Finite Group Databasebeta

← #10692817

Subgroups — #10692817

Order 768.

576 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3153C211yesmaximal, characteristic
2#82956C211yesmaximal, characteristic, complemented
3#71812C212yesmaximal, complemented
4#71719C211yesmaximal, characteristic, complemented
5#67665C212yesmaximal, complemented
6#9441—31nomaximal, Sylow
7C2×C3×#137—21no—
8C2×C3×AGL(1,ℤ/8)—21no—
9C3×#532C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C3×#497C412yescomplemented
12C3×#496C412yescomplemented
13C3×#495—22no—
14C3×C4×M16C2211yescharacteristic
15#5229—21no—
16#5220—21no—
17#4951—22no—
18Dic3×C22⋊C4—22no—
19#4932C2212yes—
20#4926C2212yes—
21#4540—22no—
22#4537—22no—
23#4537—22no—
24#3153D311yescharacteristic
25#2372—31no—
26#1533—31no—
27#1505—32no—
28#1300—32no—
29C2×C3×#41—21no—
30C22×C3×Q8D411yescharacteristic
31C22×C3×D4D411yescharacteristic
32C23×Dic3—21no—
33C3×#137—42no—
34C3×#137—42no—
35C3×AGL(1,ℤ/8)—42no—
36C3×AGL(1,ℤ/8)—42no—
37C2×C3×Q16—22no—
38C2×C3×SD16—22no—
39C2×C3×SD16—22no—
40C2×C3×D8—22no—
41C2×C3×M16—21no—
42C3×C4⋊Q8C2×C411yescharacteristic, complemented
43C3×#127C2×C411yescharacteristic, complemented
44C3×#124—21no—
45C3×#124—22no—
46C3×#124C2×C412yes—
47C3×#124D412yes—
48C3×#121—22no—
49C3×C4×D4—22no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C42C2311yescharacteristic
52C2×#217—22no—
53C2×#217—21no—
54C2×C4⋊Dic3—21no—
55C2×C4×Dic3—22no—
56C2×C4×Dic3—21no—
57C3×#97C2×C412yescomplemented
58C3×C4×C8—21no—
59C3⋊#95—22no—
60C3⋊#95—22no—
61C3⋊#95D412yes—
62C3⋊#95D412yes—
63#830—22no—
64C4⋊(C3⋊C8)—22no—
65#803—22no—
66C2×#137—21no—
67C2×AGL(1,ℤ/8)—21no—
68#532C2×D311yescharacteristic
69C2×#124C2×D311yescharacteristic
70#497Dic312yescomplemented
71#496Dic312yescomplemented
72#495—22no—
73#432—32no—
74C4×M16C2×D311yescharacteristic
75#391—61no—
76#388—62no—
77#386—61no—
78C4×C22⋊C4—62no—
79#376—32no—
80#329—62no—
81#327—62no—
82#327—62no—
83C24×C3—21no—
84C3×#41—41no—
85C3×#41—41no—
86C3×#41—41no—
87C3×#41—41no—
88C2×C3×Q8—21no—
89C2×C3×Q8—21no—
90C2×C3×Q8—41no—
91C2×C3×Q8—41no—
92C2×C3×Q8C22⋊C412yes—
93C2×C3×D4—21no—
94C2×C3×D4—22no—
95C2×C3×D4—21no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C2×C3×D4C22⋊C412yes—
100C22×C3×C4—21no—
101C22×C3×C4—41no—
102C22×C3×C4C2×D412yes—
103C22×C3×C4C2×D411yescharacteristic
104C22×Dic3—22no—
105C22×Dic3—22no—
106C22×Dic3—44no—
107C22×Dic3—81no—
108C3×Q16—44no—
109C3×SD16—44no—
110C3×SD16—44no—
111C3×D8—44no—
112C3×M16—22no—
113C3×M16—22no—
114C2×C3×C8—22no—
115C3×C4⋊C4—22no—
116C3×C22⋊C4—24no—
117C3×C22⋊C4—24no—
118C3×C22⋊C4—42no—
119C3×C42C22×C411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4112yes—
122#217—44no—
123#217—81no—
124C4⋊Dic3—81no—
125C4×Dic3—44no—
126C4×Dic3—81no—
127C2×C3⋊C8—42no—
128C2×#41—21no—
129C22×Q8#6711yescharacteristic
130C22×D4#6711yescharacteristic
131C23×C4—61no—
132#137—42no—
133#137—42no—
134AGL(1,ℤ/8)—42no—
135AGL(1,ℤ/8)—42no—
136C2×Q16—22no—
137C2×SD16—22no—
138C2×SD16—22no—
139C2×D8—22no—
140C2×M16—21no—
141C4⋊Q8C2×Dic311yescharacteristic, complemented
142#127C2×Dic311yescharacteristic, complemented
143#124—22no—
144#124—21no—
145#124C2×Dic312yes—
146#124#6712yes—
147#121—22no—
148C4×D4—22no—
149C2×C4⋊C4—61no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—61no—
153C2×C42—62no—
154C2×C42—61no—
155C2×C42C22×D311yescharacteristic
156C4⋊C8—62no—
157C22⋊C8—62no—
158#97—62no—
159#97C2×Dic312yescomplemented
160C4×C8—21no—
161#95—32no—
162#95—32no—
163#95—62no—
164#95—62no—
165C23×C3—22no—
166C23×C3—42no—
167C23×C3—42no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3#12011yescharacteristic
172C3×Q8—42no—
173C3×Q8—41no—
174C3×Q8—41no—
175C3×Q8—81no—
176C3×D4—42no—
177C3×D4—41no—
178C3×D4—41no—
179C3×D4—81no—
180C3×D4—82no—
181C3×D4—81no—
182C3×D4—81no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—22no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—81no—
191C2×C3×C4—81no—
192C2×C3×C4C2×C22⋊C411yescharacteristic
193C2×C3×C4C4×D412yes—
194C2×C3×C4#12112yes—
195C2×C3×C4#12311yescharacteristic
196C2×Dic3—24no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—84no—
203C3×C8—24no—
204C3⋊C8—82no—
205C24—21no—
206#41—41no—
207#41—41no—
208#41—41no—
209#41—41no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8#21712yes—
215C2×D4—22no—
216C2×D4—21no—
217C2×D4—21no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4#21712yes—
222C22×C4—21no—
223C22×C4—41no—
224C22×C4—62no—
225C22×C4—62no—
226C22×C4—124no—
227C22×C4—241no—
228C22×C4D3×D412yes—
229C22×C4C2×#6711yescharacteristic
230Q16—44no—
231SD16—44no—
232SD16—44no—
233D8—44no—
234M16—22no—
235M16—22no—
236C2×C8—22no—
237C2×C8—122no—
238C4⋊C4—22no—
239C4⋊C4—241no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—42no—
243C22⋊C4—124no—
244C22⋊C4—241no—
245C42—124no—
246C42—241no—
247C42#23712yes—
248C42C22×Dic311yescharacteristic
249C42C2×#6711yescharacteristic
250C22×C3—22no—
251C22×C3—42no—
252C22×C3—44no—
253C22×C3—81no—
254C22×C3—81no—
255C22×C3—82no—
256C22×C3—81no—
257C22×C3—82no—
258C22×C3#38611yescharacteristic
259C22×C3#45311yescharacteristic
260C22×C3#45311yescharacteristic
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—42no—
266C3×C4—81no—
267Dic3—44no—
268Dic3—82no—
269C23—22no—
270C23—42no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23#93711yescharacteristic
276Q8—42no—
277Q8—41no—
278Q8—41no—
279Q8—81no—
280D4—42no—
281D4—41no—
282D4—41no—
283D4—81no—
284D4—82no—
285D4—81no—
286D4—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—64no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—122no—
300C2×C4—122no—
301C2×C4—122no—
302C2×C4—244no—
303C2×C4D4×Dic312yes—
304C2×C4#93511yescharacteristic
305C2×C4#93912yes—
306C2×C4C2×#21711yescharacteristic
307C8—24no—
308C8—242no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—82no—
312C2×C3—161no—
313C2×C3#188711yescharacteristic
314C2×C3#190612yes—
315C22—22no—
316C22—42no—
317C22—44no—
318C22—81no—
319C22—81no—
320C22—82no—
321C22—81no—
322C22—82no—
323C22#507711yescharacteristic
324C22#507711yescharacteristic
325C22#522011yescharacteristic
326C4—21no—
327C4—21no—
328C4—22no—
329C4—22no—
330C4—42no—
331C4—81no—
332C4—124no—
333C4—242no—
334C3#944111yescharacteristic, complemented, Sylow
335C2—21no—
336C2—21no—
337C2—82no—
338C2—161no—
339C2#7176311yescharacteristic
340C2#7211412yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3153C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3153D3
hypocenterC3#9441
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3153

Complements

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

NH ≅ G/Ntype
2. #82956C2semidirect
3. #71812C2semidirect
4. #71719C2semidirect
5. #67665C2semidirect
11. C3×#497C4semidirect
12. C3×#496C4semidirect
42. C3×C4⋊Q8C2×C4semidirect
43. C3×#127C2×C4semidirect
57. C3×#97C2×C4semidirect
70. #497Dic3semidirect
71. #496Dic3semidirect
141. C4⋊Q8C2×Dic3semidirect
142. #127C2×Dic3semidirect
159. #97C2×Dic3semidirect
334. C3#9441semidirect

Maximal subgroups

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

subgroupindex
C3×#31532
#829562
#718122
#717192
#676652
#94413