Finite Group Databasebeta

← #10692849

Subgroups — #10692849

Order 768.

486 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 99 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#82474C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic, complemented
3C3×#1467C211yesmaximal, characteristic, complemented
4#70808C212yesmaximal
5#67750C212yesmaximal, complemented
6#9448—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#947C411yescharacteristic
9C2×#944C411yescharacteristic
10C2×#875C2211yescharacteristic
11C3×#432C2212yes—
12#5024—24no—
13#4994—21no—
14#4994—21no—
15#4988C411yescharacteristic
16#4973C411yescharacteristic
17#4940—21no—
18#4940—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4926—21no—
21#4926C2212yes—
22#4925—21no—
23C3×#324—21no—
24C3×#324—21no—
25#4464—24no—
26#2983—31no—
27#2545—31no—
28#1533—32no—
29#1467D311yescharacteristic, complemented
30#1285—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×D12C2×C411yescharacteristic
33C22×Dic6C2×C411yescharacteristic
34C3×C4×Q8—22no—
35C3×C4×Q8—22no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42#947—42no—
43#944—42no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×C4⋊Dic3—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3C2×C411yescharacteristic
54C2×C4×Dic3C2×C411yescharacteristic
55#875—22no—
56#875—22no—
57C3×C4⋊C8—22no—
58C3×C22⋊C8—22no—
59C3×#97—22no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63#830—22no—
64C4⋊(C3⋊C8)—22no—
65C4⋊(C3⋊C8)—22no—
66C4⋊(C3⋊C8)—22no—
67#803—22no—
68#803D414yes—
69C4×C3⋊C8—22no—
70C2×C4⋊Q8—31no—
71C2×#124—31no—
72C2×#124—31no—
73C2×C4×Q8C2×D311yescharacteristic
74#446—64no—
75#432—32no—
76#432—61no—
77#432C2×D312yes—
78#431—61no—
79#396—61no—
80#396—61no—
81#391—31no—
82#390—31no—
83#388—61no—
84#388—61no—
85C4×C4⋊C4—31no—
86#329—64no—
87#324—21no—
88#324—21no—
89C23×D3—21no—
90C2×C3×Q8—22no—
91C2×C3×Q8—22no—
92C2×C3×Q8—41no—
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C2×D12—22no—
104C2×D12—41no—
105C2×D12—41no—
106C2×Dic6—22no—
107C2×Dic6—41no—
108C2×Dic6—41no—
109C2×C3×C8—42no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—22no—
113C3×C42—21no—
114C3×C42—21no—
115C3×C42—21no—
116C3×C42—21no—
117C3×C42C2×D412yes—
118C3×C42#4112yes—
119(C2×D3)⋊C4—42no—
120(C2×D3)⋊C4—42no—
121(C2×D3)⋊C4—42no—
122(C2×D3)⋊C4—42no—
123(C2×D3)⋊C4—41no—
124(C2×D3)⋊C4—42no—
125(C2×D3)⋊C4—41no—
126C4⋊Dic3—42no—
127Dic3⋊C4—42no—
128Dic3⋊C4—42no—
129C4×Dic3—41no—
130C4×Dic3—41no—
131C2×C3⋊C8—24no—
132C22×Q8—31no—
133C22×Q8C4×D311yescharacteristic
134C22×D4—31no—
135C4⋊Q8—122no—
136#124—62no—
137#124—62no—
138#124—122no—
139C4×Q8—22no—
140C4×Q8—22no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4C4×D311yescharacteristic
145C2×C4⋊C4#6711yescharacteristic
146C2×C4⋊C4#6711yescharacteristic
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C42—31no—
154C2×C42—31no—
155C2×C42D1211yescharacteristic
156C2×C42D1211yescharacteristic
157C2×C42C22×D311yescharacteristic
158C4⋊C8—22no—
159C4⋊C8—62no—
160C4⋊C8—62no—
161C4⋊C8—62no—
162C22⋊C8—22no—
163C22⋊C8—62no—
164#97—22no—
165#97—34no—
166#97—62no—
167C4×C8—62no—
168#95—61no—
169#95—61no—
170#95—61no—
171C23×C3C2×C22⋊C411yescharacteristic
172C22×D3—41no—
173C22×D3—41no—
174C22×D3—41no—
175C22×D3—41no—
176C22×D3—41no—
177C22×D3—41no—
178C22×D3—41no—
179C3×Q8—44no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4#10012yes—
194C2×C3×C4C4×D412yes—
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12112yes—
197C2×C3×C4#12311yescharacteristic
198C2×Dic3—22no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207D12—82no—
208Dic6—82no—
209C3×C8—82no—
210C3⋊C8—44no—
211C24—61no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—41no—
215C2×Q8—62no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×D4—62no—
219C2×D4—121no—
220C2×D4—121no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4(C2×D3)⋊C411yescharacteristic
225C22×C4(C2×D3)⋊C411yescharacteristic
226C22×C4(C2×D3)⋊C411yescharacteristic
227C22×C4(C2×D3)⋊C411yescharacteristic
228C22×C4C2×C4×D311yescharacteristic
229C22×C4C2×D1211yescharacteristic
230C22×C4C2×#6711yescharacteristic
231C2×C8—42no—
232C2×C8—64no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—122no—
237C4⋊C4—122no—
238C4⋊C4—122no—
239C22⋊C4—122no—
240C22⋊C4—122no—
241C22⋊C4—122no—
242C22⋊C4—122no—
243C22⋊C4—121no—
244C22⋊C4—121no—
245C22⋊C4—122no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—121no—
251C42—121no—
252C42D3×D412yes—
253C42#23712yes—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3#38611yescharacteristic
257C22×C3C2×#10011yescharacteristic
258C22×C3#41311yescharacteristic
259C2×D3—81no—
260C2×D3—81no—
261C2×D3—81no—
262C2×D3—81no—
263C2×D3—81no—
264C2×D3—81no—
265C2×D3—81no—
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274Dic3—42no—
275Dic3—81no—
276Dic3—81no—
277C23—121no—
278C23—121no—
279C23—121no—
280C23—121no—
281C23—121no—
282C23—121no—
283C23—121no—
284C23C2×(C2×D3)⋊C411yescharacteristic
285Q8—44no—
286Q8—242no—
287D4—242no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—62no—
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#82312yes—
311C2×C4#67⋊C412yes—
312C2×C4#88211yescharacteristic
313C2×C4#88611yescharacteristic
314C2×C4#93912yes—
315C8—82no—
316C8—124no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3#188411yescharacteristic
320C2×C3#190011yescharacteristic
321C2×C3#191211yescharacteristic
322D3—161no—
323C22—21no—
324C22—21no—
325C22—241no—
326C22—241no—
327C22—241no—
328C22—241no—
329C22—241no—
330C22—241no—
331C22—241no—
332C22#495811yescharacteristic
333C22C2×#82311yescharacteristic
334C22#513211yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—122no—
344C4—241no—
345C4—241no—
346C3#944811yescharacteristic, complemented, Sylow
347C2—21no—
348C2—21no—
349C2—481no—
350C2#6998511yescharacteristic
351C2#7004511yescharacteristic
352C2#7010511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1467C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1467D3
hypocenterC3#9448
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #82436C2semidirect
3. C3×#1467C2semidirect
5. #67750C2semidirect
29. #1467D3semidirect
346. C3#9448semidirect

Maximal subgroups

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

subgroupindex
#824742
#824362
C3×#14672
#708082
#677502
#94483