Finite Group Databasebeta

← #10696515

Subgroups — #10696515

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×C3×#501C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic
3#83071C211yesmaximal, characteristic
4#71786C211yesmaximal, characteristic, complemented
5C2×#4543C211yesmaximal, characteristic, complemented
6C2×#4464C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic, complemented
8#10086—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C2×#875C2211yescharacteristic
12C3×#501—22no—
13C3×#501—22no—
14C2×C3×#107—21no—
15C2×C3×#106—21no—
16C2×C3×#97C2211yescharacteristic
17C2×#835—22no—
18#5089—22no—
19#5089—22no—
20#4994—22no—
21#4993—21no—
22#4992—21no—
23#4991—21no—
24#4979—21no—
25C2×#810—21no—
26C2×#809—21no—
27#4936C2211yescharacteristic
28C2×#803C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4543—22no—
31#4464—22no—
32C2×#501D311yescharacteristic, complemented
33C2×#489—31no—
34#2820—31no—
35#1960—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×#107—24no—
71C3×#106—24no—
72C3×#97—21no—
73C3×#97D412yes—
74#835—44no—
75C3⋊#95—22no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78#810—24no—
79#809—24no—
80#803—21no—
81#803—21no—
82#803—21no—
83#803D412yes—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125C2×D311yescharacteristic
86C2×#124—31no—
87#501—22no—
88#501—22no—
89#489—62no—
90#489—62no—
91C2×#107—21no—
92C2×#106—21no—
93C2×#103—61no—
94C2×#103—62no—
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#382—31no—
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—21no—
113C22×C3×C4—21no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×Dic3—22no—
119C22×Dic3—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—42no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—42no—
126C2×C3×C8—24no—
127C2×C3×C8—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—82no—
149(C2×D3)⋊C4—82no—
150C4⋊Dic3—42no—
151Dic3⋊C4—82no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—42no—
156C2×C3⋊C8—41no—
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⋊Q8#6712yes—
167#125—21no—
168#125—21no—
169#125—21no—
170#125—21no—
171#124—61no—
172#124—61no—
173#124—61no—
174#124—61no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4#6711yescharacteristic
182C2×C4⋊C4#6711yescharacteristic
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—62no—
187C2×C42C22×D311yescharacteristic
188#107—24no—
189#106—24no—
190#103—64no—
191#103—124no—
192#102—64no—
193#97—21no—
194#97—32no—
195#97—61no—
196#97—61no—
197#97—61no—
198#97#6712yes—
199#95—62no—
200#95—61no—
201#95—61no—
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—41no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—81no—
221C2×C3×C4—81no—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12711yescharacteristic
228C2×Dic3—41no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233D12—42no—
234D12—81no—
235Dic6—42no—
236Dic6—81no—
237C3×C8—24no—
238C3⋊C8—42no—
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—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—62no—
255C22×C4—61no—
256C22×C4D3×D411yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4C2×#6711yescharacteristic
259C2×C8—24no—
260C2×C8—22no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—121no—
264C2×C8—122no—
265C2×C8—121no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—22no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—122no—
279C4⋊C4—242no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—242no—
285C22⋊C4—242no—
286C42D3×D411yescharacteristic
287C42#23711yescharacteristic
288C42C2×#6711yescharacteristic
289C42C2×#6711yescharacteristic
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—81no—
307C3×C4—81no—
308C3×C4—82no—
309Dic3—82no—
310Dic3—81no—
311C23—121no—
312C23—121no—
313C23—122no—
314C23—121no—
315C23—122no—
316C23#93711yescharacteristic
317Q8—44no—
318Q8—82no—
319Q8—122no—
320Q8—241no—
321D4—122no—
322D4—241no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—121no—
339C2×C4—122no—
340C2×C4#93711yescharacteristic
341C2×C4#93811yescharacteristic
342C2×C4#93911yescharacteristic
343C2×C4#93911yescharacteristic
344C2×C4#94011yescharacteristic
345C2×C4#95311yescharacteristic
346C8—24no—
347C8—122no—
348C8—122no—
349C2×C3#166112yes—
350C2×C3#166412yes—
351C2×C3#200411yescharacteristic
352C2×C3#200511yescharacteristic
353C2×C3#200611yescharacteristic
354D3—161no—
355C22—241no—
356C22—241no—
357C22—241no—
358C22—242no—
359C22—242no—
360C22#509512yes—
361C22#519811yescharacteristic
362C22#520111yescharacteristic
363C22#520212yes—
364C22#522311yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—81no—
372C4—82no—
373C4—81no—
374C4—241no—
375C4—242no—
376C3#1008611yescharacteristic, complemented, Sylow
377C2—481no—
378C2#7139212yes—
379C2#7139412yes—
380C2#7179511yescharacteristic
381C2#7209811yescharacteristic
382C2#7210611yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#501C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#501D3
hypocenterC3#10086
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×C3×#501C2semidirect
4. #71786C2semidirect
5. C2×#4543C2semidirect
7. #67659C2semidirect
32. C2×#501D3semidirect
376. C3#10086semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5012
C2×#50892
#830712
#717862
C2×#45432
C2×#44642
#676592
#100863