Finite Group Databasebeta

← #10694901

Subgroups — #10694901

Order 768.

548 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×C3×#466C211yesmaximal, characteristic
2#71977C211yesmaximal, characteristic, complemented
3C8×#217C211yesmaximal, characteristic, complemented
4#71676C212yesmaximal, complemented
5#71673C212yesmaximal, complemented
6#9815—31nomaximal, Sylow
7C22×C3×D8C411yescharacteristic, complemented
8C23×C3×C8C2211yescharacteristic
9C2×C3×#121C2212yes—
10C3×#466—21no—
11C3×#466—21no—
12C3×#466—21no—
13C3×#466—21no—
14C2×C3×#107C411yescharacteristic, complemented
15C2×C3×#102C412yescomplemented
16C4×#217C2211yescharacteristic
17#5207C2212yes—
18C2×#830C2211yescharacteristic
19C2×#819—21no—
20C2×C8×Dic3—21no—
21#4552—21no—
22#4550—21no—
23#4537—24no—
24#4525—24no—
25C2×#466D311yescharacteristic
26#1810—31no—
27C8×C22⋊C4—31no—
28#1507—32no—
29#1471—32no—
30C22×C3×D4—22no—
31C22×C3×D4C2×C412yes—
32C23×C3×C4C2311yescharacteristic
33C2×C3×D8—21no—
34C2×C3×D8—21no—
35C2×C3×D8—42no—
36C22×C3×C8—42no—
37C22×C3×C8C2×C411yescharacteristic
38C22×C3×C8C2×C411yescharacteristic
39C22×C3×C8D411yescharacteristic
40C22×C3×C8D411yescharacteristic
41C22×C3×C8D411yescharacteristic
42C22×C3×C8D411yescharacteristic
43C3×#121—22no—
44C3×#121—22no—
45C3×#121—22no—
46C3×#121—22no—
47C2×C3×C4⋊C4C2×C412yes—
48C2×C3×C22⋊C4—22no—
49C2×#217—21no—
50C2×C4×Dic3—21no—
51C2×#208—22no—
52C22×C3⋊C8—21no—
53C3×#107—21no—
54C3×#107—21no—
55C3×#102—22no—
56C3×#102—22no—
57C3⋊#95D412yes—
58#830—21no—
59#830—22no—
60#830—22no—
61#830D412yes—
62#819—42no—
63C8×Dic3—42no—
64C22×D8Dic311yescharacteristic, complemented
65C23×C8C2×D311yescharacteristic
66C2×#121C2×D312yes—
67#466—21no—
68#466—21no—
69#466—21no—
70#466—21no—
71C2×#108—61no—
72C2×#107Dic311yescharacteristic, complemented
73C2×#102Dic312yescomplemented
74#407—32no—
75C2×C22⋊C8—31no—
76C2×C4×C8—61no—
77C4×C22⋊C4—31no—
78#341—61no—
79#336—61no—
80#327—64no—
81#323—64no—
82C24×C3—22no—
83C24×C3#4111yescharacteristic
84C2×C3×D4—22no—
85C2×C3×D4—22no—
86C2×C3×D4—42no—
87C2×C3×D4—42no—
88C2×C3×D4—42no—
89C2×C3×D4—42no—
90C2×C3×D4—84no—
91C22×C3×C4—22no—
92C22×C3×C4—41no—
93C22×C3×C4—41no—
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4111yescharacteristic
100C22×Dic3—22no—
101C3×D8—42no—
102C3×D8—42no—
103C2×C3×C8—22no—
104C2×C3×C8—22no—
105C2×C3×C8—22no—
106C2×C3×C8—41no—
107C2×C3×C8—41no—
108C2×C3×C8—41no—
109C2×C3×C8C22⋊C411yescharacteristic
110C2×C3×C8C22⋊C411yescharacteristic
111C2×C3×C8C22⋊C411yescharacteristic
112C2×C3×C8C22⋊C411yescharacteristic
113C3×C4⋊C4—22no—
114C3×C4⋊C4—22no—
115C3×C22⋊C4—42no—
116C3×C22⋊C4—42no—
117#217—24no—
118C4×Dic3—41no—
119C4×Dic3—41no—
120#208—42no—
121#208—42no—
122C2×C3⋊C8—22no—
123C2×C3⋊C8—41no—
124C2×C3⋊C8—41no—
125C2×C3⋊C8—42no—
126C22×D4—22no—
127C22×D4C2×Dic312yes—
128C23×C4C22×D311yescharacteristic
129C2×D8—21no—
130C2×D8—21no—
131C2×D8—42no—
132C2×M16—62no—
133C22×C8—42no—
134C22×C8—61no—
135C22×C8C2×Dic311yescharacteristic
136C22×C8C2×Dic311yescharacteristic
137C22×C8#6711yescharacteristic
138C22×C8#6711yescharacteristic
139C22×C8#6711yescharacteristic
140C22×C8#6711yescharacteristic
141#121—22no—
142#121—22no—
143#121—22no—
144#121—22no—
145C2×C4⋊C4C2×Dic312yes—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—61no—
148C2×C42—61no—
149#108—122no—
150#107—21no—
151#107—21no—
152#102—22no—
153#102—22no—
154C22⋊C8—32no—
155C22⋊C8—61no—
156C22⋊C8—62no—
157C22⋊C8—62no—
158C4×C8—122no—
159#95—32no—
160C23×C3—22no—
161C23×C3—22no—
162C23×C3—41no—
163C23×C3—42no—
164C23×C3—44no—
165C23×C3—82no—
166C23×C3—82no—
167C23×C3C4×D411yescharacteristic
168C23×C3#12111yescharacteristic
169C23×C3#12411yescharacteristic
170C3×D4—42no—
171C3×D4—42no—
172C3×D4—82no—
173C3×D4—84no—
174C3×D4—84no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×C3×C4—82no—
186C2×C3×C4C2×C22⋊C411yescharacteristic
187C2×C3×C4C4×D411yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12711yescharacteristic
190C2×Dic3—24no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193C3×C8—21no—
194C3×C8—21no—
195C3×C8—21no—
196C3×C8#9912yes—
197C3⋊C8—42no—
198C3⋊C8—82no—
199C24—22no—
200C24#23711yescharacteristic
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—84no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—62no—
212C22×C4D3×D411yescharacteristic
213C22×C4D3×D411yescharacteristic
214C22×C4#23711yescharacteristic
215C22×C4C22×Dic311yescharacteristic
216C22×C4C2×#6711yescharacteristic
217C22×C4C2×#6711yescharacteristic
218D8—42no—
219D8—42no—
220M16—122no—
221M16—122no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—62no—
229C2×C8—121no—
230C2×C8—121no—
231C2×C8—122no—
232C2×C8#21711yescharacteristic
233C2×C8#21711yescharacteristic
234C2×C8#21711yescharacteristic
235C2×C8#21711yescharacteristic
236C4⋊C4—22no—
237C4⋊C4—22no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—64no—
241C42—121no—
242C42—121no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—44no—
251C22×C3—82no—
252C22×C3—82no—
253C22×C3—82no—
254C22×C3—164no—
255C22×C3#39011yescharacteristic
256C22×C3#44312yes—
257C22×C3#47212yes—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—82no—
262C3×C4C2×#9911yescharacteristic
263C3×C4#41011yescharacteristic
264Dic3—44no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—42no—
269C23—44no—
270C23—82no—
271C23—82no—
272C23D4×Dic311yescharacteristic
273C23#93611yescharacteristic
274C23#93811yescharacteristic
275D4—42no—
276D4—42no—
277D4—82no—
278D4—84no—
279D4—84no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—64no—
291C2×C4—82no—
292C2×C4—122no—
293C2×C4—122no—
294C2×C4D4×Dic311yescharacteristic
295C2×C4#93811yescharacteristic
296C2×C4#94011yescharacteristic
297C2×C4C2×#21711yescharacteristic
298C8—21no—
299C8—21no—
300C8—21no—
301C8—122no—
302C8—242no—
303C8#83412yes—
304C2×C3—22no—
305C2×C3—22no—
306C2×C3—41no—
307C2×C3—162no—
308C2×C3#196411yescharacteristic
309C2×C3#197411yescharacteristic
310C2×C3#197811yescharacteristic
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—164no—
323C22#516012yes—
324C22#519612yes—
325C22#522111yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—82no—
330C4—124no—
331C4C2×#83411yescharacteristic
332C4#524011yescharacteristic
333C3#981511yescharacteristic, complemented, Sylow
334C2—22no—
335C2—22no—
336C2—41no—
337C2—162no—
338C2#7168611yescharacteristic
339C2#7204111yescharacteristic
340C2#7208411yescharacteristic
subgroupG/H
centerC22#5221
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC2×C3×#466C2
socleC22×C3#390
cosocleC23×C3×C4C23
hypercenterC2×#466D3
hypocenterC3#9815
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
2. #71977C2semidirect
3. C8×#217C2semidirect
4. #71676C2semidirect
5. #71673C2semidirect
7. C22×C3×D8C4semidirect
14. C2×C3×#107C4semidirect
15. C2×C3×#102C4semidirect
64. C22×D8Dic3semidirect
72. C2×#107Dic3semidirect
73. C2×#102Dic3semidirect
333. C3#9815semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4662
#719772
C8×#2172
#716762
#716732
#98153