yescharacteristic
261C42C2×#6711yescharacteristic
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—81no—
265C22×C3—81no—
266C22×C3—81no—
267C22×C3#39211yescharacteristic
268C22×C3#47112yes—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—81no—
275C3×C4#45412yes—
276Dic3—82no—
277Dic3—81no—
278Dic3—81no—
279C23—41no—
280C23#88411yescharacteristic
281Q8—81no—
282Q8—124no—
283Q8—242no—
284D4—42no—
285D4—42no—
286D4—81no—
287D4—82no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—82no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—242no—
298C2×C4—241no—
299C2×C4#88211yescharacteristic
300C2×C4#88211yescharacteristic
301C2×C4#88411yescharacteristic
302C2×C4#93011yescharacteristic
303C2×C4#93711yescharacteristic
304C2×C4#94011yescharacteristic
305C8—42no—
306C8—81no—
307C8—122no—
308C8—122no—
309C2×C3—22no—
310C2×C3—161no—
311C2×C3#200311yescharacteristic
312C2×C3#200911yescharacteristic
313C2×C3#201911yescharacteristic
314C22—21no—
315C22—21no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22#474812yes—
320C22#496011yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—81no—
327C4—241no—
328C4—241no—
329C4—242no—
330C4#482712yes—
331C3#1013411yescharacteristic, complemented, Sylow
332C2—22no—
333C2—161no—
334C2#7009111yescharacteristic
335C2#7038811yescharacteristic
336C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1530C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1530D3
hypocenterC3#10134
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1530

Complements

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

NH ≅ G/Ntype
3. C2×#4703C2semidirect
5. #70831C2semidirect
6. #70822C2semidirect
7. #67662C2semidirect
331. C3#10134semidirect

Maximal subgroups

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

subgroupindex
#832732
#832712
C2×#47032
C3×#15302
#708312
#708222
#676622
#101343