Finite Group Databasebeta

← #10696521

Subgroups — #10696521

Order 768.

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

normal subgroups only

This group has 500 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
2C2×#5081C211yesmaximal, characteristic
3#83055C211yesmaximal, characteristic
4#71786C211yesmaximal, characteristic
5C2×#4543C211yesmaximal, characteristic
6C2×#4465C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic
8#10087—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#501—22no—
13C3×#501—22no—
14C2×C3×#107—21no—
15C2×C3×#106—21no—
16C2×C3×#97C2211yescharacteristic
17C2×#835—22no—
18#5081—22no—
19#5081—22no—
20#4986—21no—
21#4985—21no—
22#4983—21no—
23#4982—22no—
24#4981—21no—
25C2×#808—21no—
26C2×#807—21no—
27#4936C2211yescharacteristic
28C2×#803C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4543—22no—
31#4465—22no—
32C2×#501—31no—
33C2×#501D311yescharacteristic
34#2822—31no—
35#1960—31no—
36C2×#330—31no—
37C2×#330—31no—
38#1295—31no—
39C22×C3×Q8—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D412yes—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C3×#125—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42C2311yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D412yes—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—22no—
62C22×C3⋊C8—21no—
63C22×C3⋊C8—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68C3×#107—24no—
69C3×#106—24no—
70C3×#97—21no—
71C3×#97D412yes—
72#835—44no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95—22no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78#808—24no—
79#807—24no—
80#803—21no—
81#803—21no—
82#803—21no—
83#803D412yes—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125—31no—
86C2×#125C2×D311yescharacteristic
87#501—22no—
88#501—22no—
89#501—62no—
90#501—62no—
91C2×#107—21no—
92C2×#107—61no—
93C2×#106—21no—
94C2×#106—61no—
95C2×#103—62no—
96C2×#97—31no—
97C2×#97—31no—
98C2×#97C2×D311yescharacteristic
99#400—61no—
100#400—61no—
101#398—61no—
102#398—62no—
103#398—61no—
104#382—31no—
105#330—62no—
106#330—62no—
107C2×C3×Q8—22no—
108C2×C3×Q8—42no—
109C2×C3×Q8—42no—
110C2×C3×Q8—41no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—22no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C2×C3×C8—24no—
121C2×C3×C8—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—42no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—82no—
148C2×C3⋊C8—24no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C22×Q8—21no—
154C22×C8—21no—
155C22×C8—61no—
156C22×C8—61no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8#6712yes—
161#125—21no—
162#125—21no—
163#125—21no—
164#125—21no—
165#125—61no—
166#125—61no—
167#125—61no—
168#125—61no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4#6711yescharacteristic
180C2×C4⋊C4#6711yescharacteristic
181C2×C42C22×D311yescharacteristic
182#107—24no—
183#107—64no—
184#106—24no—
185#106—64no—
186#103—124no—
187#97—21no—
188#97—32no—
189#97—61no—
190#97—61no—
191#97—61no—
192#97#6712yes—
193#95—62no—
194#95—62no—
195#95—62no—
196#95—61no—
197#95—61no—
198C23×C3#12011yescharacteristic
199C3×Q8—44no—
200C3×Q8—82no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12711yescharacteristic
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—42no—
223C2×Dic3—41no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C3×C8—24no—
228C3⋊C8—24no—
229C3⋊C8—42no—
230C2×Q8—22no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—62no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4C2×#6711yescharacteristic
243C2×C8—24no—
244C2×C8—22no—
245C2×C8—64no—
246C2×C8—62no—
247C2×C8—62no—
248C2×C8—121no—
249C2×C8—121no—
250C4⋊C4—22no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—122no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—242no—
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C42C2×#6711yescharacteristic
275C42C2×#6711yescharacteristic
276C22×C3#39211yescharacteristic
277C22×C3#45311yescharacteristic
278C22×C3#45411yescharacteristic
279C22×C3#45612yes—
280C22×C3#45612yes—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4—82no—
290Dic3—82no—
291Dic3—81no—
292Dic3—81no—
293C23#93711yescharacteristic
294Q8—44no—
295Q8—82no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—122no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—122no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4#93711yescharacteristic
317C2×C4#93811yescharacteristic
318C2×C4#93911yescharacteristic
319C2×C4#93911yescharacteristic
320C2×C4#94011yescharacteristic
321C2×C4#95311yescharacteristic
322C8—24no—
323C8—64no—
324C8—122no—
325C2×C3#166012yes—
326C2×C3#166312yes—
327C2×C3#200511yescharacteristic
328C2×C3#200511yescharacteristic
329C2×C3#200611yescharacteristic
330C22#509512yes—
331C22#519811yescharacteristic
332C22#520111yescharacteristic
333C22#520512yes—
334C22#522311yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—81no—
342C4—82no—
343C4—81no—
344C4—241no—
345C4—241no—
346C4—242no—
347C3#1008711yescharacteristic, complemented, Sylow
348C2#7139112yes—
349C2#7139312yes—
350C2#7179511yescharacteristic
351C2#7209811yescharacteristic
352C2#7210811yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#501C2
socle, prime radicalC23×C3#120
hypercenterC2×#501D3
hypocenterC3#10087
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
347. C3#10087semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5012
C2×#50812
#830552
#717862
C2×#45432
C2×#44652
#676592
#100873