Finite Group Databasebeta

← #10693079

Subgroups — #10693079

Order 768.

514 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 95 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#82284C211yesmaximal, characteristic
2#82209C211yesmaximal, characteristic, complemented
3C3×#1464C211yesmaximal, characteristic, complemented
4#70799C212yesmaximal, complemented
5#67752C212yesmaximal, complemented
6#9494—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×C4×D12C2211yescharacteristic
9C3×#432C2212yes—
10#5213—21no—
11#5207—21no—
12#5000—24no—
13#4991C411yescharacteristic
14#4989—21no—
15#4986C411yescharacteristic
16#4960C411yescharacteristic
17#4956—21no—
18#4949C411yescharacteristic
19#4940—21no—
20#4937C2211yescharacteristic
21#4924—21no—
22C4×#208C2212yes—
23C3×#324—21no—
24C3×#323—21no—
25#4483—24no—
26#2921—31no—
27#2598—31no—
28#1490—32no—
29#1464D311yescharacteristic, complemented
30#1288—32no—
31C23×C3×C4D411yescharacteristic
32C22×#67—21no—
33C22×D12C2×C411yescharacteristic
34C22×C4×D3—21no—
35C3×#117—22no—
36C3×#117—22no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C22⋊C4C2×C411yescharacteristic
40C2×C3×C22⋊C4D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×C3×C42C2311yescharacteristic
43C2×#217—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×#208—22no—
51C4×D12—22no—
52C4×D12—22no—
53C3×C4⋊C8—22no—
54C3×C22⋊C8—22no—
55C3×#97—22no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95C2×C411yescharacteristic
61C3⋊#95C2×C411yescharacteristic
62#830—24no—
63C4⋊(C3⋊C8)—24no—
64#803—22no—
65C4×C3⋊C8D414yescomplemented
66C2×C4×D4—31no—
67C2×#117C2×D311yescharacteristic
68#433—64no—
69#432C2×D312yes—
70#423—61no—
71#407—61no—
72C4×M16—32no—
73#400—31no—
74#399—31no—
75#394—31no—
76#392—31no—
77#388—61no—
78#386—61no—
79#386—61no—
80#385—61no—
81#383—31no—
82D4⋊C8—64no—
83#324—21no—
84#323—21no—
85C24×C3C22⋊C411yescharacteristic
86C23×D3—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C2×#67—42no—
98C2×#67—42no—
99C2×#67—81no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C2×D12—22no—
104C2×D12—42no—
105C2×C4×D3—42no—
106C2×C4×D3—81no—
107C2×C4×D3—81no—
108C2×C3×C8—42no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C42—21no—
114C3×C42—21no—
115C3×C42C2×D412yes—
116C3×C42#4112yes—
117#217—81no—
118(C2×D3)⋊C4—42no—
119(C2×D3)⋊C4—42no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—81no—
123C4⋊Dic3—22no—
124Dic3⋊C4—81no—
125#208—42no—
126#208—42no—
127C2×C3⋊C8—24no—
128C22×D4—31no—
129C22×D4—61no—
130C23×C4—61no—
131C23×C4D1211yescharacteristic
132C2×M16—62no—
133C4×D4—62no—
134C4×D4—62no—
135#117—22no—
136#117—22no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4C4×D311yescharacteristic
140C2×C4⋊C4#6711yescharacteristic
141C2×C22⋊C4—61no—
142C2×C22⋊C4—61no—
143C2×C22⋊C4—61no—
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4C4×D311yescharacteristic
147C2×C22⋊C4#6711yescharacteristic
148C2×C42D1211yescharacteristic
149C2×C42C22×D311yescharacteristic
150C4⋊C8—22no—
151C4⋊C8—64no—
152C22⋊C8—22no—
153C22⋊C8—64no—
154#97—22no—
155#97—62no—
156C4×C8—34no—
157#95—31no—
158#95—31no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—61no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3C2×C22⋊C411yescharacteristic
169C22×D3—22no—
170C22×D3—41no—
171C22×D3—41no—
172C22×D3—41no—
173C22×D3—41no—
174C22×D3—41no—
175C22×D3—41no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4C4×D412yes—
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12112yes—
192C2×C3×C4#12311yescharacteristic
193#67—84no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203D12—24no—
204D12—42no—
205C4×D3—84no—
206C3×C8—82no—
207C3⋊C8—44no—
208C24—61no—
209C24(C2×D3)⋊C411yescharacteristic
210C2×D4—62no—
211C2×D4—122no—
212C2×D4—122no—
213C2×D4—122no—
214C2×D4—241no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—122no—
223C22×C4—241no—
224C22×C4—241no—
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
231M16—122no—
232M16—122no—
233C2×C8—42no—
234C2×C8—64no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—62no—
238C4⋊C4—241no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—122no—
242C22⋊C4—122no—
243C22⋊C4—121no—
244C22⋊C4—121no—
245C22⋊C4—241no—
246C22⋊C4—241no—
247C42—21no—
248C42—21no—
249C42D3×D412yes—
250C42#23712yes—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—42no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—81no—
258C22×C3#38611yescharacteristic
259C22×C3#42111yescharacteristic
260C22×C3#42111yescharacteristic
261C2×D3—42no—
262C2×D3—42no—
263C2×D3—42no—
264C2×D3—81no—
265C2×D3—81no—
266C2×D3—81no—
267C2×D3—81no—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—22no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275Dic3—81no—
276Dic3—81no—
277Dic3—81no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—62no—
284C23—121no—
285C23—121no—
286C23—121no—
287C23—121no—
288C23—121no—
289C23—121no—
290C23C2×(C2×D3)⋊C411yescharacteristic
291D4—64no—
292D4—122no—
293D4—244no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—244no—
317C2×C4#67⋊C412yes—
318C2×C4#88211yescharacteristic
319C2×C4#88611yescharacteristic
320C2×C4#93912yes—
321C8—82no—
322C8—124no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—81no—
326C2×C3#189111yescharacteristic
327C2×C3#189711yescharacteristic
328C2×C3#191011yescharacteristic
329D3—82no—
330C22—21no—
331C22—21no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22—122no—
338C22—122no—
339C22—122no—
340C22—241no—
341C22—241no—
342C22—241no—
343C22—241no—
344C22#495811yescharacteristic
345C22#513911yescharacteristic
346C22#513911yescharacteristic
347C4—21no—
348C4—21no—
349C4—22no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—241no—
355C4—241no—
356C4—241no—
357C3#949411yescharacteristic, complemented, Sylow
358C2—21no—
359C2—21no—
360C2—81no—
361C2—242no—
362C2#6996611yescharacteristic
363C2#7004711yescharacteristic
364C2#7013611yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1464C2
socleC22×C3#386
prime radical#4960C4
cosocleC2×C3×C42C23
hypercenter#1464D3
hypocenterC3#9494
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #82209C2semidirect
3. C3×#1464C2semidirect
4. #70799C2semidirect
5. #67752C2semidirect
29. #1464D3semidirect
65. C4×C3⋊C8D4semidirect
357. C3#9494semidirect

Maximal subgroups

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

subgroupindex
#822842
#822092
C3×#14642
#707992
#677522
#94943