Finite Group Databasebeta

← #10698271

Subgroups — #10698271

Order 768.

392 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 89 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#74219C211yesmaximal, characteristic
2#69828C211yesmaximal, characteristic
3#69822C211yesmaximal, characteristic
4#69555C211yesmaximal, characteristic
5C3×#1295C211yesmaximal, characteristic
6#67659C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic
8#10361—31nomaximal, Sylow
9C2×#869C2211yescharacteristic
10C2×C3×#97C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C3×#382C2211yescharacteristic
13C2×#816—21no—
14C2×#816—21no—
15#4982—21no—
16#4982—21no—
17#4936C2211yescharacteristic
18#4936C2211yescharacteristic
19C2×#803C2211yescharacteristic
20#4661—21no—
21#4661—21no—
22#4657—21no—
23#4657—21no—
24#4551—21no—
25#4551—21no—
26#4529—21no—
27#4529—21no—
28#2845—31no—
29#1993—31no—
30#1993—31no—
31#1960—31no—
32#1295—31no—
33#1295—31no—
34#1295D311yescharacteristic
35C22×Dic6—21no—
36C22×C3×C8—21no—
37C22×C3×C8—21no—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3D411yescharacteristic
46C2×C4⋊Dic3D411yescharacteristic
47C2×C4⋊Dic3D411yescharacteristic
48C2×C4⋊Dic3D411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C22×C3⋊C8—21no—
54#869—21no—
55#869—21no—
56#869—21no—
57#869Q812yes—
58C3×#97—21no—
59C3×#97—21no—
60C3×#97—21no—
61C3×#97—21no—
62C3×#95—21no—
63C3×#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70#816—42no—
71#816—42no—
72#803—21no—
73#803—21no—
74C2×C4⋊Q8—31no—
75C2×#103—61no—
76C2×#103—61no—
77C2×#97—31no—
78C2×#97C2×D311yescharacteristic
79C2×#97C2×D311yescharacteristic
80#398—61no—
81#398—61no—
82#398—61no—
83#398—61no—
84#398—61no—
85#398—61no—
86#382—31no—
87#382—31no—
88#382C2×D311yescharacteristic
89#340—61no—
90#340—61no—
91#340—61no—
92#340—61no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—41no—
103C2×Dic6—22no—
104C2×Dic6—41no—
105C2×Dic6—41no—
106C2×Dic6—41no—
107C2×Dic6—41no—
108C2×Dic6—41no—
109C2×C3×C8—22no—
110C2×C3×C8—22no—
111C2×C3×C8—41no—
112C2×C3×C8—41no—
113C2×C3×C8—41no—
114C2×C3×C8—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C42C2×Q811yescharacteristic
118C3×C42#4111yescharacteristic
119C3×C42#4111yescharacteristic
120C3×C42#4111yescharacteristic
121C4⋊Dic3—22no—
122C4⋊Dic3—22no—
123C4⋊Dic3—41no—
124C4⋊Dic3—41no—
125C4⋊Dic3—41no—
126C4⋊Dic3—41no—
127C4⋊Dic3—41no—
128C4⋊Dic3—41no—
129C4⋊Dic3—81no—
130C4⋊Dic3—81no—
131Dic3⋊C4—81no—
132Dic3⋊C4—81no—
133Dic3⋊C4—81no—
134Dic3⋊C4—81no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—41no—
137C2×C3⋊C8—41no—
138C22×Q8—61no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—61no—
142C4⋊Q8—32no—
143C4⋊Q8—61no—
144C4⋊Q8—61no—
145C4⋊Q8—61no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4D1211yescharacteristic
159C2×C4⋊C4D1211yescharacteristic
160C2×C42C22×D311yescharacteristic
161#103—122no—
162#103—122no—
163#97—21no—
164#97—21no—
165#97—21no—
166#97—21no—
167#97—61no—
168#97—61no—
169#95—21no—
170#95—21no—
171#95—61no—
172#95—61no—
173#95—61no—
174#95—61no—
175#95—61no—
176#95—61no—
177C23×C3#12011yescharacteristic
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4C22⋊Q811yescharacteristic
191C2×C3×C4C22⋊Q811yescharacteristic
192C2×C3×C4C22⋊Q811yescharacteristic
193C2×C3×C4#12411yescharacteristic
194C2×C3×C4#12411yescharacteristic
195C2×C3×C4#12411yescharacteristic
196C2×Dic3—42no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—81no—
210C2×Dic3—81no—
211Dic6—42no—
212Dic6—42no—
213Dic6—81no—
214Dic6—81no—
215C3×C8—42no—
216C3×C8—42no—
217C3⋊C8—42no—
218C2×Q8—62no—
219C2×Q8—121no—
220C2×Q8—121no—
221C2×Q8—121no—
222C2×Q8—121no—
223C2×Q8—121no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—121no—
231C22×C4C2×D1211yescharacteristic
232C22×C4D3×D411yescharacteristic
233C22×C4D3×D411yescharacteristic
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—62no—
241C2×C8—121no—
242C2×C8—121no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—62no—
246C4⋊C4—62no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—241no—
254C4⋊C4—241no—
255C4⋊C4—241no—
256C4⋊C4—241no—
257C4⋊C4—241no—
258C4⋊C4—241no—
259C42#23511yescharacteristic
260C42#23511yescharacteristic
261C42#23711yescharacteristic
262C42D3×Q811yescharacteristic
263C22×C3#39311yescharacteristic
264C22×C3#45311yescharacteristic
265C22×C3#45411yescharacteristic
266C22×C3#45511yescharacteristic
267C22×C3#45511yescharacteristic
268C22×C3#45611yescharacteristic
269C22×C3#45611yescharacteristic
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—81no—
277C3×C4—81no—
278Dic3—81no—
279Dic3—81no—
280Dic3—81no—
281Dic3—81no—
282Dic3—82no—
283C23#88211yescharacteristic
284Q8—122no—
285Q8—122no—
286Q8—241no—
287Q8—241no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—122no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—241no—
314C2×C4—241no—
315C2×C4#87511yescharacteristic
316C2×C4#88511yescharacteristic
317C2×C4#88511yescharacteristic
318C2×C4#89611yescharacteristic
319C2×C4#89611yescharacteristic
320C2×C4#89711yescharacteristic
321C8—42no—
322C8—42no—
323C8—122no—
324C2×C3#166111yescharacteristic
325C2×C3#166111yescharacteristic
326C2×C3#166211yescharacteristic
327C2×C3#166311yescharacteristic
328C2×C3#202711yescharacteristic
329C2×C3#202811yescharacteristic
330C2×C3#202811yescharacteristic
331C22#467311yescharacteristic
332C22#474611yescharacteristic
333C22#475211yescharacteristic
334C22#482211yescharacteristic
335C22#482511yescharacteristic
336C22#482611yescharacteristic
337C22#482711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—81no—
345C4—81no—
346C4—241no—
347C4—241no—
348C4—241no—
349C4—241no—
350C4—242no—
351C3#1036111yescharacteristic, complemented, Sylow
352C2#6835711yescharacteristic
353C2#6835811yescharacteristic
354C2#6837311yescharacteristic
355C2#6837611yescharacteristic
356C2#6876311yescharacteristic
357C2#6876911yescharacteristic
358C2#6899811yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1295C2
socle, prime radicalC23×C3#120
hypercenter#1295D3
hypocenterC3#10361
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
351. C3#10361semidirect

Maximal subgroups

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

subgroupindex
#742192
#698282
#698222
#695552
C3×#12952
#676592
#103613