Finite Group Databasebeta

← #10696516

Subgroups — #10696516

Order 768.

542 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 95 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×#5081C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic
3#74255C211yesmaximal, characteristic
4C3×#1960C211yesmaximal, characteristic, complemented
5C2×#4465C211yesmaximal, characteristic, complemented
6C2×#4464C211yesmaximal, characteristic
7#67630C211yesmaximal, characteristic, complemented
8#10086—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#875C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C2×C3×#103—22no—
13C2×C3×#97C2211yescharacteristic
14C3×#382C2211yescharacteristic
15#5089—22no—
16#5089—22no—
17#5081—22no—
18#5081—22no—
19#4994—22no—
20C2×#810—21no—
21C2×#809—21no—
22C2×#808—21no—
23C2×#807—21no—
24C2×#803C2211yescharacteristic
25C2×#803C2211yescharacteristic
26#4677—21no—
27#4674—21no—
28#4672—21no—
29#4656—21no—
30#4465—22no—
31#4464—22no—
32C2×#501—31no—
33C2×#489—31no—
34#2820—31no—
35#1960D311yescharacteristic, complemented
36C2×#330—31no—
37C2×#329—31no—
38#1295—31no—
39C22×C3×Q8—21no—
40C22×D12D411yescharacteristic
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D412yes—
50C2×C3×C42C2311yescharacteristic
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—22no—
54C2×(C2×D3)⋊C4—22no—
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C22×C3⋊C8—21no—
61C22×C3⋊C8—21no—
62#875—21no—
63#875—21no—
64#875—21no—
65#875—21no—
66#870—21no—
67#870—21no—
68#870—21no—
69#870—21no—
70C3×#103—44no—
71C3×#97—21no—
72C3×#97—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3⋊#95—22no—
76#810—24no—
77#809—24no—
78#808—24no—
79#807—24no—
80#803—21no—
81#803—21no—
82#803D412yes—
83#803D412yes—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125—31no—
86C2×#124—31no—
87#501—62no—
88#501—62no—
89#489—62no—
90#489—62no—
91C2×#107—61no—
92C2×#106—61no—
93C2×#103—22no—
94C2×#103—61no—
95C2×#102—61no—
96C2×#97—31no—
97C2×#97—31no—
98C2×#97C2×D311yescharacteristic
99#399—61no—
100#397—61no—
101#396—62no—
102#395—61no—
103#394—61no—
104#382C2×D311yescharacteristic
105#330—62no—
106#329—62no—
107C23×D3—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—42no—
110C2×C3×Q8—42no—
111C2×C3×Q8—41no—
112C22×C3×C4—22no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—42no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—42no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—42no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—82no—
143(C2×D3)⋊C4—82no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—82no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153C2×C3⋊C8—24no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—42no—
157C22×Q8—21no—
158C22×Q8—31no—
159C22×D4—31no—
160C22×C8—21no—
161C22×C8—61no—
162C22×C8—61no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8D1212yes—
167#125—61no—
168#125—61no—
169#125—61no—
170#125—61no—
171#124—61no—
172#124—61no—
173#124—61no—
174#124—61no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4#6712yes—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—62no—
187C2×C42C22×D311yescharacteristic
188#107—64no—
189#106—64no—
190#103—44no—
191#103—64no—
192#102—64no—
193#97—21no—
194#97—21no—
195#97—32no—
196#97—32no—
197#97—61no—
198#97—61no—
199#95—21no—
200#95—21no—
201#95—62no—
202C23×C3#12011yescharacteristic
203C22×D3—41no—
204C22×D3—41no—
205C22×D3—41no—
206C22×D3—42no—
207C22×D3—42no—
208C3×Q8—44no—
209C3×Q8—82no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—81no—
219C2×C3×C4—81no—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12711yescharacteristic
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233D12—42no—
234D12—81no—
235Dic6—42no—
236Dic6—81no—
237C3×C8—42no—
238C3⋊C8—24no—
239C3⋊C8—42no—
240C24—61no—
241C2×Q8—22no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—41no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—122no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—122no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4D3×D411yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4C2×#6711yescharacteristic
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—64no—
263C2×C8—62no—
264C2×C8—62no—
265C2×C8—122no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C4⋊C4—242no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—242no—
285C22⋊C4—242no—
286C42C2×D1211yescharacteristic
287C42#23511yescharacteristic
288C42D3×D411yescharacteristic
289C42D3×D411yescharacteristic
290C22×C3#39211yescharacteristic
291C22×C3#45311yescharacteristic
292C22×C3#45411yescharacteristic
293C22×C3#45512yes—
294C22×C3#45612yes—
295C2×D3—82no—
296C2×D3—82no—
297C2×D3—81no—
298C2×D3—81no—
299C2×D3—81no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—82no—
307C3×C4—82no—
308Dic3—81no—
309Dic3—81no—
310Dic3—81no—
311C23—122no—
312C23—122no—
313C23—121no—
314C23—121no—
315C23—121no—
316C23#93711yescharacteristic
317Q8—44no—
318Q8—82no—
319Q8—122no—
320Q8—241no—
321D4—122no—
322D4—241no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4#88211yescharacteristic
341C2×C4#88211yescharacteristic
342C2×C4#88411yescharacteristic
343C2×C4#88411yescharacteristic
344C2×C4#93011yescharacteristic
345C2×C4#94011yescharacteristic
346C8—42no—
347C8—64no—
348C8—122no—
349C2×C3#166112yes—
350C2×C3#166412yes—
351C2×C3#200411yescharacteristic
352C2×C3#200511yescharacteristic
353C2×C3#200611yescharacteristic
354D3—161no—
355C22—241no—
356C22—242no—
357C22—242no—
358C22—241no—
359C22—241no—
360C22#496011yescharacteristic
361C22#506211yescharacteristic
362C22#507711yescharacteristic
363C22#520212yes—
364C22#520512yes—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—82no—
373C4—241no—
374C4—241no—
375C4—241no—
376C3#1008611yescharacteristic, complemented, Sylow
377C2—481no—
378C2#6966411yescharacteristic
379C2#7034411yescharacteristic
380C2#7034611yescharacteristic
381C2#7125312yes—
382C2#7133112yes—
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1960C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1960D3
hypocenterC3#10086
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. C2×#5081C2semidirect
4. C3×#1960C2semidirect
5. C2×#4465C2semidirect
7. #67630C2semidirect
35. #1960D3semidirect
376. C3#10086semidirect

Maximal subgroups

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

subgroupindex
C2×#50812
C2×#50892
#742552
C3×#19602
C2×#44652
C2×#44642
#676302
#100863