Finite Group Databasebeta

← #10696524

Subgroups — #10696524

Order 768.

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

normal subgroups only

This group has 500 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
2C2×#5064C211yesmaximal, characteristic
3C2×#4702C211yesmaximal, characteristic
4C2×C3×#330C211yesmaximal, characteristic
5#69822C211yesmaximal, characteristic
6C2×#4465C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic
8#10087—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14C2×#818—21no—
15C2×#817—21no—
16C2×#816—22no—
17#5064—22no—
18#5064—22no—
19#4986—21no—
20#4985—21no—
21#4983—21no—
22#4982—22no—
23#4981—21no—
24C2×#808—21no—
25C2×#807—21no—
26#4936C2211yescharacteristic
27C2×#803C2211yescharacteristic
28#4702—22no—
29#4702—22no—
30C3×#330—22no—
31#4465—22no—
32C2×#501—31no—
33C2×#501—31no—
34#2822—31no—
35#1960—31no—
36C2×#330—31no—
37C2×#330D311yescharacteristic
38#1295—31no—
39C22×Dic6—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C3×#125—21no—
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42C2311yescharacteristic
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
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#870—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869D412yes—
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#818—24no—
78#817—24no—
79#816—44no—
80#808—24no—
81#807—24no—
82#803—21no—
83#803D412yes—
84C2×C4⋊Q8—31no—
85C2×#125—31no—
86C2×#125C2×D311yescharacteristic
87#501—62no—
88#501—62no—
89#501—62no—
90#501—62no—
91C2×#107—61no—
92C2×#107—61no—
93C2×#106—61no—
94C2×#106—61no—
95C2×#103—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—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×Dic3—22no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—41no—
116C2×Dic6—22no—
117C2×Dic6—42no—
118C2×Dic6—42no—
119C2×Dic6—41no—
120C2×C3×C8—24no—
121C2×C3×C8—22no—
122C2×C3×C8—22no—
123C2×C3×C8—41no—
124C2×C3×C8—41no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C4⋊Dic3—22no—
138C4⋊Dic3—21no—
139C4⋊Dic3—22no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—42no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—82no—
151C2×C3⋊C8—24no—
152C2×C3⋊C8—22no—
153C22×Q8—61no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—61no—
157C4⋊Q8—32no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
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—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—32no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4D1211yescharacteristic
180C2×C4⋊C4D1211yescharacteristic
181C2×C42C22×D311yescharacteristic
182#107—64no—
183#107—64no—
184#106—64no—
185#106—64no—
186#103—124no—
187#97—21no—
188#97—21no—
189#97—21no—
190#97—32no—
191#97—61no—
192#97#6712yes—
193#95—62no—
194#95—62no—
195#95—62no—
196#95—61no—
197#95—61no—
198C23×C3#12011yescharacteristic
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12711yescharacteristic
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—42no—
217C2×Dic3—41no—
218C2×Dic3—42no—
219C2×Dic3—41no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—81no—
224C2×Dic3—81no—
225Dic6—44no—
226Dic6—82no—
227C3×C8—24no—
228C3×C8—42no—
229C3⋊C8—24no—
230C2×Q8—62no—
231C2×Q8—122no—
232C2×Q8—122no—
233C2×Q8—121no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—62no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—121no—
240C22×C4C2×D1211yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4D3×D411yescharacteristic
243C2×C8—24no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—64no—
249C2×C8—62no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—62no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—62no—
263C4⋊C4—61no—
264C4⋊C4—122no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—242no—
272C42#23511yescharacteristic
273C42D3×D411yescharacteristic
274C42D3×D411yescharacteristic
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—
289Dic3—82no—
290Dic3—81no—
291Dic3—82no—
292Dic3—81no—
293C23#88211yescharacteristic
294Q8—124no—
295Q8—242no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—122no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4—121no—
312C2×C4—122no—
313C2×C4—121no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4#88211yescharacteristic
317C2×C4#88411yescharacteristic
318C2×C4#88411yescharacteristic
319C2×C4#93011yescharacteristic
320C2×C4#93711yescharacteristic
321C2×C4#94011yescharacteristic
322C8—24no—
323C8—42no—
324C8—64no—
325C2×C3#166012yes—
326C2×C3#166312yes—
327C2×C3#200511yescharacteristic
328C2×C3#200511yescharacteristic
329C2×C3#200611yescharacteristic
330C22#475212yes—
331C22#482211yescharacteristic
332C22#482612yes—
333C22#482711yescharacteristic
334C22#496011yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—81no—
342C4—81no—
343C4—242no—
344C4—241no—
345C4—241no—
346C4—242no—
347C3#1008711yescharacteristic, complemented, Sylow
348C2#6837212yes—
349C2#6837612yes—
350C2#7011911yescharacteristic
351C2#7038911yescharacteristic
352C2#7039211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#330C2
socle, prime radicalC23×C3#120
hypercenterC2×#330D3
hypocenterC3#10087
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
347. C3#10087semidirect

Maximal subgroups

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

subgroupindex
#830552
C2×#50642
C2×#47022
C2×C3×#3302
#698222
C2×#44652
#676592
#100873