Finite Group Databasebeta

← #10698380

Subgroups — #10698380

Order 768.

390 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 73 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#73405C211yesmaximal, characteristic, complemented
2#70835C211yesmaximal, characteristic
3#69867C211yesmaximal, characteristic
4#69432C211yesmaximal, characteristic
5C3×#1300C211yesmaximal, characteristic, complemented
6#67749C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#10380—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C3×#432C2211yescharacteristic
11C3×C4×M16C2211yescharacteristic
12C3×#376C2211yescharacteristic
13C2×#825—21no—
14C2×#825—21no—
15#4937C2211yescharacteristic
16C4×C4⋊Dic3C2211yescharacteristic
17#4933—21no—
18#4928—21no—
19#4926C2211yescharacteristic
20#4661—21no—
21#4657—21no—
22#4654—21no—
23#4647—21no—
24#4470—21no—
25#4470—21no—
26#4451—21no—
27#4451—21no—
28#2606—31no—
29#1980—31no—
30#1514—31no—
31#1505—31no—
32#1300D311yescharacteristic, complemented
33#1288—31no—
34#1285—31no—
35C2×#235—21no—
36C2×C3×M16—21no—
37C2×C3×C42—21no—
38C2×C3×C42C2311yescharacteristic
39#930—21no—
40#930—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3D411yescharacteristic
49C2×C4×Dic3D411yescharacteristic
50#875—21no—
51#874Q811yescharacteristic
52C4×D12—21no—
53C4×D12—21no—
54#869Q811yescharacteristic
55C3×C4⋊C8—21no—
56C3×C22⋊C8—21no—
57C3×#97—21no—
58C3×#97—21no—
59C3×C4×C8—21no—
60C3×#95—21no—
61C3×#95D411yescharacteristic
62C3×#95D411yescharacteristic
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95D411yescharacteristic
69C3⋊#95D411yescharacteristic
70#830—21no—
71#825—42no—
72#825—42no—
73C4⋊(C3⋊C8)—21no—
74#803—21no—
75#532—31no—
76#432—31no—
77#432C2×D311yescharacteristic
78C2×#104—61no—
79C2×#104—61no—
80C4×M16C2×D311yescharacteristic
81#398—61no—
82#398—61no—
83#389—61no—
84#389—61no—
85#387—61no—
86#387—61no—
87#383—31no—
88C4×C4⋊C4—31no—
89#376C2×D311yescharacteristic
90#327—61no—
91#327—61no—
92#327—61no—
93#327—61no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C2×#67—41no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104#235—41no—
105#235—41no—
106#235—41no—
107#235—41no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×C4×D3—41no—
112C2×Dic6—21no—
113C3×M16—22no—
114C3×M16—41no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C3×C42—42no—
118C3×C42C2×Q811yescharacteristic
119C3×C42#4111yescharacteristic
120C3×C42#4111yescharacteristic
121C3×C42#4111yescharacteristic
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—41no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—42no—
127C4⋊Dic3—42no—
128C4⋊Dic3—42no—
129Dic3⋊C4—42no—
130Dic3⋊C4—42no—
131Dic3⋊C4—81no—
132Dic3⋊C4—81no—
133C4×Dic3—41no—
134C4×Dic3—41no—
135C2×C3⋊C8—41no—
136C2×#41—61no—
137C2×M16—21no—
138C4⋊Q8—31no—
139#127—31no—
140#124—61no—
141#121—61no—
142#121—61no—
143C4×D4—61no—
144C4×D4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C42—21no—
153C2×C42—31no—
154C2×C42—31no—
155C2×C42C22×D311yescharacteristic
156C4⋊C8—21no—
157C4⋊C8—61no—
158#104—122no—
159#104—122no—
160C22⋊C8—21no—
161C22⋊C8—61no—
162#97—21no—
163#97—21no—
164#97—61no—
165C4×C8—21no—
166#95—21no—
167#95—31no—
168#95—31no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95—61no—
174#95D1211yescharacteristic
175#95D1211yescharacteristic
176C23×C3#12011yescharacteristic
177C22×D3—41no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—22no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4C22⋊Q811yescharacteristic
189C2×C3×C4C22⋊Q811yescharacteristic
190C2×C3×C4C22⋊Q811yescharacteristic
191C2×C3×C4#12411yescharacteristic
192C2×C3×C4#12411yescharacteristic
193C2×C3×C4#12411yescharacteristic
194#67—81no—
195#67—81no—
196C2×Dic3—22no—
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—
209D12—41no—
210D12—41no—
211D12—81no—
212D12—81no—
213C4×D3—81no—
214C4×D3—81no—
215Dic6—41no—
216Dic6—41no—
217C3×C8—42no—
218C3×C8—81no—
219C3⋊C8—81no—
220#41—121no—
221#41—121no—
222#41—121no—
223#41—121no—
224C2×Q8—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—121no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—121no—
236C22×C4C2×D1211yescharacteristic
237C22×C4D3×D411yescharacteristic
238C22×C4D3×D411yescharacteristic
239M16—22no—
240M16—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—121no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—122no—
247C4⋊C4—122no—
248C4⋊C4—122no—
249C4⋊C4—122no—
250C4⋊C4—122no—
251C4⋊C4—241no—
252C4⋊C4—241no—
253C22⋊C4—121no—
254C22⋊C4—121no—
255C42—42no—
256C42—121no—
257C42—121no—
258C42#23511yescharacteristic
259C42#23511yescharacteristic
260C42#23711yescharacteristic
261C42D3×Q811yescharacteristic
262C22×C3—21no—
263C22×C3—21no—
264C22×C3#39311yescharacteristic
265C22×C3#45311yescharacteristic
266C22×C3#45411yescharacteristic
267C2×D3—81no—
268C2×D3—81no—
269C2×D3—81no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—42no—
277C3×C4—81no—
278Dic3—42no—
279Dic3—81no—
280Dic3—81no—
281Dic3—81no—
282Dic3—81no—
283C23—121no—
284C23#88211yescharacteristic
285Q8—121no—
286Q8—121no—
287D4—121no—
288D4—121no—
289D4—241no—
290D4—241no—
291D4—241no—
292D4—241no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—62no—
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—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—241no—
317C2×C4—241no—
318C2×C4#87511yescharacteristic
319C2×C4#88511yescharacteristic
320C2×C4#88511yescharacteristic
321C2×C4#89611yescharacteristic
322C2×C4#89611yescharacteristic
323C2×C4#89711yescharacteristic
324C8—42no—
325C8—81no—
326C8—241no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3#202711yescharacteristic
330C2×C3#203111yescharacteristic
331C2×C3#203111yescharacteristic
332D3—161no—
333C22—21no—
334C22—21no—
335C22—241no—
336C22—241no—
337C22—241no—
338C22#467311yescharacteristic
339C22#482211yescharacteristic
340C22#482711yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—81no—
349C4—122no—
350C4—241no—
351C4—241no—
352C4—241no—
353C4—241no—
354C3#1038011yescharacteristic, complemented, Sylow
355C2—21no—
356C2—21no—
357C2—481no—
358C2#6876311yescharacteristic
359C2#6877911yescharacteristic
360C2#6904211yescharacteristic
subgroupG/H
centerC22#4673
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#393
prime radical#5477C22
hypercenter#1300D3
hypocenterC3#10380
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
1. #73405C2semidirect
5. C3×#1300C2semidirect
6. #67749C2semidirect
7. #67666C2semidirect
32. #1300D3semidirect
354. C3#10380semidirect

Maximal subgroups

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

subgroupindex
#734052
#708352
#698672
#694322
C3×#13002
#677492
#676662
#103803