Finite Group Databasebeta

← #10804129

Subgroups — #10804129

Order 768.

1570 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 1570 conjugacy classes of subgroups — too many to draw. Only its 243 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
1C22×#886C211yesmaximal, characteristic
2#73817C212yesmaximal, complemented
3C3×#1819C211yesmaximal, characteristic, complemented
4#71542C211yesmaximal, characteristic, complemented
5#69498C212yesmaximal
6#68954C218yesmaximal, complemented
7#21747—31nomaximal, Sylow
8C23×#67C2211yescharacteristic
9C24×Dic3C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×(C2×D3)⋊C4C412yescomplemented
12C22×C4⋊Dic3C412yescomplemented
13C2×#886—24no—
14C2×#886—24no—
15C2×#886C414yescomplemented
16C2×#886C2214yes—
17C2×#877C2214yescomplemented
18C3×#379C2212yes—
19#5247C2212yes—
20C2×#806C2218yes—
21#4943C2214yescomplemented
22#4942—24no—
23Dic3×C22⋊C4—24no—
24C3×#342C2214yescomplemented
25#4526C2214yescomplemented
26C22×#123—31no—
27#2280—32no—
28#1899—38no—
29#1819—31no—
30#1819D311yescharacteristic, complemented
31#1781—32no—
32C25×C3C2311yescharacteristic
33C24×D3C2×C411yescharacteristic
34C23×C3×C4C2×C412yes—
35C22×#67—24no—
36C22×#67—44no—
37C22×#67C2×C412yes—
38C22×#67C2312yes—
39C23×Dic3—44no—
40C23×Dic3C2×C411yescharacteristic
41C23×Dic3C2×C412yes—
42C23×Dic3C2×C412yes—
43C23×Dic3D414yes—
44C23×Dic3D414yes—
45C23×Dic3C2312yes—
46C2×C3×C22⋊C4—24no—
47C2×C3×C22⋊C4—24no—
48C2×C3×C22⋊C4C2×C412yes—
49C2×C3×C22⋊C4C2314yes—
50C2×C3×C22⋊C4C2312yes—
51C2×#217—24no—
52C2×#217C2314yes—
53C2×(C2×D3)⋊C4—22no—
54C2×(C2×D3)⋊C4—24no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4C2×C418yes—
57C2×C4⋊Dic3—22no—
58C2×C4⋊Dic3—24no—
59C2×C4⋊Dic3—22no—
60C2×C4⋊Dic3C2×C418yes—
61C2×Dic3⋊C4—24no—
62C2×C4×Dic3—24no—
63#886—24no—
64#886—28no—
65#886—24no—
66#886—28no—
67#886—24no—
68#886—24no—
69#877—48no—
70C3⋊#95—28no—
71#806—48no—
72C23×D4—31no—
73C24×C4—31no—
74C2×#123—34no—
75C2×#123—34no—
76C2×#123—64no—
77C2×#123—64no—
78C2×#117—34no—
79C22×C4⋊C4—32no—
80C22×C22⋊C4—32no—
81C22×C22⋊C4C2×D311yescharacteristic
82C2×#99—38no—
83#381—34no—
84#380—64no—
85#379—32no—
86#379C2×D312yes—
87C4×C22⋊C4—64no—
88#342—34no—
89#342C2×D314yescomplemented
90C24×C3—24no—
91C24×C3—44no—
92C24×C3C22×C411yescharacteristic
93C24×C3C22×C411yescharacteristic
94C24×C3C2×D414yes—
95C24×C3C2411yescharacteristic
96C23×D3—22no—
97C23×D3—42no—
98C23×D3—44no—
99C23×D3C22×C412yes—
100C22×C3×C4—22no—
101C22×C3×C4—22no—
102C22×C3×C4—24no—
103C22×C3×C4—44no—
104C22×C3×C4C22×C414yes—
105C2×#67—24no—
106C2×#67—24no—
107C2×#67—28no—
108C2×#67—44no—
109C2×#67—416no—
110C22×Dic3—28no—
111C22×Dic3—22no—
112C22×Dic3—24no—
113C22×Dic3—44no—
114C22×Dic3—44no—
115C22×Dic3—48no—
116C22×Dic3—48no—
117C22×Dic3—42no—
118C22×Dic3C22⋊C412yes—
119C22×Dic3C22⋊C414yes—
120C22×Dic3C22⋊C412yes—
121C22×Dic3C22⋊C412yes—
122C22×Dic3C22⋊C414yes—
123C22×Dic3C22⋊C412yes—
124C22×Dic3C22×C412yes—
125C22×Dic3C22×C414yes—
126C22×Dic3C2×D414yes—
127C22×Dic3C2×D412yes—
128C22×Dic3C2×D412yes—
129C3×C22⋊C4—24no—
130C3×C22⋊C4—24no—
131C3×C22⋊C4—28no—
132C3×C22⋊C4—48no—
133#217—28no—
134#217—48no—
135(C2×D3)⋊C4—28no—
136(C2×D3)⋊C4—28no—
137C4⋊Dic3—28no—
138C4⋊Dic3—28no—
139Dic3⋊C4—48no—
140C4×Dic3—48no—
141C25—31no—
142C25C22×D311yescharacteristic
143C22×D4—32no—
144C22×D4—32no—
145C22×D4—64no—
146C22×D4—124no—
147C23×C4—32no—
148C23×C4—32no—
149C23×C4—32no—
150C23×C4—34no—
151C23×C4—34no—
152C23×C4—31no—
153C23×C4—124no—
154C23×C4C4×D312yes—
155#123—64no—
156#123—68no—
157#123—64no—
158#123—68no—
159#123—64no—
160#123—64no—
161#117—128no—
162C2×C4⋊C4—38no—
163C2×C4⋊C4—64no—
164C2×C4⋊C4—62no—
165C2×C4⋊C4—62no—
166C2×C4⋊C4—64no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—38no—
170C2×C22⋊C4—34no—
171C2×C22⋊C4—62no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—64no—
174C2×C22⋊C4—64no—
175C2×C22⋊C4C4×D312yes—
176C2×C22⋊C4C22×D314yes—
177C2×C22⋊C4C22×D312yes—
178C2×C42—64no—
179#99—128no—
180#95—68no—
181C23×C3—22no—
182C23×C3—24no—
183C23×C3—22no—
184C23×C3—28no—
185C23×C3—28no—
186C23×C3—42no—
187C23×C3—44no—
188C23×C3—42no—
189C23×C3—416no—
190C23×C3C2×C22⋊C412yes—
191C23×C3C2×C22⋊C412yes—
192C23×C3C23×C411yescharacteristic
193C23×C3C22×D412yes—
194C23×C3H(5,𝔽2)14yes—
195C22×D3—22no—
196C22×D3—24no—
197C22×D3—44no—
198C22×D3—44no—
199C22×D3—44no—
200C22×D3—44no—
201C22×D3—88no—
202C2×C3×C4—24no—
203C2×C3×C4—44no—
204C2×C3×C4—44no—
205C2×C3×C4—44no—
206C2×C3×C4—48no—
207C2×C3×C4—44no—
208#67—48no—
209#67—48no—
210C2×Dic3—28no—
211C2×Dic3—24no—
212C2×Dic3—24no—
213C2×Dic3—28no—
214C2×Dic3—22no—
215C2×Dic3—24no—
216C2×Dic3—42no—
217C2×Dic3—44no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220C2×Dic3—44no—
221C2×Dic3—42no—
222C2×Dic3—44no—
223C2×Dic3—48no—
224C2×Dic3—44no—
225C2×Dic3C2×C22⋊C414yes—
226C2×Dic3C2×C22⋊C414yes—
227C24—24no—
228C24—32no—
229C24—44no—
230C24—62no—
231C24—122no—
232C24—124no—
233C24C2×C4×D311yescharacteristic
234C24C2×C4×D311yescharacteristic
235C24D3×D414yes—
236C24C23×D311yescharacteristic
237C2×D4—64no—
238C2×D4—64no—
239C2×D4—68no—
240C2×D4—124no—
241C2×D4—1216no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—34no—
246C22×C4—34no—
247C22×C4—32no—
248C22×C4—32no—
249C22×C4—32no—
250C22×C4—34no—
251C22×C4—32no—
252C22×C4—32no—
253C22×C4—34no—
254C22×C4—32no—
255C22×C4—32no—
256C22×C4—44no—
257C22×C4—68no—
258C22×C4—62no—
259C22×C4—64no—
260C22×C4—128no—
261C22×C4—128no—
262C22×C4—124no—
263C22×C4—124no—
264C22×C4—122no—
265C22×C4C2×C4×D314yes—
266C4⋊C4—68no—
267C4⋊C4—68no—
268C4⋊C4—128no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—28no—
272C22⋊C4—48no—
273C22⋊C4—68no—
274C22⋊C4—68no—
275C22⋊C4—68no—
276C22⋊C4—128no—
277C42—128no—
278C22×C3—28no—
279C22×C3—22no—
280C22×C3—24no—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—22no—
284C22×C3—24no—
285C22×C3—44no—
286C22×C3—416no—
287C22×C3—44no—
288C22×C3#41014yes—
289C22×C3C22×C22⋊C411yescharacteristic
290C22×C3#51812yes—
291C22×C3#53414yes—
292C2×D3—42no—
293C2×D3—44no—
294C2×D3—88no—
295C2×D3—84no—
296C3×C4—44no—
297C3×C4—84no—
298Dic3—24no—
299Dic3—24no—
300Dic3—42no—
301Dic3—84no—
302C23—24no—
303C23—22no—
304C23—28no—
305C23—28no—
306C23—22no—
307C23—42no—
308C23—44no—
309C23—42no—
310C23—416no—
311C23—62no—
312C23—64no—
313C23—124no—
314C23—124no—
315C23—124no—
316C23—124no—
317C23—248no—
318C23D3×C22⋊C412yes—
319C23D3×C22⋊C412yes—
320C23C22×C4×D311yescharacteristic
321C23C2×D3×D412yes—
322C23#100914yes—
323D4—128no—
324D4—128no—
325C2×C4—24no—
326C2×C4—34no—
327C2×C4—34no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—48no—
332C2×C4—44no—
333C2×C4—64no—
334C2×C4—64no—
335C2×C4—68no—
336C2×C4—68no—
337C2×C4—62no—
338C2×C4—64no—
339C2×C4—122no—
340C2×C4—124no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—124no—
344C2×C4—122no—
345C2×C4—124no—
346C2×C4—128no—
347C2×C4—124no—
348C2×C3—24no—
349C2×C3—22no—
350C2×C3—24no—
351C2×C3—42no—
352C2×C3#230811yescharacteristic
353C2×C3C2×#41012yes—
354D3—82no—
355C22—28no—
356C22—24no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—24no—
361C22—22no—
362C22—44no—
363C22—416no—
364C22—44no—
365C22—124no—
366C22—122no—
367C22—244no—
368C22—248no—
369C22#474014yes—
370C22C2×D3×C22⋊C411yescharacteristic
371C22#552112yes—
372C22#565314yes—
373C4—44no—
374C4—64no—
375C4—64no—
376C4—84no—
377C4—122no—
378C4—244no—
379C3#2174711yescharacteristic, complemented, Sylow
380C2—24no—
381C2—24no—
382C2—22no—
383C2—42no—
384C2—242no—
385C2#7411911yescharacteristic
386C2C2×#474012yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1819C2
socleC22×C3C22×C22⋊C4
prime radicalC23×#67C22
cosocleC24×C3C24
hypercenter#1819D3
hypocenterC3#21747
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
2. #73817C2semidirect
3. C3×#1819C2semidirect
4. #71542C2semidirect
6. #68954C2semidirect
11. C22×(C2×D3)⋊C4C4semidirect
12. C22×C4⋊Dic3C4semidirect
15. C2×#886C4semidirect
17. C2×#877C22semidirect
21. #4943C22semidirect
24. C3×#342C22semidirect
25. #4526C22semidirect
30. #1819D3semidirect
89. #342C2×D3semidirect
379. C3#21747semidirect

Maximal subgroups

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

subgroupindex
C22×#8862
#738172
C3×#18192
#715422
#694982
#689542
#217473