Finite Group Databasebeta

← #10706114

Subgroups — #10706114

Order 768.

368 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 69 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#78206C211yesmaximal, characteristic
2#68991C211yesmaximal, characteristic
3#68124C211yesmaximal, characteristic
4C3×#1353C211yesmaximal, characteristic, complemented
5#67652C211yesmaximal, characteristic, complemented
6#67495C211yesmaximal, characteristic, complemented
7#67458C211yesmaximal, characteristic, complemented
8#11853—31nomaximal, Sylow
9C2×#892C2211yescharacteristic
10C2×#890C411yescharacteristic
11C3×#400C2211yescharacteristic
12#5126—21no—
13#4993C411yescharacteristic
14#4981C2211yescharacteristic
15#4718—21no—
16#4674C411yescharacteristic
17#4669—21no—
18#4668—21no—
19#4655—21no—
20#4648C411yescharacteristic
21#4647—21no—
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4554—21no—
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4468—21no—
28#2611—31no—
29#1822—31no—
30#1458—31no—
31#1353—31no—
32#1353D311yescharacteristic, complemented
33#1346—31no—
34#1346—31no—
35C22×C4×D3D411yescharacteristic
36C2×C3×M16—21no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4C2311yescharacteristic
39C2×(C2×D3)⋊C4—21no—
40C2×(C2×D3)⋊C4—21no—
41C2×(C2×D3)⋊C4C2×C411yescharacteristic
42C2×(C2×D3)⋊C4D411yescharacteristic
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3D411yescharacteristic
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3C2×C411yescharacteristic
51C2×C4×Dic3D411yescharacteristic
52C2×#208—21no—
53#892—41no—
54#892—41no—
55#890—41no—
56#890—41no—
57C3×C22⋊C8—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95C2×C411yescharacteristic
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95C2×C411yescharacteristic
65#830—21no—
66(C2×D3)⋊C8—41no—
67(C2×D3)⋊C8—41no—
68Dic3⋊C8—41no—
69C8×Dic3—41no—
70C2×#125—31no—
71C2×#117—31no—
72#431—61no—
73#407—61no—
74#400C2×D311yescharacteristic
75#399—31no—
76#398—31no—
77#397—31no—
78#389—61no—
79#388—61no—
80#387—61no—
81#385—61no—
82#376—31no—
83#343—31no—
84#343—61no—
85#343C2×D311yescharacteristic
86#324—31no—
87#324C2×D311yescharacteristic
88#323—61no—
89C23×D3C22⋊C411yescharacteristic
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×Dic3—21no—
96C22×Dic3—21no—
97C22×Dic3C22⋊C411yescharacteristic
98C22×Dic3C22⋊C411yescharacteristic
99C22×Dic3C22⋊C411yescharacteristic
100C2×C4×D3—21no—
101C2×C4×D3—21no—
102C2×C4×D3—21no—
103C2×C4×D3—21no—
104C2×C4×D3—41no—
105C3×M16—41no—
106C3×M16—41no—
107C2×C3×C8—41no—
108C2×C3×C8—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111(C2×D3)⋊C4—41no—
112(C2×D3)⋊C4—41no—
113(C2×D3)⋊C4—42no—
114(C2×D3)⋊C4—42no—
115C4⋊Dic3—41no—
116C4⋊Dic3—41no—
117C4⋊Dic3—41no—
118C4⋊Dic3—41no—
119Dic3⋊C4—42no—
120Dic3⋊C4—42no—
121Dic3⋊C4—41no—
122Dic3⋊C4—41no—
123C4×Dic3—21no—
124C4×Dic3—21no—
125C4×Dic3—21no—
126C4×Dic3—21no—
127C4×Dic3—42no—
128#208—41no—
129#208—41no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8—41no—
132C23×C4—31no—
133C2×M16—21no—
134C2×M16—61no—
135#125—121no—
136#125—121no—
137#117—121no—
138#117—121no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4C4×D311yescharacteristic
146C2×C4⋊C4C22×D311yescharacteristic
147C2×C22⋊C4—31no—
148C2×C22⋊C4—31no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C42—31no—
152C2×C42—31no—
153C2×C42—61no—
154C4⋊C8—121no—
155C22⋊C8—21no—
156C22⋊C8—61no—
157C22⋊C8—121no—
158C22⋊C8—121no—
159C4×C8—121no—
160#95—21no—
161#95—21no—
162#95—31no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95C4×D311yescharacteristic
167C23×C3C2×C22⋊C411yescharacteristic
168C22×D3—21no—
169C22×D3—21no—
170C22×D3—21no—
171C22×D3—41no—
172C22×D3—41no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4#13511yescharacteristic
186C2×C3×C4#13511yescharacteristic
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—22no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3#9911yescharacteristic
200C2×Dic3#9911yescharacteristic
201C4×D3—41no—
202C4×D3—41no—
203C4×D3—41no—
204C4×D3—41no—
205C3×C8—81no—
206C3×C8—81no—
207C3⋊C8—81no—
208C3⋊C8—81no—
209C24—31no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—121no—
222C22×C4C2×C4×D311yescharacteristic
223C22×C4D3×D411yescharacteristic
224C22×C4D3×D411yescharacteristic
225M16—41no—
226M16—41no—
227M16—121no—
228M16—121no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—121no—
232C2×C8—121no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—122no—
240C4⋊C4—122no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C22⋊C4—121no—
244C22⋊C4—121no—
245C22⋊C4—122no—
246C22⋊C4—122no—
247C42—61no—
248C42—61no—
249C42—61no—
250C42—61no—
251C42—122no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3C2×#9911yescharacteristic
255C22×C3#41611yescharacteristic
256C22×C3#42111yescharacteristic
257C2×D3—42no—
258C2×D3—41no—
259C2×D3—41no—
260C2×D3—41no—
261C2×D3—81no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—81no—
267C3×C4—81no—
268Dic3—21no—
269Dic3—21no—
270Dic3—42no—
271Dic3—41no—
272Dic3—41no—
273Dic3—81no—
274C23—61no—
275C23—61no—
276C23—61no—
277C23—121no—
278C23—121no—
279C23D3×C22⋊C411yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—62no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4#91611yescharacteristic
311C2×C4#91911yescharacteristic
312C8—81no—
313C8—81no—
314C8—241no—
315C8—241no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3#151611yescharacteristic
319C2×C3#212311yescharacteristic
320C2×C3#213211yescharacteristic
321D3—81no—
322C22—21no—
323C22—21no—
324C22—122no—
325C22—121no—
326C22—121no—
327C22—121no—
328C22—241no—
329C22D3×#9911yescharacteristic
330C22#480411yescharacteristic
331C22#482111yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—61no—
337C4—61no—
338C4—81no—
339C4—81no—
340C4—122no—
341C4—121no—
342C4—121no—
343C4—241no—
344C3#1185311yescharacteristic, complemented, Sylow
345C2—21no—
346C2—21no—
347C2—241no—
348C2#6833811yescharacteristic
349C2#6914511yescharacteristic
350C2#6916511yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1353C2
socleC22×C3C2×#99
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1353D3
hypocenterC3#11853
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1353

Complements

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

NH ≅ G/Ntype
4. C3×#1353C2semidirect
5. #67652C2semidirect
6. #67495C2semidirect
7. #67458C2semidirect
32. #1353D3semidirect
344. C3#11853semidirect

Maximal subgroups

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

subgroupindex
#782062
#689912
#681242
C3×#13532
#676522
#674952
#674582
#118533