Finite Group Databasebeta

← #10696479

Subgroups — #10696479

Order 768.

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

normal subgroups only

This group has 552 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
1#83055C211yesmaximal, characteristic, complemented
2C2×#5067C211yesmaximal, characteristic
3C2×#4713C211yesmaximal, characteristic
4C2×C3×#330C211yesmaximal, characteristic, complemented
5#69823C211yesmaximal, characteristic
6C2×#4465C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#10080—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×#874C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14C2×#821—21no—
15C2×#821—21no—
16C2×#821—22no—
17#5067—22no—
18#5067—22no—
19#4986—21no—
20#4985—21no—
21#4983—21no—
22#4982—22no—
23#4981—21no—
24C2×#809—21no—
25C2×#809—21no—
26#4936C2211yescharacteristic
27C2×#803C2211yescharacteristic
28#4713—22no—
29#4713—22no—
30C3×#330—22no—
31#4465—22no—
32C2×#490—31no—
33C2×#490—31no—
34#2822—31no—
35#1961—31no—
36C2×#330—31no—
37C2×#330D311yescharacteristic, complemented
38#1295—31no—
39C22×D12—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C42C2311yescharacteristic
53C2×C4⋊Dic3D411yescharacteristic
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4⋊Dic3D412yes—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—22no—
59C22×C3⋊C8—21no—
60#874—21no—
61#874—21no—
62#874—21no—
63#874D412yes—
64#870—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68C3×#97—21no—
69C3×#97—21no—
70C3×#97—21no—
71C3×#97D412yes—
72C3⋊#95—22no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#821—24no—
78#821—24no—
79#821—44no—
80#809—24no—
81#809—24no—
82#803—21no—
83#803D412yes—
84C2×#127—31no—
85C2×#125—31no—
86C2×#125C2×D311yescharacteristic
87#490—62no—
88#490—62no—
89#490—62no—
90#490—62no—
91C2×#102—61no—
92C2×#102—61no—
93C2×#102—61no—
94C2×#102—61no—
95C2×#102—62no—
96C2×#97—31no—
97C2×#97C2×D311yescharacteristic
98C2×#97C2×D311yescharacteristic
99#400—61no—
100#400—61no—
101#398—61no—
102#398—62no—
103#398—61no—
104#382—31no—
105#330—22no—
106#330—62no—
107C23×D3—41no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—22no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C2×D12—22no—
117C2×D12—22no—
118C2×D12—22no—
119C2×D12—42no—
120C2×D12—44no—
121C2×D12—44no—
122C2×D12—42no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×C3×C8—22no—
127C2×C3×C8—22no—
128C2×C3×C8—42no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—42no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—82no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—42no—
155C22×D4—61no—
156C22×D4—61no—
157C22×D4—61no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—61no—
161#127—32no—
162#127—61no—
163#127—61no—
164#127—61no—
165#125—21no—
166#125—21no—
167#125—21no—
168#125—21no—
169#125—61no—
170#125—61no—
171#125—61no—
172#125—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4D1211yescharacteristic
182C2×C4⋊C4D1211yescharacteristic
183C2×C42C22×D311yescharacteristic
184#102—64no—
185#102—64no—
186#102—64no—
187#102—64no—
188#102—124no—
189#97—21no—
190#97—21no—
191#97—21no—
192#97—32no—
193#97—61no—
194#97#6712yes—
195#95—62no—
196#95—62no—
197#95—62no—
198#95—61no—
199#95—61no—
200C23×C3#12011yescharacteristic
201C22×D3—42no—
202C22×D3—81no—
203C22×D3—82no—
204C22×D3—82no—
205C22×D3—81no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—21no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
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—41no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—41no—
229D12—44no—
230D12—44no—
231D12—44no—
232D12—82no—
233D12—82no—
234D12—82no—
235C3×C8—42no—
236C3×C8—42no—
237C3⋊C8—42no—
238C24—121no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—62no—
242C2×D4—122no—
243C2×D4—124no—
244C2×D4—124no—
245C2×D4—122no—
246C2×D4—121no—
247C2×D4—121no—
248C2×D4—121no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—62no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4C2×D1211yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4D3×D411yescharacteristic
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—62no—
263C2×C8—122no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—122no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—242no—
282C42#23511yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C42C2×#6711yescharacteristic
286C22×C3#39211yescharacteristic
287C22×C3#45311yescharacteristic
288C22×C3#45412yes—
289C22×C3#45412yes—
290C22×C3#45411yescharacteristic
291C2×D3—82no—
292C2×D3—82no—
293C2×D3—82no—
294C2×D3—162no—
295C2×D3—162no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—81no—
303C3×C4—81no—
304Dic3—82no—
305Dic3—81no—
306Dic3—81no—
307C23—122no—
308C23—242no—
309C23—241no—
310C23—242no—
311C23—241no—
312C23#88211yescharacteristic
313D4—124no—
314D4—124no—
315D4—124no—
316D4—242no—
317D4—242no—
318D4—242no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—121no—
334C2×C4—122no—
335C2×C4—121no—
336C2×C4#88211yescharacteristic
337C2×C4#88411yescharacteristic
338C2×C4#88411yescharacteristic
339C2×C4#93011yescharacteristic
340C2×C4#93711yescharacteristic
341C2×C4#94011yescharacteristic
342C8—42no—
343C8—42no—
344C8—122no—
345C2×C3#165712yes—
346C2×C3#165812yes—
347C2×C3#200311yescharacteristic
348C2×C3#200311yescharacteristic
349C2×C3#200611yescharacteristic
350D3—162no—
351C22—242no—
352C22—242no—
353C22—242no—
354C22—482no—
355C22—482no—
356C22#475312yes—
357C22#482211yescharacteristic
358C22#482712yes—
359C22#482711yescharacteristic
360C22#496011yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—81no—
368C4—81no—
369C4—242no—
370C4—241no—
371C4—241no—
372C3#1008011yescharacteristic, complemented, Sylow
373C2—482no—
374C2#6837112yes—
375C2#6837512yes—
376C2#7011611yescharacteristic
377C2#7038811yescharacteristic
378C2#7039211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#330C2
socleC23×C3#120
prime radicalC2×#874C22
hypercenterC2×#330D3
hypocenterC3#10080
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
1. #83055C2semidirect
4. C2×C3×#330C2semidirect
6. C2×#4465C2semidirect
7. #67659C2semidirect
37. C2×#330D3semidirect
372. C3#10080semidirect

Maximal subgroups

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

subgroupindex
#830552
C2×#50672
C2×#47132
C2×C3×#3302
#698232
C2×#44652
#676592
#100803