Finite Group Databasebeta

← #10697868

Subgroups — #10697868

Order 768.

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

normal subgroups only

This group has 554 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#83252C211yesmaximal, characteristic, complemented
2#83213C211yesmaximal, characteristic
3#79268C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic, complemented
5#70512C211yesmaximal, characteristic
6#70506C211yesmaximal, characteristic, complemented
7#67736C211yesmaximal, characteristic, complemented
8#10288—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5229—22no—
15#5227—22no—
16C2×#825—22no—
17#5085—21no—
18#5083—22no—
19#5079—21no—
20#4977—22no—
21#4973—22no—
22C42×Dic3C2211yescharacteristic
23#4925C2211yescharacteristic
24#4846—22no—
25#4736—22no—
26#4694—21no—
27#4682—21no—
28C3×#333—22no—
29#4482—22no—
30#3226—31no—
31#3206—31no—
32#2862—31no—
33#1962—31no—
34#1548D311yescharacteristic, complemented
35#1546—31no—
36#1276—31no—
37C22×C3×Q8D412yes—
38C2×#235—21no—
39C22×Dic6—21no—
40C2×C3×M16—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#215—24no—
49#930—21no—
50#930—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3D412yes—
53C2×C4×Dic3—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3D412yes—
56#875—21no—
57#874D411yescharacteristic
58C4×D12—21no—
59C4×D12—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8D412yes—
68C3×C22⋊C8—21no—
69C3×C4×C8—21no—
70#835—24no—
71C3⋊#95—28no—
72#830—21no—
73#825—44no—
74#821—24no—
75#818—24no—
76C4⋊(C3⋊C8)D412yes—
77C4×C3⋊C8—21no—
78#532—31no—
79C2×C4⋊Q8—31no—
80C2×C4⋊Q8C2×D311yescharacteristic
81#500—61no—
82#492—61no—
83#487—61no—
84#486—61no—
85#480—62no—
86#474—62no—
87#461—62no—
88#448—62no—
89#431—31no—
90#431C2×D311yescharacteristic
91#423C2×D311yescharacteristic
92C2×#104—62no—
93#391—62no—
94#391—62no—
95#382—62no—
96C43—31no—
97#333—22no—
98#332—62no—
99C2×C3×Q8—22no—
100C2×C3×Q8—22no—
101C2×C3×Q8—44no—
102C22×C3×C4—22no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C2×#67—41no—
107C22×Dic3—22no—
108C22×Dic3—22no—
109#235—42no—
110#235—42no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×C4×D3—41no—
115C2×Dic6—21no—
116C2×Dic6—41no—
117C2×Dic6—41no—
118C2×Dic6—42no—
119C2×Dic6—42no—
120C3×M16—42no—
121C2×C3×C8—22no—
122C2×C3×C8—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131#215—84no—
132(C2×D3)⋊C4—41no—
133(C2×D3)⋊C4—41no—
134C4⋊Dic3—21no—
135C4⋊Dic3—22no—
136C4⋊Dic3—21no—
137C4⋊Dic3—22no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140C4×Dic3—24no—
141C4×Dic3—22no—
142C4×Dic3—22no—
143C4×Dic3—42no—
144C2×C3⋊C8—22no—
145C2×#41—61no—
146C22×Q8—61no—
147C22×Q8D1212yes—
148C2×SD16—64no—
149C2×M16—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—31no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159#127—31no—
160#124—61no—
161#121—61no—
162#121—61no—
163C4×D4—61no—
164C4×D4—61no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—32no—
168C2×C4⋊C4—61no—
169C2×C42—32no—
170C2×C42—61no—
171C2×C42—61no—
172C2×C42C22×D311yescharacteristic
173#107—64no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—32no—
177C4⋊C8#6712yes—
178#104—124no—
179#103—64no—
180#102—64no—
181C22⋊C8—21no—
182C22⋊C8—61no—
183C4×C8—21no—
184C4×C8—61no—
185#95—68no—
186C23×C3#12011yescharacteristic
187C22×D3—41no—
188C3×Q8—44no—
189C3×Q8—82no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—44no—
193C2×C3×C4—42no—
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12711yescharacteristic
200C2×C3×C4#13512yes—
201C2×C3×C4AGL(1,ℤ/8)12yes—
202#67—82no—
203C2×Dic3—24no—
204C2×Dic3—42no—
205C2×Dic3—42no—
206C2×Dic3—42no—
207C2×Dic3—41no—
208C2×Dic3—44no—
209D12—42no—
210D12—42no—
211D12—42no—
212C4×D3—81no—
213C4×D3—81no—
214Dic6—42no—
215Dic6—82no—
216Dic6—81no—
217Dic6—81no—
218C3×C8—42no—
219C3×C8—81no—
220C3⋊C8—42no—
221#41—122no—
222#41—122no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—44no—
226C2×Q8—61no—
227C2×Q8—121no—
228C2×Q8—122no—
229C2×Q8—122no—
230C2×Q8—121no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—121no—
235C22×C4—22no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4—121no—
239C22×C4C2×D1211yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242SD16—244no—
243M16—42no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—62no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—61no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—61no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C22⋊C4—121no—
258C22⋊C4—121no—
259C42—64no—
260C42—62no—
261C42—62no—
262C42—122no—
263C42#23511yescharacteristic
264C42D3×D411yescharacteristic
265C42D3×D411yescharacteristic
266C42C2×#6711yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3#39211yescharacteristic
270C22×C3#44912yes—
271C2×D3—81no—
272C2×D3—81no—
273C2×D3—81no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—82no—
277C3×C4#45512yes—
278C3×C4#45612yes—
279C3×C4#46312yes—
280C3×C4#49511yescharacteristic
281C3×C4#49611yescharacteristic
282Dic3—44no—
283Dic3—81no—
284Dic3—82no—
285C23—121no—
286C23#88211yescharacteristic
287Q8—44no—
288Q8—82no—
289Q8—122no—
290Q8—241no—
291Q8—241no—
292Q8—242no—
293D4—122no—
294D4—122no—
295D4—122no—
296D4—242no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—64no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4—121no—
305C2×C4—122no—
306C2×C4—124no—
307C2×C4—241no—
308C2×C4—241no—
309C2×C4#88211yescharacteristic
310C2×C4#88411yescharacteristic
311C2×C4#88411yescharacteristic
312C2×C4#91412yes—
313C2×C4#91912yes—
314C2×C4#93011yescharacteristic
315C2×C4#93711yescharacteristic
316C2×C4#94011yescharacteristic
317C8—42no—
318C8—81no—
319C8—122no—
320C2×C3—22no—
321C2×C3#200411yescharacteristic
322C2×C3#200511yescharacteristic
323C2×C3#201211yescharacteristic
324D3—161no—
325C22—21no—
326C22—21no—
327C22—241no—
328C22—241no—
329C22—241no—
330C22#481012yes—
331C22#496011yescharacteristic
332C4—21no—
333C4—21no—
334C4—82no—
335C4—124no—
336C4—242no—
337C4—241no—
338C4#474612yes—
339C4#481912yes—
340C4#517611yescharacteristic
341C4#519111yescharacteristic
342C4#520512yes—
343C3#1028811yescharacteristic, complemented, Sylow
344C2—22no—
345C2—481no—
346C2#7008411yescharacteristic
347C2#7027711yescharacteristic
348C2#7034611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1548D3
hypocenterC3#10288
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. #83252C2semidirect
4. C3×#1548C2semidirect
6. #70506C2semidirect
7. #67736C2semidirect
34. #1548D3semidirect
343. C3#10288semidirect

Maximal subgroups

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

subgroupindex
#832522
#832132
#792682
C3×#15482
#705122
#705062
#677362
#102883