Finite Group Databasebeta

← #10697039

Subgroups — #10697039

Order 768.

556 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 107 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#83207C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic
3#79305C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic, complemented
5#70521C211yesmaximal, characteristic
6#70501C211yesmaximal, characteristic, complemented
7#67739C211yesmaximal, characteristic, complemented
8#10161—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#874C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×#823—22no—
15#5088—22no—
16#5088—22no—
17#5081—22no—
18#5080—21no—
19#5078—21no—
20C2×#809—22no—
21#4964—22no—
22C4×#208C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4848—22no—
25#4731—22no—
26#4694—21no—
27#4693—21no—
28C3×#333—22no—
29#4484—22no—
30#3208—31no—
31#3165—31no—
32C2×#486—31no—
33#1965—31no—
34#1548D311yescharacteristic, complemented
35#1493—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×D12D412yes—
41C2×C3×M16—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49#926—21no—
50#924—21no—
51#924—21no—
52C2×#208D412yes—
53C22×C3⋊C8—21no—
54C2×#204—24no—
55#874—21no—
56#874—21no—
57#874—21no—
58#874—21no—
59#870—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D412yes—
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69#823—44no—
70#821—24no—
71#809—28no—
72#808—24no—
73#807—24no—
74#803—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8D412yes—
78#533—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×#127—31no—
81#501—62no—
82#500—61no—
83#498—61no—
84#492—61no—
85#486—61no—
86#486—62no—
87#486—62no—
88#460—62no—
89#450—62no—
90#431C2×D311yescharacteristic
91#428—62no—
92#423C2×D311yescharacteristic
93C2×#102—62no—
94C2×#100—62no—
95C4×M16—31no—
96C2×C4×C8—31no—
97#333—22no—
98Q8⋊C8—62no—
99C23×D3—22no—
100C2×C3×Q8—22no—
101C2×C3×Q8—41no—
102C2×C3×Q8—82no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×D12—22no—
108C2×D12—22no—
109C2×D12—44no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—42no—
115C2×D12—42no—
116C2×D12—82no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C3×M16—42no—
120C2×C3×C8—22no—
121C2×C3×C8—41no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130#217—41no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135Dic3⋊C4—41no—
136Dic3⋊C4—41no—
137C4×Dic3—41no—
138#208—22no—
139#208SD1614yescomplemented
140C2×C3⋊C8—22no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—42no—
143#204—84no—
144C22×Q8—21no—
145C22×D4—32no—
146C22×D4—61no—
147C22×D4—61no—
148C2×SD16—64no—
149C2×M16—21no—
150C2×M16—32no—
151C22×C8—61no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—31no—
156C4⋊Q8—31no—
157C4⋊Q8D1212yes—
158#127—61no—
159#127—61no—
160#127—61no—
161#127—61no—
162#125—61no—
163C22⋊Q8—61no—
164C22⋊Q8—61no—
165C4×Q8—61no—
166C4×Q8—61no—
167#117—61no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C42C22×D311yescharacteristic
171#107—64no—
172#106—64no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8#6712yes—
176#102—68no—
177#102—64no—
178#100—124no—
179C22⋊C8—21no—
180#97—61no—
181C4×C8—21no—
182C4×C8—32no—
183C4×C8—61no—
184C4×C8—61no—
185C23×C3#12111yescharacteristic
186C22×D3—42no—
187C22×D3—44no—
188C22×D3—82no—
189C22×D3—82no—
190C3×Q8—82no—
191C3×Q8—82no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—42no—
195C2×C3×C4—82no—
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12711yescharacteristic
202C2×C3×C4C2×SD1612yes—
203C2×C3×C4#13712yes—
204C2×Dic3—41no—
205C2×Dic3—41no—
206D12—44no—
207D12—44no—
208D12—82no—
209D12—82no—
210D12—82no—
211D12—81no—
212D12—81no—
213Dic6—42no—
214Dic6—42no—
215C3×C8—42no—
216C3×C8—81no—
217C3⋊C8—24no—
218C3⋊C8—42no—
219C24—62no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—82no—
225C2×D4—62no—
226C2×D4—62no—
227C2×D4—124no—
228C2×D4—121no—
229C2×D4—121no—
230C2×D4—122no—
231C2×D4—122no—
232C2×D4—121no—
233C2×D4—121no—
234C2×D4—242no—
235C22×C4—41no—
236C22×C4#23511yescharacteristic
237C22×C4D3×D411yescharacteristic
238C22×C4D3×D411yescharacteristic
239SD16—244no—
240M16—34no—
241M16—42no—
242M16—62no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—62no—
246C2×C8—62no—
247C2×C8—122no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C22⋊C4—121no—
259C42—121no—
260C42C2×D1211yescharacteristic
261C42D3×D411yescharacteristic
262C42D3×D411yescharacteristic
263C42C2×#6711yescharacteristic
264C22×C3—21no—
265C22×C3—21no—
266C22×C3#39211yescharacteristic
267C22×C3#46112yes—
268C2×D3—82no—
269C2×D3—82no—
270C2×D3—82no—
271C2×D3—164no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—82no—
275C3×C4#44812yes—
276C3×C4#45512yes—
277C3×C4#47212yes—
278C3×C4#49211yescharacteristic
279C3×C4#49711yescharacteristic
280Dic3—81no—
281Dic3—81no—
282C23—122no—
283C23—124no—
284C23—242no—
285C23—242no—
286C23#88411yescharacteristic
287Q8—82no—
288Q8—82no—
289Q8—122no—
290Q8—122no—
291D4—124no—
292D4—124no—
293D4—242no—
294D4—241no—
295D4—241no—
296D4—242no—
297D4—242no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—82no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4#88211yescharacteristic
305C2×C4#88211yescharacteristic
306C2×C4#88411yescharacteristic
307C2×C4D3×SD1612yes—
308C2×C4#91812yes—
309C2×C4#93011yescharacteristic
310C2×C4#93711yescharacteristic
311C2×C4#94011yescharacteristic
312C8—42no—
313C8—64no—
314C8—81no—
315C8—122no—
316C2×C3—22no—
317C2×C3#200411yescharacteristic
318C2×C3#201311yescharacteristic
319C2×C3#201811yescharacteristic
320D3—162no—
321C22—21no—
322C22—21no—
323C22—242no—
324C22—242no—
325C22—242no—
326C22—484no—
327C22#481812yes—
328C22#496011yescharacteristic
329C4—21no—
330C4—21no—
331C4—82no—
332C4—241no—
333C4—241no—
334C4#474912yes—
335C4#480612yes—
336C4#517511yescharacteristic
337C4#519011yescharacteristic
338C4#520212yes—
339C3#1016111yescharacteristic, complemented, Sylow
340C2—22no—
341C2—482no—
342C2#7009011yescharacteristic
343C2#7028211yescharacteristic
344C2#7034411yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#392
prime radicalC2×#874C22
hypercenter#1548D3
hypocenterC3#10161
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #83207C2semidirect
4. C3×#1548C2semidirect
6. #70501C2semidirect
7. #67739C2semidirect
34. #1548D3semidirect
139. #208SD16semidirect
339. C3#10161semidirect

Maximal subgroups

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

subgroupindex
#832072
C2×#50882
#793052
C3×#15482
#705212
#705012
#677392
#101613