Finite Group Databasebeta

← #10697104

Subgroups — #10697104

Order 768.

574 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 111 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#83213C211yesmaximal, characteristic
2C2×#5079C211yesmaximal, characteristic, complemented
3#78538C211yesmaximal, characteristic
4C3×#1997C211yesmaximal, characteristic, complemented
5#70506C211yesmaximal, characteristic, complemented
6#70498C211yesmaximal, characteristic
7#67667C211yesmaximal, characteristic, complemented
8#10170—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#909—22no—
11#5477C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5227—22no—
16C2×#835—22no—
17#5085—21no—
18#5083—22no—
19#5079—21no—
20#5079—22no—
21#5079—22no—
22#4925C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4715—22no—
25#4699—21no—
26#4693—21no—
27C3×#343—22no—
28#4483—22no—
29#4482—22no—
30#3206—31no—
31#3152—31no—
32C2×#487—31no—
33#1997D311yescharacteristic, complemented
34#1546—31no—
35#1492—31no—
36#1278—31no—
37C22×C3×Q8D412yes—
38C2×#235—21no—
39C22×Dic6—21no—
40C2×C3×M16—21no—
41C2×C3×M16D412yes—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#215—24no—
50#930—21no—
51#930—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C22×C3⋊C8—21no—
55#909—44no—
56#909—44no—
57C2×Dic12—24no—
58C2×#204—24no—
59#875—21no—
60#874D411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869D411yescharacteristic
67#869D412yes—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×#97—21no—
71C3×C4×C8—21no—
72#835—28no—
73#835—24no—
74#830—21no—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8D412yes—
79C2×#137—62no—
80#532—31no—
81C2×C4⋊Q8—31no—
82C2×C4⋊Q8C2×D311yescharacteristic
83#497—62no—
84#494—61no—
85#492—61no—
86#492—61no—
87#487—62no—
88#487—62no—
89#487—61no—
90#461—62no—
91#448—62no—
92#431—31no—
93#423C2×D311yescharacteristic
94C2×#103—62no—
95C4×M16C2×D311yescharacteristic
96C2×C4×C8—31no—
97#343—22no—
98#332—62no—
99D4⋊C8—62no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—44no—
103C22×C3×C4—22no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×#67—41no—
108C22×Dic3—41no—
109#235—42no—
110#235—42no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×C4×D3—41no—
115C2×Dic6—22no—
116C2×Dic6—21no—
117C2×Dic6—42no—
118C2×Dic6—42no—
119C2×Dic6—41no—
120C3×M16—22no—
121C3×M16—42no—
122C3×M16SD1614yescomplemented
123C2×C3×C8—22no—
124C2×C3×C8—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133#215—84no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—22no—
137C4⋊Dic3—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—42no—
145Dic12—48no—
146#204—48no—
147C2×#41—61no—
148C22×Q8—61no—
149C22×Q8D1212yes—
150#137—124no—
151#137—124no—
152C2×Q16—64no—
153C2×SD16—64no—
154C2×SD16—64no—
155C2×M16—21no—
156C2×M16#6712yes—
157C22×C8—61no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—32no—
163C4⋊Q8—31no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167#127—31no—
168#124—61no—
169#121—61no—
170#121—61no—
171C4×D4—61no—
172C4×D4—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—32no—
181#103—68no—
182#103—64no—
183C22⋊C8—61no—
184#97—21no—
185C4×C8—21no—
186C4×C8—32no—
187C4×C8—61no—
188C4×C8—61no—
189C23×C3#12111yescharacteristic
190C22×D3—41no—
191C3×Q8—44no—
192C3×Q8—44no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—42no—
196C2×C3×C4—82no—
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12711yescharacteristic
203C2×C3×C4C2×SD1612yes—
204C2×C3×C4AGL(1,ℤ/8)12yes—
205#67—82no—
206C2×Dic3—42no—
207C2×Dic3—41no—
208C2×Dic3—82no—
209D12—42no—
210D12—42no—
211D12—42no—
212C4×D3—81no—
213C4×D3—81no—
214Dic6—44no—
215Dic6—42no—
216Dic6—81no—
217Dic6—81no—
218C3×C8—24no—
219C3×C8—81no—
220C3⋊C8—42no—
221C3⋊C8—42no—
222#41—122no—
223#41—122no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—44no—
227C2×Q8—61no—
228C2×Q8—62no—
229C2×Q8—121no—
230C2×Q8—122no—
231C2×Q8—122no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—121no—
236C22×C4—22no—
237C22×C4—121no—
238C22×C4—121no—
239C22×C4C2×D1211yescharacteristic
240C22×C4#23511yescharacteristic
241C22×C4C2×#6711yescharacteristic
242Q16—128no—
243SD16—128no—
244SD16—244no—
245M16—22no—
246M16—42no—
247M16#21514yescomplemented
248C2×C8—22no—
249C2×C8—41no—
250C2×C8—62no—
251C2×C8—62no—
252C2×C8—122no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—62no—
258C4⋊C4—61no—
259C4⋊C4—62no—
260C4⋊C4—61no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C42D3×D411yescharacteristic
266C42D3×D411yescharacteristic
267C42D3×D411yescharacteristic
268C42D3×D411yescharacteristic
269C22×C3—21no—
270C22×C3—21no—
271C22×C3#39211yescharacteristic
272C22×C3#46012yes—
273C2×D3—81no—
274C2×D3—81no—
275C2×D3—81no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—82no—
279C3×C4#45012yes—
280C3×C4#45512yes—
281C3×C4#47312yes—
282C3×C4#49211yescharacteristic
283C3×C4#49611yescharacteristic
284Dic3—82no—
285Dic3—81no—
286C23—121no—
287C23#93011yescharacteristic
288Q8—44no—
289Q8—44no—
290Q8—124no—
291Q8—122no—
292Q8—241no—
293Q8—241no—
294D4—122no—
295D4—122no—
296D4—122no—
297D4—242no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—82no—
302C2×C4—121no—
303C2×C4—122no—
304C2×C4—242no—
305C2×C4—241no—
306C2×C4—241no—
307C2×C4#88211yescharacteristic
308C2×C4#88211yescharacteristic
309C2×C4#88411yescharacteristic
310C2×C4#88411yescharacteristic
311C2×C4#93711yescharacteristic
312C2×C4#94011yescharacteristic
313C2×C4C2×#21512yes—
314C2×C4#94912yes—
315C8—24no—
316C8—81no—
317C8—122no—
318C8—122no—
319C2×C3—22no—
320C2×C3#200411yescharacteristic
321C2×C3#201411yescharacteristic
322C2×C3#202011yescharacteristic
323D3—161no—
324C22—21no—
325C22—21no—
326C22—241no—
327C22—241no—
328C22—241no—
329C22#496011yescharacteristic
330C22#502712yes—
331C4—21no—
332C4—21no—
333C4—82no—
334C4—241no—
335C4—242no—
336C4#474612yes—
337C4#475512yes—
338C4#504612yes—
339C4#505811yescharacteristic
340C4#508511yescharacteristic
341C3#1017011yescharacteristic, complemented, Sylow
342C2—22no—
343C2—481no—
344C2#6968811yescharacteristic
345C2#7008411yescharacteristic
346C2#7012511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1997D3
hypocenterC3#10170
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
2. C2×#5079C2semidirect
4. C3×#1997C2semidirect
5. #70506C2semidirect
7. #67667C2semidirect
33. #1997D3semidirect
122. C3×M16SD16semidirect
247. M16#215semidirect
341. C3#10170semidirect

Maximal subgroups

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

subgroupindex
#832132
C2×#50792
#785382
C3×#19972
#705062
#704982
#676672
#101703