Finite Group Databasebeta

← #10699224

Subgroups — #10699224

Order 768.

390 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 99 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#79271C211yesmaximal, characteristic
2C3×#1565C211yesmaximal, characteristic
3#70460C211yesmaximal, characteristic
4#70438C211yesmaximal, characteristic
5#69877C211yesmaximal, characteristic
6#69868C211yesmaximal, characteristic
7#67746C211yesmaximal, characteristic
8#10507—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C2×#824—21no—
15C2×#824—21no—
16C4×C4⋊Dic3C2211yescharacteristic
17#4924C2211yescharacteristic
18#4851—21no—
19#4851—21no—
20#4721—22no—
21#4681—21no—
22#4680—21no—
23C3×#335—21no—
24C3×#334—21no—
25#4555—21no—
26#4555—21no—
27#4551—22no—
28#4482—21no—
29#4481—21no—
30#3229—31no—
31#1988—31no—
32#1986—31no—
33#1966—31no—
34#1565D311yescharacteristic
35#1550—31no—
36#1303—31no—
37C22×Dic6—21no—
38C2×C3×M16D411yescharacteristic
39C2×C3×M16D411yescharacteristic
40C22×C3×C8—21no—
41C2×C3×C42C2311yescharacteristic
42#926—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3D412yes—
46#924—21no—
47#924—21no—
48C2×C4×Dic3—21no—
49C2×#208—21no—
50#870—21no—
51#869—21no—
52#869—21no—
53#869—21no—
54#869—21no—
55#869D411yescharacteristic
56#869D411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8D411yescharacteristic
62C3×C4⋊C8D411yescharacteristic
63C3×C22⋊C8—21no—
64C3×#97—21no—
65C3×C4×C8—21no—
66C3×C4×C8—21no—
67C3⋊#95—21no—
68#824—42no—
69#824—42no—
70#817—22no—
71#817—22no—
72#816—22no—
73#816—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76#533—31no—
77C2×C4⋊Q8—31no—
78#498—61no—
79#487—61no—
80#483—62no—
81#475—61no—
82#475—61no—
83#431C2×D311yescharacteristic
84#423—31no—
85C2×C4⋊C8C2×D311yescharacteristic
86C2×#101—61no—
87C2×#101—61no—
88C4×M16C2×D311yescharacteristic
89C4×C4⋊C4—31no—
90#340—62no—
91#339—61no—
92#339—61no—
93#335—21no—
94#334—21no—
95#333—61no—
96#332—61no—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4111yescharacteristic
100C22×Dic3—22no—
101C22×Dic3—41no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C2×Dic6—41no—
105C2×Dic6—41no—
106C2×Dic6—41no—
107C2×Dic6—41no—
108C2×Dic6—41no—
109C2×Dic6—41no—
110C3×M16—21no—
111C3×M16—21no—
112C3×M16—21no—
113C3×M16SD1612yes—
114C2×C3×C8—21no—
115C2×C3×C8—21no—
116C2×C3×C8—21no—
117C2×C3×C8—21no—
118C2×C3×C8—42no—
119C2×C3×C8—41no—
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123C3×C42#4111yescharacteristic
124#217—41no—
125C4⋊Dic3—22no—
126C4⋊Dic3—22no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—22no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135Dic3⋊C4—41no—
136Dic3⋊C4—41no—
137C4×Dic3—41no—
138C4×Dic3—41no—
139C4×Dic3—41no—
140#208—41no—
141#208—41no—
142C2×C3⋊C8—41no—
143C22×Q8—61no—
144C2×M16—61no—
145C2×M16D1211yescharacteristic
146C2×M16D1211yescharacteristic
147C22×C8—21no—
148C4⋊Q8—31no—
149C4⋊Q8—31no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154#125—61no—
155C22⋊Q8—61no—
156C22⋊Q8—61no—
157C4×Q8—61no—
158C4×Q8—61no—
159#117—61no—
160C2×C4⋊C4—32no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C42—61no—
164C2×C42C22×D311yescharacteristic
165#106—62no—
166#106—62no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8#6711yescharacteristic
172C4⋊C8#6711yescharacteristic
173#103—62no—
174#103—62no—
175#101—122no—
176#101—122no—
177C22⋊C8—21no—
178#97—21no—
179C4×C8—21no—
180C4×C8—21no—
181#95—61no—
182C23×C3#12111yescharacteristic
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12111yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12311yescharacteristic
190C2×C3×C4#12311yescharacteristic
191C2×C3×C4#12411yescharacteristic
192C2×C3×C4C2×SD1611yescharacteristic
193C2×C3×C4#13511yescharacteristic
194C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
195C2×C3×C4#13711yescharacteristic
196C2×Dic3—42no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204Dic6—41no—
205Dic6—41no—
206Dic6—81no—
207Dic6—81no—
208Dic6—81no—
209Dic6—81no—
210Dic6—81no—
211C3×C8—22no—
212C3×C8—41no—
213C3×C8—41no—
214C3×C8—41no—
215C3×C8—81no—
216C3⋊C8—81no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×Q8—121no—
220C2×Q8—121no—
221C2×Q8—121no—
222C2×Q8—121no—
223C2×Q8—121no—
224C2×Q8—121no—
225C22×C4—62no—
226C22×C4—121no—
227C22×C4C2×D1211yescharacteristic
228C22×C4D3×D411yescharacteristic
229C22×C4#23911yescharacteristic
230M16—21no—
231M16—21no—
232M16—21no—
233M16—121no—
234M16—121no—
235M16#20412yes—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—42no—
241C2×C8—41no—
242C2×C8—121no—
243C4⋊C4—62no—
244C4⋊C4—62no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—62no—
249C4⋊C4—61no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C22⋊C4—121no—
256C42—121no—
257C42—121no—
258C42—121no—
259C42#23511yescharacteristic
260C42D3×D411yescharacteristic
261C42#23911yescharacteristic
262C42C2×#6711yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3#39411yescharacteristic
266C22×C3#46111yescharacteristic
267C22×C3#46311yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4#45411yescharacteristic
272C3×C4#45611yescharacteristic
273C3×C4#46011yescharacteristic
274C3×C4#46411yescharacteristic
275C3×C4#46511yescharacteristic
276C3×C4#46811yescharacteristic
277Dic3—82no—
278Dic3—81no—
279Dic3—82no—
280Dic3—81no—
281C23#89511yescharacteristic
282Q8—121no—
283Q8—121no—
284Q8—241no—
285Q8—241no—
286Q8—241no—
287Q8—241no—
288Q8—241no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—122no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4—241no—
299C2×C4—241no—
300C2×C4#89411yescharacteristic
301C2×C4#89411yescharacteristic
302C2×C4#89511yescharacteristic
303C2×C4C2×#20411yescharacteristic
304C2×C4#90411yescharacteristic
305C2×C4#90811yescharacteristic
306C2×C4#90911yescharacteristic
307C2×C4#93011yescharacteristic
308C2×C4#93711yescharacteristic
309C2×C4#94711yescharacteristic
310C8—22no—
311C8—41no—
312C8—41no—
313C8—41no—
314C8—81no—
315C8—241no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3#203811yescharacteristic
319C2×C3#204911yescharacteristic
320C2×C3#205511yescharacteristic
321C22—21no—
322C22—21no—
323C22#484111yescharacteristic
324C22#484411yescharacteristic
325C22#499111yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—242no—
330C4—241no—
331C4—241no—
332C4—242no—
333C4#484511yescharacteristic
334C4#484811yescharacteristic
335C4#511411yescharacteristic
336C4#513411yescharacteristic
337C4#520111yescharacteristic
338C4#520511yescharacteristic
339C3#1050711yescharacteristic, complemented, Sylow
340C2—21no—
341C2—21no—
342C2#7048211yescharacteristic
343C2#7067811yescharacteristic
344C2#7073711yescharacteristic
subgroupG/H
centerC22#4991
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1565C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1565D3
hypocenterC3#10507
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
339. C3#10507semidirect

Maximal subgroups

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

subgroupindex
#792712
C3×#15652
#704602
#704382
#698772
#698682
#677462
#105073