Finite Group Databasebeta

← #10710814

Subgroups — #10710814

Order 768.

810 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C3×#3508C211yesmaximal, characteristic
2#71860C212yesmaximal, complemented
3C2×#4545C212yesmaximal, complemented
4#71653C212yesmaximal, complemented
5#12727—31nomaximal, Sylow
6C3×#559—22no—
7C3×#550—22no—
8C3×#546—24no—
9C3×#540—22no—
10C3×#535C411yescharacteristic, complemented
11C3×#535C2211yescharacteristic
12C3×#533C411yescharacteristic, complemented
13C3×#532—22no—
14C2×C3×#127C2211yescharacteristic
15C3×#518C411yescharacteristic, complemented
16C3×#518C2211yescharacteristic
17C3×C4×#41C411yescharacteristic, complemented
18C2×#834C2214yes—
19#4545—22no—
20#4545—22no—
21#4545—22no—
22#4541—22no—
23#4527—24no—
24#3508D311yescharacteristic
25#2127—32no—
26C2×#353—32no—
27#1828—32no—
28C2×C3×#41—24no—
29C2×C3×#41C2×C411yescharacteristic
30C22×C3×D4—22no—
31C22×C3×D4C2×C411yescharacteristic
32C22×C3×D4C2311yescharacteristic
33C3×C4⋊Q8—21no—
34C3×#127—21no—
35C3×#127—22no—
36C3×#127D412yes—
37C3×#127D412yes—
38C3×#127D412yes—
39C3×#125—21no—
40C3×#124—24no—
41C3×#124—21no—
42C3×#124D412yes—
43C3×#123—24no—
44C3×#123—24no—
45C3×C22⋊Q8—22no—
46C3×#121—24no—
47C3×#121—24no—
48C3×#121—24no—
49C3×#121—22no—
50C3×#121—22no—
51C3×#120—22no—
52C3×#120—22no—
53C3×C4×Q8—21no—
54C3×C4×Q8—22no—
55C3×C4×D4—24no—
56C3×C4×D4—21no—
57C3×C4×D4—22no—
58C3×C4×D4—22no—
59C3×C4×D4—22no—
60C3×C4×D4—22no—
61C3×C4×D4—22no—
62C3×#117—21no—
63C3×#117C2×C411yescharacteristic
64C3×#117C2×C411yescharacteristic
65C3×#117C2×C411yescharacteristic
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4D414yes—
68C2×C3×C42—21no—
69C2×C3×C42C2×C411yescharacteristic
70C2×#217—24no—
71#834—24no—
72#834—24no—
73#559—22no—
74#550—22no—
75#546—24no—
76#540—22no—
77#535Dic311yescharacteristic, complemented
78#535C2×D311yescharacteristic
79#533Dic311yescharacteristic, complemented
80#532—22no—
81C2×#127C2×D311yescharacteristic
82#518Dic311yescharacteristic, complemented
83#518C2×D311yescharacteristic
84C4×#41Dic311yescharacteristic, complemented
85C2×#99—34no—
86#354—62no—
87#353—62no—
88#353—62no—
89#353—62no—
90#342—64no—
91C24×C3—21no—
92C24×C3C2×D412yes—
93C3×#41—42no—
94C3×#41—88no—
95C2×C3×Q8—22no—
96C2×C3×Q8C22⋊C411yescharacteristic
97C2×C3×D4—21no—
98C2×C3×D4—24no—
99C2×C3×D4—24no—
100C2×C3×D4—24no—
101C2×C3×D4—24no—
102C2×C3×D4—21no—
103C2×C3×D4—42no—
104C2×C3×D4—42no—
105C2×C3×D4—84no—
106C2×C3×D4C22⋊C412yes—
107C2×C3×D4C22⋊C411yescharacteristic
108C2×C3×D4C2×D412yes—
109C2×C3×D4C2×D412yes—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—24no—
113C22×C3×C4—24no—
114C22×C3×C4—21no—
115C22×C3×C4C22×C411yescharacteristic
116C22×Dic3—44no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C22⋊C4—24no—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—24no—
127C3×C22⋊C4—21no—
128C3×C22⋊C4—24no—
129C3×C22⋊C4—24no—
130C3×C22⋊C4C22⋊C412yes—
131C3×C22⋊C4C22⋊C412yes—
132C3×C42—22no—
133C3×C42—21no—
134C3×C42—21no—
135C3×C42—21no—
136C3×C42C22⋊C411yescharacteristic
137C3×C42C22⋊C411yescharacteristic
138C3×C42C22⋊C412yes—
139#217—44no—
140#217—44no—
141C2×#41—24no—
142C2×#41C2×Dic311yescharacteristic
143C22×D4—22no—
144C22×D4C2×Dic311yescharacteristic
145C22×D4C22×D311yescharacteristic
146C4⋊Q8—21no—
147#127—21no—
148#127—22no—
149#127#6712yes—
150#127#6712yes—
151#127#6712yes—
152#125—21no—
153#124—24no—
154#124—21no—
155#124#6712yes—
156#123—24no—
157#123—24no—
158C22⋊Q8—22no—
159#121—24no—
160#121—24no—
161#121—24no—
162#121—22no—
163#121—22no—
164#120—22no—
165#120—22no—
166C4×Q8—21no—
167C4×Q8—22no—
168C4×D4—24no—
169C4×D4—21no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175#117—21no—
176#117C2×Dic311yescharacteristic
177#117C2×Dic311yescharacteristic
178#117C2×Dic311yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—64no—
181C2×C22⋊C4#6714yes—
182C2×C42—21no—
183C2×C42C2×Dic311yescharacteristic
184#99—64no—
185#99—64no—
186C23×C3—24no—
187C23×C3—22no—
188C23×C3—21no—
189C23×C3—42no—
190C23×C3—82no—
191C23×C3—82no—
192C23×C3C2×C22⋊C411yescharacteristic
193C23×C3#12014yes—
194C3×Q8—41no—
195C3×Q8—44no—
196C3×D4—42no—
197C3×D4—42no—
198C3×D4—42no—
199C3×D4—41no—
200C3×D4—42no—
201C3×D4—48no—
202C3×D4—42no—
203C3×D4—84no—
204C3×D4—84no—
205C3×D4—84no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—21no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—44no—
216C2×C3×C4—84no—
217C2×C3×C4—84no—
218C2×C3×C4C2×C22⋊C411yescharacteristic
219C2×C3×C4C2×C22⋊C411yescharacteristic
220C2×Dic3—84no—
221C2×Dic3—164no—
222C24—21no—
223C24C2×#6712yes—
224#41—42no—
225#41—88no—
226C2×Q8—22no—
227C2×Q8#21711yescharacteristic
228C2×D4—21no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—84no—
237C2×D4#21712yes—
238C2×D4#21711yescharacteristic
239C2×D4C2×#6712yes—
240C2×D4C2×#6712yes—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—124no—
247C22×C4C22×Dic311yescharacteristic
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C22⋊C4—24no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—21no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—124no—
262C22⋊C4—124no—
263C22⋊C4#21712yes—
264C22⋊C4#21712yes—
265C42—22no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42#21711yescharacteristic
270C42#21711yescharacteristic
271C42#21712yes—
272C22×C3—24no—
273C22×C3—41no—
274C22×C3—44no—
275C22×C3—42no—
276C22×C3—44no—
277C22×C3—41no—
278C22×C3—42no—
279C22×C3—84no—
280C22×C3—162no—
281C22×C3#37911yescharacteristic
282C22×C3#41012yes—
283C3×C4—22no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—42no—
287C3×C4—41no—
288C3×C4—42no—
289C3×C4—42no—
290C3×C4—41no—
291Dic3—164no—
292C23—24no—
293C23—22no—
294C23—21no—
295C23—42no—
296C23—82no—
297C23—82no—
298C23C2×#21711yescharacteristic
299C23#95314yes—
300Q8—41no—
301Q8—44no—
302D4—42no—
303D4—42no—
304D4—42no—
305D4—41no—
306D4—42no—
307D4—48no—
308D4—42no—
309D4—84no—
310D4—84no—
311D4—84no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—244no—
325C2×C4—484no—
326C2×C4C2×#21711yescharacteristic
327C2×C4C2×#21711yescharacteristic
328C2×C3—21no—
329C2×C3—22no—
330C2×C3—44no—
331C2×C3—81no—
332C2×C3—82no—
333C2×C3#178211yescharacteristic
334C22—24no—
335C22—41no—
336C22—44no—
337C22—42no—
338C22—44no—
339C22—41no—
340C22—42no—
341C22—84no—
342C22—162no—
343C22#524012yes—
344C22#524711yescharacteristic
345C4—22no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—41no—
350C4—42no—
351C4—42no—
352C4—41no—
353C4—484no—
354C3#1272711yescharacteristic, complemented, Sylow
355C2—21no—
356C2—22no—
357C2—44no—
358C2—82no—
359C2—81no—
360C2#7213911yescharacteristic
subgroupG/H
centerC2#72139
commutatorC22×C3×C4C22×C4
FrattiniC22×D4C22×D3
Fitting, prime radicalC3×#3508C2
socleC2×C3#1782
cosocleC22×C3×D4C23
hypercenter#3508D3
hypocenterC3#12727
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ #518 ◁ #3508

Complements

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

NH ≅ G/Ntype
2. #71860C2semidirect
3. C2×#4545C2semidirect
4. #71653C2semidirect
10. C3×#535C4semidirect
12. C3×#533C4semidirect
15. C3×#518C4semidirect
17. C3×C4×#41C4semidirect
77. #535Dic3semidirect
79. #533Dic3semidirect
82. #518Dic3semidirect
84. C4×#41Dic3semidirect
354. C3#12727semidirect

Maximal subgroups

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

subgroupindex
C3×#35082
#718602
C2×#45452
#716532
#127273