Finite Group Databasebeta

← #10696497

Subgroups — #10696497

Order 768.

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

normal subgroups only

This group has 526 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×#5082C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic
3#74255C211yesmaximal, characteristic
4C3×#1993C211yesmaximal, characteristic, complemented
5C2×#4465C211yesmaximal, characteristic, complemented
6C2×#4464C211yesmaximal, characteristic
7#67630C211yesmaximal, characteristic, complemented
8#10083—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#875C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C3×#382C2211yescharacteristic
14C2×#835—21no—
15C2×#835—21no—
16#5089—22no—
17#5089—22no—
18#5082—22no—
19#5082—22no—
20#4994—22no—
21C2×#810—21no—
22C2×#809—21no—
23C2×#803C2211yescharacteristic
24C2×#803C2211yescharacteristic
25#4677—21no—
26#4674—21no—
27#4672—21no—
28#4656—21no—
29C3×#340—22no—
30#4465—22no—
31#4464—22no—
32C2×#491—31no—
33C2×#489—31no—
34#2820—31no—
35#1993D311yescharacteristic, 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×#97—21no—
71C3×#97—21no—
72C3×#95—21no—
73C3×#95—21no—
74#835—24no—
75#835—24no—
76C3⋊#95—22no—
77#810—24no—
78#809—24no—
79#803—21no—
80#803—21no—
81#803D412yes—
82#803D412yes—
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×#125—31no—
85C2×#124—31no—
86#491—62no—
87#491—62no—
88#489—62no—
89#489—62no—
90C2×#103—61no—
91C2×#103—61no—
92C2×#103—61no—
93C2×#102—61no—
94C2×#97—31no—
95C2×#97—31no—
96C2×#97C2×D311yescharacteristic
97#399—61no—
98#397—61no—
99#396—62no—
100#395—61no—
101#394—61no—
102#382C2×D311yescharacteristic
103#340—22no—
104#330—62no—
105#329—62no—
106C23×D3—21no—
107C2×C3×Q8—22no—
108C2×C3×Q8—44no—
109C2×C3×Q8—41no—
110C22×C3×C4—22no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—42no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—42no—
124C2×C3×C8—22no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—42no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—82no—
141(C2×D3)⋊C4—82no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—82no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—42no—
154C2×C3⋊C8—42no—
155C22×Q8—21no—
156C22×Q8—31no—
157C22×D4—31no—
158C22×C8—21no—
159C22×C8—61no—
160C22×C8—61no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8D1212yes—
165#125—61no—
166#125—61no—
167#125—61no—
168#125—61no—
169#124—61no—
170#124—61no—
171#124—61no—
172#124—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4#6712yes—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—62no—
185C2×C42C22×D311yescharacteristic
186#103—64no—
187#103—64no—
188#103—64no—
189#102—64no—
190#97—21no—
191#97—21no—
192#97—32no—
193#97—32no—
194#97—61no—
195#97—61no—
196#95—21no—
197#95—21no—
198#95—62no—
199C23×C3#12011yescharacteristic
200C22×D3—41no—
201C22×D3—42no—
202C22×D3—41no—
203C22×D3—42no—
204C22×D3—41no—
205C3×Q8—44no—
206C3×Q8—82no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—82no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12711yescharacteristic
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229D12—42no—
230D12—81no—
231Dic6—42no—
232Dic6—81no—
233C3×C8—42no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236C24—61no—
237C2×Q8—22no—
238C2×Q8—44no—
239C2×Q8—41no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—122no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—122no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4C2×#6711yescharacteristic
254C2×C8—22no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—62no—
258C2×C8—62no—
259C2×C8—122no—
260C2×C8—122no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—242no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—242no—
280C22⋊C4—242no—
281C42C2×D1211yescharacteristic
282C42#23511yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C22×C3#39211yescharacteristic
286C22×C3#45412yes—
287C22×C3#45512yes—
288C22×C3#45511yescharacteristic
289C22×C3#45611yescharacteristic
290C2×D3—82no—
291C2×D3—82no—
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4—82no—
303Dic3—81no—
304Dic3—81no—
305Dic3—81no—
306C23—122no—
307C23—122no—
308C23—121no—
309C23—121no—
310C23—121no—
311C23#93711yescharacteristic
312Q8—44no—
313Q8—82no—
314Q8—122no—
315Q8—241no—
316D4—122no—
317D4—241no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4#88211yescharacteristic
335C2×C4#88211yescharacteristic
336C2×C4#88411yescharacteristic
337C2×C4#88411yescharacteristic
338C2×C4#93011yescharacteristic
339C2×C4#94011yescharacteristic
340C8—42no—
341C8—122no—
342C8—122no—
343C2×C3#166212yes—
344C2×C3#166412yes—
345C2×C3#200311yescharacteristic
346C2×C3#200411yescharacteristic
347C2×C3#200711yescharacteristic
348D3—161no—
349C22—241no—
350C22—242no—
351C22—241no—
352C22—241no—
353C22—242no—
354C22#496011yescharacteristic
355C22#506111yescharacteristic
356C22#509511yescharacteristic
357C22#520112yes—
358C22#520212yes—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—82no—
367C4—241no—
368C4—241no—
369C4—241no—
370C3#1008311yescharacteristic, complemented, Sylow
371C2—481no—
372C2#6969411yescharacteristic
373C2#7032611yescharacteristic
374C2#7034411yescharacteristic
375C2#7124612yes—
376C2#7139412yes—
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1993C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1993D3
hypocenterC3#10083
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
1. C2×#5082C2semidirect
4. C3×#1993C2semidirect
5. C2×#4465C2semidirect
7. #67630C2semidirect
35. #1993D3semidirect
370. C3#10083semidirect

Maximal subgroups

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

subgroupindex
C2×#50822
C2×#50892
#742552
C3×#19932
C2×#44652
C2×#44642
#676302
#100833