Finite Group Databasebeta

← #10697354

Subgroups — #10697354

Order 768.

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

normal subgroups only

This group has 560 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#83223C211yesmaximal, characteristic, complemented
2#83203C211yesmaximal, characteristic
3#79027C211yesmaximal, characteristic
4C3×#1988C211yesmaximal, characteristic, complemented
5#69493C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic
7#67706C211yesmaximal, characteristic, complemented
8#10206—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#942—22no—
11#5477C2211yescharacteristic
12#5470C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5114—22no—
16#5094—22no—
17#5093—21no—
18#5092—21no—
19#5086—22no—
20#5085—21no—
21#5080—21no—
22#4971—22no—
23#4925C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4851—21no—
26#4848—21no—
27#4842—21no—
28#4841—21no—
29C3×#340—22no—
30#4523—22no—
31#4451—22no—
32#3246—31no—
33#3229—31no—
34#3152—31no—
35#1988D311yescharacteristic, complemented
36#1986—31no—
37#1502—31no—
38#1283—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C22×C3×C8D412yes—
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×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#216—24no—
52#942—44no—
53#942—44no—
54C2×#215—24no—
55#930—21no—
56#930—21no—
57#926—21no—
58#924—21no—
59#924—21no—
60C2×#208D412yes—
61C2×#204—22no—
62#875—21no—
63#874D411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69#869D411yescharacteristic
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×#95—21no—
75#830D412yes—
76#821—22no—
77#817—22no—
78#816—22no—
79#810—24no—
80#808—24no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85C2×#137—62no—
86#533—31no—
87#532—31no—
88C2×C4⋊Q8C2×D311yescharacteristic
89#498—61no—
90#497—62no—
91#494—61no—
92#492—61no—
93#487—61no—
94#483—62no—
95#476—61no—
96#475—61no—
97#475—62no—
98#465—62no—
99#461—61no—
100#460—61no—
101#431—31no—
102C2×C4⋊C8C2×D311yescharacteristic
103C4×M16—31no—
104C4×C4⋊C4C2×D311yescharacteristic
105#340—22no—
106#339—62no—
107#327—62no—
108C2×C3×Q8—22no—
109C2×C3×Q8—41no—
110C2×C3×Q8—42no—
111C2×C3×Q8—42no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C2×#67—41no—
118#235—42no—
119#235—42no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×C4×D3—41no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×C3×C8—22no—
128C2×C3×C8—42no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C42—42no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#217—41no—
141#216—48no—
142#215—48no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153C4×Dic3—41no—
154#208—22no—
155#208SD1614yescomplemented
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158#204—82no—
159C2×#41—61no—
160C22×Q8—21no—
161#137—124no—
162#137—124no—
163C2×Q16—64no—
164C2×SD16—62no—
165C2×SD16—64no—
166C2×M16—32no—
167C22×C8#6712yes—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—31no—
172C4⋊Q8—31no—
173C4⋊Q8—31no—
174C4⋊Q8D1212yes—
175#127—31no—
176#125—61no—
177#124—61no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180#121—61no—
181#121—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×D4—61no—
185C4×D4—61no—
186#117—61no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C42—21no—
191C2×C42C22×D311yescharacteristic
192#106—64no—
193#106—62no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—61no—
197#103—64no—
198#103—62no—
199#102—62no—
200C22⋊C8—32no—
201#97—61no—
202C4×C8—61no—
203C4×C8—61no—
204#95—21no—
205C23×C3#12711yescharacteristic
206C22×D3—41no—
207C3×Q8—44no—
208C3×Q8—82no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—81no—
215C2×C3×C4—81no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4C2×SD1612yes—
225C2×C3×C4AGL(1,ℤ/8)12yes—
226#67—81no—
227#67—81no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231D12—42no—
232D12—41no—
233D12—41no—
234D12—81no—
235C4×D3—82no—
236Dic6—42no—
237Dic6—42no—
238Dic6—41no—
239Dic6—41no—
240C3×C8—42no—
241C3⋊C8—24no—
242C3⋊C8—42no—
243#41—122no—
244#41—122no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—121no—
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4C2×#6711yescharacteristic
262Q16—128no—
263SD16—128no—
264SD16—242no—
265M16—34no—
266M16—62no—
267C2×C8—22no—
268C2×C8—42no—
269C2×C8—62no—
270C2×C8—62no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C42—42no—
289C42—121no—
290C42C2×D1211yescharacteristic
291C42#23511yescharacteristic
292C42D3×D411yescharacteristic
293C42D3×D411yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3#39211yescharacteristic
297C22×C3#49211yescharacteristic
298C22×C3#49611yescharacteristic
299C2×D3—81no—
300C2×D3—81no—
301C2×D3—81no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—82no—
306C3×C4—82no—
307C3×C4#45012yes—
308C3×C4#45511yescharacteristic
309C3×C4#45611yescharacteristic
310C3×C4#46012yes—
311Dic3—81no—
312Dic3—81no—
313Dic3—81no—
314C23—121no—
315C23#94011yescharacteristic
316Q8—44no—
317Q8—82no—
318Q8—122no—
319Q8—122no—
320Q8—121no—
321Q8—121no—
322D4—122no—
323D4—121no—
324D4—121no—
325D4—241no—
326D4—241no—
327D4—241no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—242no—
341C2×C4#88211yescharacteristic
342C2×C4#88211yescharacteristic
343C2×C4#88411yescharacteristic
344C2×C4#88411yescharacteristic
345C2×C4#91112yes—
346C2×C4D3×SD1612yes—
347C2×C4#93011yescharacteristic
348C2×C4#93711yescharacteristic
349C8—42no—
350C8—64no—
351C8—122no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3#200711yescharacteristic
355C2×C3#200911yescharacteristic
356C2×C3#201411yescharacteristic
357D3—161no—
358C22—21no—
359C22—21no—
360C22—241no—
361C22—241no—
362C22—241no—
363C22#496011yescharacteristic
364C22#515411yescharacteristic
365C22#517511yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—82no—
370C4—82no—
371C4—241no—
372C4—241no—
373C4—241no—
374C4#476212yes—
375C4#477612yes—
376C4#506111yescharacteristic
377C4#509511yescharacteristic
378C3#1020611yescharacteristic, complemented, Sylow
379C2—21no—
380C2—21no—
381C2—481no—
382C2#6969411yescharacteristic
383C2#7021011yescharacteristic
384C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1988C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1988D3
hypocenterC3#10206
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
1. #83223C2semidirect
4. C3×#1988C2semidirect
5. #69493C2semidirect
7. #67706C2semidirect
35. #1988D3semidirect
155. #208SD16semidirect
378. C3#10206semidirect

Maximal subgroups

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

subgroupindex
#832232
#832032
#790272
C3×#19882
#694932
#694692
#677062
#102063