Finite Group Databasebeta

← #10697095

Subgroups — #10697095

Order 768.

556 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 103 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#83198C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic
3#79305C211yesmaximal, characteristic
4C3×#1546C211yesmaximal, characteristic, complemented
5#70521C211yesmaximal, characteristic
6#70501C211yesmaximal, characteristic, complemented
7#67739C211yesmaximal, characteristic, complemented
8#10169—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#874C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5225—22no—
15C2×#823—22no—
16#5088—22no—
17#5088—22no—
18#5082—22no—
19#5079—21no—
20#5079—21no—
21C2×#809—22no—
22C4×#208C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4848—22no—
25#4731—22no—
26#4694—21no—
27#4693—21no—
28C3×#332—22no—
29#4484—22no—
30#3208—31no—
31#3139—31no—
32C2×#486—31no—
33#1965—31no—
34#1546D311yescharacteristic, 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#835—24no—
70#835—24no—
71#823—44no—
72#821—24no—
73#809—28no—
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#492—61no—
82#491—62no—
83#487—61no—
84#487—61no—
85#486—61no—
86#486—62no—
87#486—62no—
88#460—62no—
89#450—62no—
90#431C2×D311yescharacteristic
91#423C2×D311yescharacteristic
92#419—62no—
93C2×#102—62no—
94C2×#100—62no—
95C4×M16—31no—
96C2×C4×C8—31no—
97#332—22no—
98Q8⋊C8—62no—
99C23×D3—22no—
100C2×C3×Q8—22no—
101C2×C3×Q8—41no—
102C2×C3×Q8—42no—
103C2×C3×Q8—42no—
104C22×C3×C4—41no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C2×D12—22no—
109C2×D12—22no—
110C2×D12—44no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—42no—
116C2×D12—42no—
117C2×D12—82no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C3×M16—42no—
121C2×C3×C8—22no—
122C2×C3×C8—41no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131#217—41no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138C4×Dic3—41no—
139#208—22no—
140#208—22no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—42no—
144#204—84no—
145C22×Q8—21no—
146C22×D4—32no—
147C22×D4—61no—
148C22×D4—61no—
149C2×SD16—64no—
150C2×M16—21no—
151C2×M16—32no—
152C22×C8—61no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—31no—
157C4⋊Q8—31no—
158C4⋊Q8D1212yes—
159#127—61no—
160#127—61no—
161#127—61no—
162#127—61no—
163#125—61no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166C4×Q8—61no—
167C4×Q8—61no—
168#117—61no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C42C22×D311yescharacteristic
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8#6712yes—
175#103—64no—
176#103—64no—
177#102—68no—
178#102—64no—
179#100—124no—
180C22⋊C8—21no—
181#97—61no—
182C4×C8—21no—
183C4×C8—32no—
184C4×C8—61no—
185C4×C8—61no—
186C23×C3#12111yescharacteristic
187C22×D3—42no—
188C22×D3—44no—
189C22×D3—82no—
190C22×D3—82no—
191C3×Q8—44no—
192C3×Q8—81no—
193C3×Q8—81no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—42no—
197C2×C3×C4—82no—
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12711yescharacteristic
204C2×C3×C4#13512yes—
205C2×C3×C4AGL(1,ℤ/8)12yes—
206C2×Dic3—41no—
207C2×Dic3—41no—
208D12—44no—
209D12—44no—
210D12—82no—
211D12—82no—
212D12—82no—
213D12—81no—
214D12—81no—
215Dic6—42no—
216Dic6—42no—
217C3×C8—42no—
218C3×C8—81no—
219C3⋊C8—42no—
220C3⋊C8—42no—
221C24—62no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—61no—
227C2×Q8—61no—
228C2×D4—62no—
229C2×D4—62no—
230C2×D4—124no—
231C2×D4—121no—
232C2×D4—121no—
233C2×D4—122no—
234C2×D4—122no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4—242no—
238C22×C4—41no—
239C22×C4#23511yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242SD16—244no—
243M16—42no—
244M16—62no—
245M16—62no—
246C2×C8—22no—
247C2×C8—41no—
248C2×C8—62no—
249C2×C8—62no—
250C2×C8—122no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C22⋊C4—121no—
262C42—121no—
263C42C2×D1211yescharacteristic
264C42D3×D411yescharacteristic
265C42D3×D411yescharacteristic
266C42C2×#6711yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3#39211yescharacteristic
270C22×C3#46312yes—
271C2×D3—82no—
272C2×D3—82no—
273C2×D3—82no—
274C2×D3—164no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—82no—
278C3×C4#44912yes—
279C3×C4#45512yes—
280C3×C4#47112yes—
281C3×C4#49511yescharacteristic
282C3×C4#49611yescharacteristic
283Dic3—81no—
284Dic3—81no—
285C23—122no—
286C23—124no—
287C23—242no—
288C23—242no—
289C23#88411yescharacteristic
290Q8—44no—
291Q8—81no—
292Q8—81no—
293Q8—122no—
294Q8—122no—
295D4—124no—
296D4—124no—
297D4—242no—
298D4—241no—
299D4—241no—
300D4—242no—
301D4—242no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—42no—
305C2×C4—82no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4#88211yescharacteristic
309C2×C4#88211yescharacteristic
310C2×C4#88411yescharacteristic
311C2×C4#91112yes—
312C2×C4#91612yes—
313C2×C4#93011yescharacteristic
314C2×C4#93711yescharacteristic
315C2×C4#94011yescharacteristic
316C8—42no—
317C8—81no—
318C8—122no—
319C8—122no—
320C2×C3—22no—
321C2×C3#200411yescharacteristic
322C2×C3#201211yescharacteristic
323C2×C3#201911yescharacteristic
324D3—162no—
325C22—21no—
326C22—21no—
327C22—242no—
328C22—242no—
329C22—242no—
330C22—484no—
331C22#478712yes—
332C22#496011yescharacteristic
333C4—21no—
334C4—21no—
335C4—82no—
336C4—241no—
337C4—241no—
338C4#475412yes—
339C4#478112yes—
340C4#515411yescharacteristic
341C4#516811yescharacteristic
342C4#520212yes—
343C3#1016911yescharacteristic, complemented, Sylow
344C2—22no—
345C2—482no—
346C2#7011711yescharacteristic
347C2#7021411yescharacteristic
348C2#7034411yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#392
prime radicalC2×#874C22
hypercenter#1546D3
hypocenterC3#10169
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
1. #83198C2semidirect
4. C3×#1546C2semidirect
6. #70501C2semidirect
7. #67739C2semidirect
34. #1546D3semidirect
343. C3#10169semidirect

Maximal subgroups

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

subgroupindex
#831982
C2×#50882
#793052
C3×#15462
#705212
#705012
#677392
#101693