Finite Group Databasebeta

← #10696495

Subgroups — #10696495

Order 768.

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

normal subgroups only

This group has 526 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×C3×#491C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic
3#83071C211yesmaximal, characteristic
4#71788C211yesmaximal, characteristic, complemented
5C2×#4543C211yesmaximal, characteristic, complemented
6C2×#4464C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic, complemented
8#10083—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C2×#875C2211yescharacteristic
12C3×#491—22no—
13C3×#491—22no—
14C2×C3×#103—21no—
15C2×C3×#103—21no—
16C2×C3×#97C2211yescharacteristic
17#5089—22no—
18#5089—22no—
19#4994—22no—
20#4993—21no—
21#4992—21no—
22#4991—21no—
23#4979—21no—
24C2×#810—21no—
25C2×#809—21no—
26#4936C2211yescharacteristic
27C2×#803C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4543—22no—
30#4529—22no—
31#4464—22no—
32C2×#491D311yescharacteristic, complemented
33C2×#489—31no—
34#2820—31no—
35#1993—31no—
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—
47C3×#125—21no—
48C3×#125—21no—
49C3×#125—21no—
50C3×#125—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C42C2311yescharacteristic
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×C4⋊Dic3D412yes—
63C2×Dic3⋊C4—22no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8—21no—
66#875—21no—
67#875—21no—
68#875—21no—
69#875—21no—
70C3×#103—24no—
71C3×#103—24no—
72C3×#97—21no—
73C3×#97D412yes—
74C3⋊#95—22no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#810—24no—
78#809—24no—
79#803—21no—
80#803—21no—
81#803—21no—
82#803D412yes—
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×#125C2×D311yescharacteristic
85C2×#124—31no—
86#491—22no—
87#491—22no—
88#489—62no—
89#489—62no—
90C2×#103—21no—
91C2×#103—21no—
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#382—31no—
103#340—62no—
104#330—62no—
105#329—62no—
106C23×D3—21no—
107C2×C3×Q8—22no—
108C2×C3×Q8—44no—
109C2×C3×Q8—41no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×Dic3—22no—
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—42no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—82no—
147(C2×D3)⋊C4—82no—
148C4⋊Dic3—42no—
149Dic3⋊C4—82no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—42no—
154C2×C3⋊C8—41no—
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⋊Q8#6712yes—
165#125—21no—
166#125—21no—
167#125—21no—
168#125—21no—
169#124—61no—
170#124—61no—
171#124—61no—
172#124—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4#6711yescharacteristic
180C2×C4⋊C4#6711yescharacteristic
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—62no—
185C2×C42C22×D311yescharacteristic
186#103—24no—
187#103—24no—
188#103—64no—
189#102—64no—
190#97—21no—
191#97—32no—
192#97—61no—
193#97—61no—
194#97—61no—
195#97#6712yes—
196#95—62no—
197#95—61no—
198#95—61no—
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—21no—
209C2×C3×C4—22no—
210C2×C3×C4—21no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—82no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12711yescharacteristic
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
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—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4C2×#6711yescharacteristic
254C2×C8—22no—
255C2×C8—42no—
256C2×C8—62no—
257C2×C8—62no—
258C2×C8—121no—
259C2×C8—122no—
260C2×C8—121no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—122no—
274C4⋊C4—242no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—242no—
280C22⋊C4—242no—
281C42D3×D411yescharacteristic
282C42#23711yescharacteristic
283C42C2×#6711yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3#39211yescharacteristic
286C22×C3#45412yes—
287C22×C3#45511yescharacteristic
288C22×C3#45512yes—
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—81no—
303C3×C4—81no—
304Dic3—82no—
305Dic3—81no—
306C23—121no—
307C23—122no—
308C23—121no—
309C23—121no—
310C23—122no—
311C23#93711yescharacteristic
312Q8—44no—
313Q8—82no—
314Q8—122no—
315Q8—241no—
316D4—122no—
317D4—241no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—82no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—121no—
333C2×C4—122no—
334C2×C4#93711yescharacteristic
335C2×C4#93811yescharacteristic
336C2×C4#93911yescharacteristic
337C2×C4#93911yescharacteristic
338C2×C4#94011yescharacteristic
339C2×C4#95311yescharacteristic
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—242no—
352C22—241no—
353C22—241no—
354C22#506212yes—
355C22#520211yescharacteristic
356C22#520212yes—
357C22#520511yescharacteristic
358C22#522311yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—81no—
367C4—81no—
368C4—242no—
369C4—241no—
370C3#1008311yescharacteristic, complemented, Sylow
371C2—481no—
372C2#7125112yes—
373C2#7125312yes—
374C2#7173011yescharacteristic
375C2#7210611yescharacteristic
376C2#7211011yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#491C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#491D3
hypocenterC3#10083
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
1. C2×C3×#491C2semidirect
4. #71788C2semidirect
5. C2×#4543C2semidirect
7. #67659C2semidirect
32. C2×#491D3semidirect
370. C3#10083semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4912
C2×#50892
#830712
#717882
C2×#45432
C2×#44642
#676592
#100833