Finite Group Databasebeta

← #10697330

Subgroups — #10697330

Order 768.

512 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 101 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#79272C211yesmaximal, characteristic, complemented
2#79027C211yesmaximal, characteristic
3#78528C211yesmaximal, characteristic
4#70789C211yesmaximal, characteristic
5#70456C211yesmaximal, characteristic, complemented
6#69868C211yesmaximal, characteristic, complemented
7C3×#1283C211yesmaximal, characteristic, complemented
8#10202—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5133—22no—
16#5114—22no—
17C2×#816—22no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4851—21no—
20#4850—22no—
21#4848—21no—
22#4842—21no—
23#4841—21no—
24#4722—22no—
25#4714—22no—
26#4694—21no—
27#4682—21no—
28#4681—21no—
29#4681—21no—
30#4555—22no—
31#4470—22no—
32#3246—31no—
33#3230—31no—
34#3141—31no—
35#1986—31no—
36#1981—31no—
37#1502—31no—
38#1283D311yescharacteristic, complemented
39C2×#235—21no—
40C22×Dic6—21no—
41C2×C3×M16D412yes—
42C22×C3×C8D412yes—
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930—21no—
46#926—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50#924—21no—
51#924—21no—
52C2×C4×Dic3—21no—
53C2×#204—22no—
54#875—21no—
55#874D411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#870—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869D411yescharacteristic
63#869D411yescharacteristic
64#869D411yescharacteristic
65#869D412yes—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C22⋊C8D412yes—
72C3×#97—21no—
73C3×C4×C8—21no—
74C3×C4×C8—21no—
75C3⋊#95—21no—
76#821—24no—
77#821—22no—
78#818—24no—
79#817—22no—
80#816—24no—
81#816—24no—
82#816—22no—
83#816—44no—
84#533—31no—
85#532—31no—
86C2×C4⋊Q8—31no—
87#500—61no—
88#489—62no—
89#487—61no—
90#487—61no—
91#486—61no—
92#484—62no—
93#477—62no—
94#476—61no—
95#475—61no—
96#465—62no—
97#461—61no—
98#460—61no—
99#431C2×D311yescharacteristic
100C2×C4⋊C8C2×D311yescharacteristic
101#417—62no—
102C2×#103—62no—
103C4×M16C2×D311yescharacteristic
104C4×C4⋊C4—31no—
105#339—62no—
106#327—62no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C2×#67—41no—
111C22×Dic3—41no—
112C22×Dic3—41no—
113#235—42no—
114#235—42no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—22no—
123C2×Dic6—42no—
124C2×Dic6—42no—
125C2×Dic6—41no—
126C3×M16—22no—
127C3×M16—22no—
128C2×C3×C8—22no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C2×C3×C8—42no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#217—41no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139C4⋊Dic3—22no—
140C4⋊Dic3—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153C4×Dic3—41no—
154C4×Dic3—42no—
155#204—82no—
156C2×#41—61no—
157C22×Q8—61no—
158C2×SD16—62no—
159C2×M16D1212yes—
160C22×C8D1212yes—
161C4⋊Q8—32no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8—31no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168#127—31no—
169#125—61no—
170#124—61no—
171C22⋊Q8—61no—
172C22⋊Q8—61no—
173#121—61no—
174#121—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×D4—61no—
178C4×D4—61no—
179#117—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C42—61no—
184C2×C42C22×D311yescharacteristic
185#107—64no—
186#106—62no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190#103—64no—
191#103—62no—
192#103—64no—
193#103—124no—
194#102—64no—
195#102—62no—
196C22⋊C8D1212yes—
197#97—21no—
198C4×C8—21no—
199C4×C8—21no—
200#95—61no—
201C23×C3#12711yescharacteristic
202C22×D3—41no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#13512yes—
213C2×C3×C4#13712yes—
214#67—81no—
215#67—81no—
216C2×Dic3—42no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—42no—
220C2×Dic3—41no—
221C2×Dic3—82no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224D12—42no—
225D12—41no—
226D12—41no—
227D12—81no—
228C4×D3—82no—
229Dic6—42no—
230Dic6—44no—
231Dic6—42no—
232Dic6—41no—
233Dic6—41no—
234Dic6—81no—
235Dic6—81no—
236C3×C8—42no—
237C3×C8—42no—
238C3×C8—42no—
239#41—122no—
240#41—122no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—62no—
245C2×Q8—122no—
246C2×Q8—122no—
247C2×Q8—121no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—121no—
252C22×C4—121no—
253C22×C4—121no—
254C22×C4—121no—
255C22×C4C2×D1211yescharacteristic
256C22×C4C2×D1211yescharacteristic
257C22×C4C2×D1211yescharacteristic
258SD16—242no—
259M16—22no—
260M16—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—42no—
265C4⋊C4—62no—
266C4⋊C4—62no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—62no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C42—121no—
283C42—122no—
284C42D3×D411yescharacteristic
285C42D3×D411yescharacteristic
286C42D3×D411yescharacteristic
287C42#23911yescharacteristic
288C22×C3—21no—
289C22×C3—21no—
290C22×C3#39211yescharacteristic
291C22×C3#49511yescharacteristic
292C22×C3#49711yescharacteristic
293C2×D3—81no—
294C2×D3—81no—
295C2×D3—81no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4#45212yes—
300C3×C4#45311yescharacteristic
301C3×C4#45411yescharacteristic
302C3×C4#46412yes—
303Dic3—82no—
304Dic3—82no—
305Dic3—81no—
306Dic3—81no—
307Dic3—81no—
308C23—121no—
309C23#87411yescharacteristic
310Q8—124no—
311Q8—122no—
312Q8—122no—
313Q8—121no—
314Q8—121no—
315Q8—241no—
316Q8—241no—
317D4—122no—
318D4—121no—
319D4—121no—
320D4—241no—
321D4—241no—
322D4—241no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—122no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—242no—
332C2×C4—241no—
333C2×C4—241no—
334C2×C4—242no—
335C2×C4#88211yescharacteristic
336C2×C4#88211yescharacteristic
337C2×C4#88211yescharacteristic
338C2×C4#89511yescharacteristic
339C2×C4#89511yescharacteristic
340C2×C4#89511yescharacteristic
341C2×C4#90412yes—
342C2×C4#90912yes—
343C8—42no—
344C8—42no—
345C8—42no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3#200611yescharacteristic
349C2×C3#200911yescharacteristic
350C2×C3#201511yescharacteristic
351D3—161no—
352C22—21no—
353C22—21no—
354C22—241no—
355C22—241no—
356C22—241no—
357C22#467111yescharacteristic
358C22#469611yescharacteristic
359C22#471511yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—242no—
364C4—241no—
365C4—242no—
366C4—241no—
367C4—241no—
368C4#473312yes—
369C4#474711yescharacteristic
370C4#475311yescharacteristic
371C4#484512yes—
372C3#1020211yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—481no—
376C2#6867711yescharacteristic
377C2#6900811yescharacteristic
378C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1283D3
hypocenterC3#10202
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
1. #79272C2semidirect
5. #70456C2semidirect
6. #69868C2semidirect
7. C3×#1283C2semidirect
38. #1283D3semidirect
372. C3#10202semidirect

Maximal subgroups

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

subgroupindex
#792722
#790272
#785282
#707892
#704562
#698682
C3×#12832
#102023