Finite Group Databasebeta

← #10848788

Subgroups — #10848788

Order 768.

1152 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 191 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
1C2×#5731C211yesmaximal, characteristic, complemented
2C2×#5085C211yesmaximal, characteristic, complemented
3C2×#4713C211yesmaximal, characteristic, complemented
4#74244C211yesmaximal, characteristic
5C3×#1960C211yesmaximal, characteristic, complemented
6#70272C214yesmaximal, complemented
7#70240C214yesmaximal, complemented
8#69830C211yesmaximal, characteristic, complemented
9#69554C211yesmaximal, characteristic, complemented
10#25607—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×#946C2212yes—
13C2×Q8×Dic3C2212yes—
14C22×#215C2214yes—
15#5731—24no—
16#5731—24no—
17C2×#898—22no—
18C2×#895C2212yes—
19C2×#889—22no—
20C2×#874C2211yescharacteristic
21C2×#872C2211yescharacteristic
22C2×#870C2211yescharacteristic
23C2×C3×#103C2214yes—
24C2×C3×#97C2211yescharacteristic
25C3×#382C2211yescharacteristic
26C2×#821C2214yes—
27C2×(C2×D3)⋊C8C2212yes—
28#5085—24no—
29#4988C2212yes—
30C2×C4×C3⋊C8C2211yescharacteristic
31#4713—24no—
32#4672—22no—
33#4671—22no—
34#4550C2212yes—
35#4529C2214yes—
36C2×#541—31no—
37C2×#492—31no—
38C2×#490—31no—
39#2619—31no—
40#2000—34no—
41#1991—31no—
42#1960D311yescharacteristic, complemented
43#1880—34no—
44#1750—31no—
45C22×C3×Q8C2312yes—
46C22×D12—22no—
47C22×D12—22no—
48C22×D12C2312yes—
49C22×C4×D3C2311yescharacteristic
50C22×C3×C8C2312yes—
51C3×C4⋊Q8—22no—
52C3×C4⋊Q8D414yescomplemented
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4C2312yes—
55C2×C3×C42C2311yescharacteristic
56#946—24no—
57#946—24no—
58Q8×Dic3—24no—
59Q8×Dic3—24no—
60C2×#215—28no—
61C2×#215—28no—
62C2×#215—44no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—24no—
66C2×(C2×D3)⋊C4D412yes—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3C2312yes—
69C2×Dic3⋊C4—22no—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4D412yes—
72C2×C4×Dic3—22no—
73C2×C4×Dic3C2311yescharacteristic
74C22×C3⋊C8C2312yes—
75#898—216no—
76#895—44no—
77#889—216no—
78#874—22no—
79#874—22no—
80#872—22no—
81#872—22no—
82#870—22no—
83#870D414yescomplemented
84C3×#103—44no—
85C3×#97—22no—
86C3×#95—22no—
87#821—28no—
88(C2×D3)⋊C8—24no—
89C4×C3⋊C8—22no—
90C22×SD16—34no—
91#541—64no—
92#541—64no—
93C2×C4⋊Q8C2×D311yescharacteristic
94C2×#127—31no—
95C2×#126—62no—
96C2×#125—31no—
97C2×#125—62no—
98C2×C22⋊Q8—32no—
99C2×#121—32no—
100C2×C4×Q8—32no—
101C2×#117—31no—
102#492—64no—
103#490—64no—
104C2×#103C2×D314yes—
105C2×#102—34no—
106C2×C22⋊C8—32no—
107C2×#97C2×D311yescharacteristic
108C2×C4×C8—31no—
109#394—62no—
110#392—62no—
111#390—32no—
112#382C2×D311yescharacteristic
113#340—34no—
114#336—32no—
115C23×D3—22no—
116C23×D3C2×D411yescharacteristic
117C2×C3×Q8—22no—
118C2×C3×Q8—24no—
119C2×C3×Q8—24no—
120C2×C3×Q8C2×D414yes—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4C2×D412yes—
124C22×C3×C4C2411yescharacteristic
125C22×Dic3—22no—
126C22×Dic3C2×D412yes—
127C22×Dic3C2×D411yescharacteristic
128C2×D12—22no—
129C2×D12—22no—
130C2×D12—24no—
131C2×D12—24no—
132C2×D12—44no—
133C2×D12—44no—
134C2×D12—84no—
135C2×D12—84no—
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×C4×D3—22no—
139C2×C4×D3—22no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C3×C4⋊C4—28no—
144C3×C4⋊C4—24no—
145C3×C42C2×D412yes—
146C3×C42C2×D412yes—
147#215—416no—
148(C2×D3)⋊C4—28no—
149(C2×D3)⋊C4—24no—
150(C2×D3)⋊C4—44no—
151(C2×D3)⋊C4—84no—
152C4⋊Dic3—28no—
153C4⋊Dic3—22no—
154C4⋊Dic3C2×D414yes—
155Dic3⋊C4—28no—
156Dic3⋊C4—28no—
157Dic3⋊C4—24no—
158C4×Dic3—28no—
159C4×Dic3—21no—
160C4×Dic3—21no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8#4114yes—
164C22×Q8C22×D312yes—
165C22×D4—32no—
166C22×D4—62no—
167C22×D4—62no—
168C23×C4—31no—
169C2×SD16—68no—
170C2×SD16—68no—
171C2×SD16—124no—
172C22×C8—32no—
173C22×C8C22×D312yes—
174C4⋊Q8—22no—
175C4⋊Q8D1214yescomplemented
176#127—62no—
177#127—62no—
178#126—616no—
179#125—34no—
180#125—616no—
181#125—62no—
182C22⋊Q8—64no—
183C22⋊Q8—64no—
184#121—124no—
185C4×Q8—64no—
186C4×Q8—64no—
187#117—62no—
188#117—62no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4C22×D312yes—
196C2×C22⋊C4—32no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—64no—
200C2×C42—31no—
201C2×C42—62no—
202C2×C42C22×D311yescharacteristic
203#103—44no—
204#102—68no—
205C22⋊C8—64no—
206#97—22no—
207C4×C8—62no—
208#95—22no—
209C23×C3C22×D411yescharacteristic
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—21no—
213C22×D3—24no—
214C22×D3—42no—
215C22×D3—42no—
216C22×D3—42no—
217C22×D3—44no—
218C22×D3—44no—
219C3×Q8—28no—
220C3×Q8—44no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—44no—
227C2×C3×C4#12014yes—
228C2×C3×C4C22×D411yescharacteristic
229C2×C3×C4C22×D411yescharacteristic
230C2×C3×C4C2×#4112yes—
231C2×C3×C4H(5,𝔽2)12yes—
232C2×Dic3—22no—
233C2×Dic3—24no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—24no—
238C2×Dic3—44no—
239D12—44no—
240D12—44no—
241D12—88no—
242D12—88no—
243C4×D3—42no—
244C4×D3—42no—
245C3×C8—42no—
246C3⋊C8—24no—
247C24—31no—
248C24—62no—
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8C2×D1214yes—
253C2×D4—62no—
254C2×D4—62no—
255C2×D4—64no—
256C2×D4—64no—
257C2×D4—124no—
258C2×D4—124no—
259C2×D4—244no—
260C2×D4—244no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—32no—
264C22×C4—31no—
265C22×C4—62no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4D3×D412yes—
271C22×C4C23×D311yescharacteristic
272SD16—1216no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—34no—
277C2×C8—62no—
278C2×C8—62no—
279C4⋊C4—28no—
280C4⋊C4—24no—
281C4⋊C4—34no—
282C4⋊C4—68no—
283C4⋊C4—68no—
284C4⋊C4—64no—
285C4⋊C4—68no—
286C4⋊C4—62no—
287C22⋊C4—68no—
288C22⋊C4—64no—
289C22⋊C4—124no—
290C22⋊C4—244no—
291C42—68no—
292C42—61no—
293C42—61no—
294C42C2×D1212yes—
295C42D3×D412yes—
296C22×C3C2×#12011yescharacteristic
297C22×C3#53211yescharacteristic
298C22×C3#53511yescharacteristic
299C22×C3#57714yes—
300C2×D3—42no—
301C2×D3—41no—
302C2×D3—42no—
303C2×D3—42no—
304C2×D3—84no—
305C2×D3—82no—
306C2×D3—82no—
307C2×D3—82no—
308C2×D3—84no—
309C3×C4—24no—
310C3×C4—44no—
311C3×C4—82no—
312C3×C4#54612yes—
313C3×C4#55012yes—
314Dic3—41no—
315Dic3—42no—
316Dic3—44no—
317C23—61no—
318C23—61no—
319C23—61no—
320C23—64no—
321C23—122no—
322C23—122no—
323C23—122no—
324C23—124no—
325C23—124no—
326C23C2×D3×D411yescharacteristic
327Q8—28no—
328Q8—44no—
329D4—124no—
330D4—124no—
331D4—248no—
332D4—248no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—62no—
342C2×C4—64no—
343C2×C4—61no—
344C2×C4—64no—
345C2×C4—124no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4#88214yes—
349C2×C4C22×D1211yescharacteristic
350C2×C4C2×D3×D411yescharacteristic
351C2×C4#100412yes—
352C2×C4D3×#4112yes—
353C8—42no—
354C8—64no—
355C2×C3#243111yescharacteristic
356C2×C3#301012yes—
357C2×C3#317412yes—
358C2×C3#322212yes—
359D3—81no—
360D3—162no—
361C22—121no—
362C22—122no—
363C22—122no—
364C22—122no—
365C22—242no—
366C22—244no—
367C22—244no—
368C22—242no—
369C22—242no—
370C22C2×#88211yescharacteristic
371C22#566811yescharacteristic
372C22#571511yescharacteristic
373C22#577814yes—
374C4—24no—
375C4—44no—
376C4—82no—
377C4—121no—
378C4—122no—
379C4—124no—
380C4#555412yes—
381C4#559912yes—
382C3#2560711yescharacteristic, complemented, Sylow
383C2—241no—
384C2—482no—
385C2#7631311yescharacteristic
386C2#7896012yes—
387C2#8172912yes—
388C2#8187812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1960C2
socleC23×C3C22×D4
prime radical#74244C2
hypercenter#1960D3
hypocenterC3#25607
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. C2×#5731C2semidirect
2. C2×#5085C2semidirect
3. C2×#4713C2semidirect
5. C3×#1960C2semidirect
6. #70272C2semidirect
7. #70240C2semidirect
8. #69830C2semidirect
9. #69554C2semidirect
42. #1960D3semidirect
52. C3×C4⋊Q8D4semidirect
83. #870D4semidirect
175. C4⋊Q8D12semidirect
382. C3#25607semidirect

Maximal subgroups

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

subgroupindex
C2×#57312
C2×#50852
C2×#47132
#742442
C3×#19602
#702722
#702402
#698302
#695542
#256073