Finite Group Databasebeta

← #10696864

Subgroups — #10696864

Order 768.

506 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 83 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#83213C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic, complemented
3#79349C211yesmaximal, characteristic
4C3×#1556C211yesmaximal, characteristic, complemented
5#70828C211yesmaximal, characteristic, complemented
6#70806C211yesmaximal, characteristic, complemented
7#67751C211yesmaximal, characteristic, complemented
8#10136—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#875C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5227—22no—
15#5089—22no—
16#5089—22no—
17#5085—21no—
18#5083—22no—
19#5079—21no—
20C2×#810—21no—
21C2×#809—21no—
22#4925C2211yescharacteristic
23C2×#803C2211yescharacteristic
24#4844—22no—
25#4738—21no—
26#4732—21no—
27#4714—21no—
28#4712—21no—
29C3×#332—22no—
30#4468—22no—
31#4464—22no—
32#3213—31no—
33#3206—31no—
34C2×#489—31no—
35#1556D311yescharacteristic, complemented
36#1530—31no—
37#1469—31no—
38#1302—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C22×D12D411yescharacteristic
42C22×Dic6D411yescharacteristic
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#215—24no—
51#930—21no—
52#930—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C22×C3⋊C8—21no—
56C2×D24—22no—
57C2×#204—22no—
58#875—21no—
59#875—21no—
60#875—21no—
61#875—21no—
62#875—21no—
63#874D411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#97—21no—
72C3×#97—21no—
73#835—24no—
74#830D412yes—
75#821—22no—
76#816—22no—
77#810—24no—
78#809—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#532—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#124—31no—
86#496—61no—
87#492—61no—
88#489—61no—
89#489—62no—
90#489—62no—
91#487—61no—
92#463—62no—
93#461—62no—
94#448—61no—
95#448—62no—
96#447—61no—
97#432C2×D311yescharacteristic
98#432C2×D311yescharacteristic
99#431—31no—
100C2×#103—61no—
101C2×#102—61no—
102C2×#97—31no—
103#332—22no—
104#329—62no—
105#323—62no—
106C23×D3—21no—
107C2×C3×Q8—22no—
108C2×C3×Q8—41no—
109C2×C3×Q8—82no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C2×#67—41no—
115C22×Dic3—21no—
116#235—82no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—42no—
123C2×C4×D3—41no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—42no—
128C2×C3×C8—22no—
129C2×C3×C8—41no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138#215—84no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—42no—
150D24—82no—
151#204—82no—
152C2×#41—61no—
153C22×Q8—21no—
154C22×Q8—31no—
155C22×D4—31no—
156C2×SD16—62no—
157C2×SD16—64no—
158C2×D8—62no—
159C22×C8—61no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—31no—
164C4⋊Q8D1212yes—
165#127—31no—
166#124—61no—
167#124—61no—
168#124—61no—
169#124—61no—
170#124—61no—
171#121—61no—
172#121—61no—
173C4×D4—61no—
174C4×D4—61no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—61no—
179C2×C42C22×D311yescharacteristic
180C4⋊C8—21no—
181C4⋊C8—61no—
182C4⋊C8#6712yes—
183#103—64no—
184#103—62no—
185#103—64no—
186#102—64no—
187#102—62no—
188C22⋊C8—21no—
189C22⋊C8—21no—
190C22⋊C8—32no—
191#97—21no—
192#97—21no—
193#97—32no—
194#97—61no—
195C4×C8—61no—
196C23×C3#12111yescharacteristic
197C22×D3—41no—
198C22×D3—41no—
199C22×D3—42no—
200C22×D3—81no—
201C22×D3—81no—
202C3×Q8—44no—
203C3×Q8—82no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—82no—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12711yescharacteristic
215#67—82no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—81no—
219D12—42no—
220D12—42no—
221D12—42no—
222D12—42no—
223D12—81no—
224C4×D3—82no—
225Dic6—42no—
226Dic6—42no—
227Dic6—81no—
228C3×C8—42no—
229C3×C8—81no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232C24—61no—
233#41—242no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8—61no—
237C2×Q8—61no—
238C2×Q8—61no—
239C2×Q8—82no—
240C2×Q8—122no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—122no—
247C2×D4—121no—
248C22×C4—41no—
249C22×C4—61no—
250C22×C4—121no—
251C22×C4#23511yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4D3×D411yescharacteristic
254SD16—242no—
255SD16—244no—
256D8—242no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—62no—
260C2×C8—62no—
261C2×C8—122no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C22⋊C4—121no—
269C22⋊C4—121no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C42C2×D1211yescharacteristic
275C42D3×D411yescharacteristic
276C42D3×D411yescharacteristic
277C42C2×#6711yescharacteristic
278C22×C3—21no—
279C22×C3—21no—
280C22×C3#39211yescharacteristic
281C22×C3#47112yes—
282C2×D3—81no—
283C2×D3—81no—
284C2×D3—81no—
285C2×D3—81no—
286C2×D3—81no—
287C2×D3—81no—
288C2×D3—162no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—82no—
295C3×C4#45512yes—
296Dic3—81no—
297Dic3—81no—
298C23—121no—
299C23—121no—
300C23—122no—
301C23—241no—
302C23—241no—
303C23#88411yescharacteristic
304Q8—44no—
305Q8—82no—
306Q8—122no—
307Q8—122no—
308Q8—241no—
309D4—122no—
310D4—122no—
311D4—122no—
312D4—122no—
313D4—241no—
314D4—242no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—242no—
323C2×C4—241no—
324C2×C4#88211yescharacteristic
325C2×C4#88211yescharacteristic
326C2×C4#88411yescharacteristic
327C2×C4#93011yescharacteristic
328C2×C4#93711yescharacteristic
329C2×C4#94011yescharacteristic
330C8—42no—
331C8—81no—
332C8—122no—
333C8—122no—
334C2×C3—22no—
335C2×C3#200411yescharacteristic
336C2×C3#200811yescharacteristic
337C2×C3#201911yescharacteristic
338D3—161no—
339D3—161no—
340C22—21no—
341C22—21no—
342C22—241no—
343C22—241no—
344C22—241no—
345C22—241no—
346C22—241no—
347C22—241no—
348C22—482no—
349C22#475412yes—
350C22#496011yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—82no—
357C4—241no—
358C4—241no—
359C4#520212yes—
360C3#1013611yescharacteristic, complemented, Sylow
361C2—22no—
362C2—481no—
363C2—481no—
364C2#7011711yescharacteristic
365C2#7033111yescharacteristic
366C2#7034411yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#392
prime radical#79349C2
hypercenter#1556D3
hypocenterC3#10136
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
1. #83213C2semidirect
2. C2×#5089C2semidirect
4. C3×#1556C2semidirect
5. #70828C2semidirect
6. #70806C2semidirect
7. #67751C2semidirect
35. #1556D3semidirect
360. C3#10136semidirect

Maximal subgroups

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

subgroupindex
#832132
C2×#50892
#793492
C3×#15562
#708282
#708062
#677512
#101363