Finite Group Databasebeta

← #10696520

Subgroups — #10696520

Order 768.

542 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 97 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×#4714C211yesmaximal, characteristic
2C2×#4702C211yesmaximal, characteristic, complemented
3#74255C211yesmaximal, characteristic
4C2×#4465C211yesmaximal, characteristic, complemented
5C2×#4464C211yesmaximal, characteristic
6#69552C211yesmaximal, characteristic, complemented
7C3×#1295C211yesmaximal, characteristic, complemented
8#10086—31nomaximal, Sylow
9C2×#875C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14C3×#382C2211yescharacteristic
15C2×#821—21no—
16C2×#818—21no—
17C2×#817—21no—
18C2×#816—21no—
19#4994—22no—
20C2×#810—22no—
21C2×#803C2211yescharacteristic
22#4714—22no—
23#4714—22no—
24#4702—22no—
25#4702—22no—
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#1960—31no—
36C2×#330—31no—
37C2×#329—31no—
38#1295D311yescharacteristic, complemented
39C22×D12D411yescharacteristic
40C22×Dic6—21no—
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C2×C3×C4⋊C4D412yes—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—22no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—22no—
53C2×C4⋊Dic3D411yescharacteristic
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C22×C3⋊C8—21no—
58#875—21no—
59#875—21no—
60#875—21no—
61#875—21no—
62#870—21no—
63#870—21no—
64#870—21no—
65#870—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869D412yes—
70C3×#97—21no—
71C3×#97—21no—
72C3×#97D412yes—
73C3×#97D412yes—
74C3×#95—21no—
75C3×#95—21no—
76C3⋊#95—22no—
77#821—24no—
78#818—24no—
79#817—24no—
80#816—24no—
81#810—44no—
82#803—21no—
83#803—21no—
84C2×C4⋊Q8—31no—
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—61no—
94C2×#103—62no—
95C2×#102—61no—
96C2×#97—31no—
97C2×#97C2×D311yescharacteristic
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—
108C22×C3×C4—22no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—41no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—42no—
119C2×Dic6—21no—
120C2×Dic6—22no—
121C2×Dic6—21no—
122C2×Dic6—42no—
123C2×Dic6—42no—
124C2×Dic6—42no—
125C2×Dic6—41no—
126C2×C3×C8—24no—
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C2×C3×C8—42no—
130C3×C4⋊C4—42no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135(C2×D3)⋊C4—41no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—82no—
140(C2×D3)⋊C4—82no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—22no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—82no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C22×Q8—31no—
158C22×Q8—61no—
159C22×D4—31no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—61no—
163C4⋊Q8—32no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
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—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4D1212yes—
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—64no—
191#103—124no—
192#102—64no—
193#97—21no—
194#97—21no—
195#97—61no—
196#97—61no—
197#97D1212yes—
198#97D1212yes—
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—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12711yescharacteristic
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—81no—
230C2×Dic3—81no—
231D12—42no—
232D12—81no—
233Dic6—42no—
234Dic6—44no—
235Dic6—82no—
236Dic6—81no—
237C3×C8—24no—
238C3×C8—42no—
239C3⋊C8—42no—
240C24—61no—
241C2×Q8—61no—
242C2×Q8—62no—
243C2×Q8—61no—
244C2×Q8—122no—
245C2×Q8—122no—
246C2×Q8—122no—
247C2×Q8—121no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—122no—
251C22×C4—22no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—121no—
256C22×C4C2×D1211yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259C2×C8—24no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—42no—
263C2×C8—62no—
264C2×C8—121no—
265C2×C8—121no—
266C4⋊C4—42no—
267C4⋊C4—62no—
268C4⋊C4—61no—
269C4⋊C4—62no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
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
287C42C2×D1211yescharacteristic
288C42D3×D411yescharacteristic
289C42#23911yescharacteristic
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—
307Dic3—81no—
308Dic3—81no—
309Dic3—82no—
310Dic3—81no—
311C23—121no—
312C23—121no—
313C23—121no—
314C23—122no—
315C23—122no—
316C23#88211yescharacteristic
317Q8—124no—
318Q8—122no—
319Q8—241no—
320Q8—242no—
321D4—122no—
322D4—241no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—122no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—241no—
339C2×C4—241no—
340C2×C4#87411yescharacteristic
341C2×C4#88211yescharacteristic
342C2×C4#88211yescharacteristic
343C2×C4#89511yescharacteristic
344C2×C4#89511yescharacteristic
345C2×C4#89511yescharacteristic
346C8—24no—
347C8—42no—
348C8—122no—
349C2×C3#166112yes—
350C2×C3#166412yes—
351C2×C3#200411yescharacteristic
352C2×C3#200511yescharacteristic
353C2×C3#200611yescharacteristic
354D3—161no—
355C22—242no—
356C22—242no—
357C22—241no—
358C22—241no—
359C22—241no—
360C22#467111yescharacteristic
361C22#474711yescharacteristic
362C22#475311yescharacteristic
363C22#482512yes—
364C22#482612yes—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—241no—
373C4—242no—
374C4—241no—
375C4—241no—
376C3#1008611yescharacteristic, complemented, Sylow
377C2—481no—
378C2#6835612yes—
379C2#6837912yes—
380C2#6876211yescharacteristic
381C2#6876811yescharacteristic
382C2#6900811yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1295C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1295D3
hypocenterC3#10086
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
2. C2×#4702C2semidirect
4. C2×#4465C2semidirect
6. #69552C2semidirect
7. C3×#1295C2semidirect
38. #1295D3semidirect
376. C3#10086semidirect

Maximal subgroups

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

subgroupindex
C2×#47142
C2×#47022
#742552
C2×#44652
C2×#44642
#695522
C3×#12952
#100863