Finite Group Databasebeta

← #10691946

Subgroups — #10691946

Order 768.

436 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 113 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#83367C211yesmaximal, characteristic
2#79263C211yesmaximal, characteristic
3C3×#1518C211yesmaximal, characteristic
4#71824C211yesmaximal, characteristic
5#69559C211yesmaximal, characteristic
6#67745C211yesmaximal, characteristic
7#67631C211yesmaximal, characteristic
8#9304—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#382C2211yescharacteristic
13#5094C412yescomplemented
14#5091C412yescomplemented
15#5063—22no—
16#5004—21no—
17#4995—21no—
18C2×#810—21no—
19#4925C2211yescharacteristic
20#4924C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22#4830—22no—
23#4718—22no—
24C8×Dic6—22no—
25C3×#328—22no—
26#4535—22no—
27#4529—21no—
28#3078—31no—
29#2982—31no—
30#1805—31no—
31#1518D311yescharacteristic
32#1481—31no—
33#1301—31no—
34#1286—31no—
35C22×Dic6D411yescharacteristic
36C3×C4⋊Q8C2×C411yescharacteristic
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×#125—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×#117—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46C2×C4⋊Dic3D411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C2×#208—21no—
50C22×C3⋊C8—21no—
51C4×Dic6—21no—
52C4×Dic6—21no—
53C4×Dic6C2×C412yes—
54C4×Dic6D412yes—
55C3×C4⋊C8—21no—
56C3×C22⋊C8D412yes—
57C3×C4×C8—21no—
58C3×#95—21no—
59C3×#95—21no—
60#830—21no—
61Dic3⋊C8—24no—
62C8×Dic3—24no—
63#810—22no—
64#810—22no—
65#810—42no—
66#808—22no—
67#807—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)C2×C412yes—
73C4×C3⋊C8D412yes—
74#533C2×D311yescharacteristic
75C2×C4×Q8—31no—
76#479—62no—
77#477—32no—
78#475—32no—
79C8×Q8—62no—
80C8×Q8—62no—
81#431—31no—
82#431—62no—
83#431C2×D311yescharacteristic
84#428—61no—
85#423—31no—
86C2×C4⋊C8—31no—
87#419—61no—
88C2×#103—61no—
89#382C2×D311yescharacteristic
90#340—61no—
91#328—22no—
92Q8⋊C8—62no—
93C2×C3×Q8—21no—
94C2×C3×Q8—21no—
95C22×C3×C4—41no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4#4111yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C2×Dic6—21no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C2×Dic6C22⋊C412yes—
105C2×Dic6SD1612yes—
106C2×Dic6Q1612yes—
107C2×C3×C8—22no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C42—41no—
118C3×C42C22×C411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122C4⋊Dic3—21no—
123C4⋊Dic3C22⋊C412yes—
124Dic3⋊C4—22no—
125Dic3⋊C4—22no—
126C4×Dic3—22no—
127C4×Dic3—22no—
128#208—41no—
129#208—41no—
130C2×C3⋊C8—22no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—41no—
135C22×Q8—31no—
136C2×M16—61no—
137C22×C8—61no—
138C4⋊Q8C4×D311yescharacteristic
139C4⋊Q8C4×D311yescharacteristic
140#125—21no—
141C22⋊Q8—21no—
142C22⋊Q8—21no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—32no—
146C4×Q8—32no—
147C4×Q8—61no—
148C4×Q8—61no—
149#117—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—61no—
153C2×C42—61no—
154C2×C42C22×D311yescharacteristic
155#107—62no—
156#106—62no—
157C4⋊C8—21no—
158C4⋊C8—32no—
159C4⋊C8—64no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164#103—62no—
165#103—62no—
166#103—122no—
167C22⋊C8—61no—
168C22⋊C8#6712yes—
169C4×C8—21no—
170C4×C8—32no—
171C4×C8—64no—
172#95—21no—
173#95—21no—
174C23×C3#12111yescharacteristic
175C3×Q8—42no—
176C3×Q8—81no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—81no—
182C2×C3×C4—81no—
183C2×C3×C4—81no—
184C2×C3×C4C2×C22⋊C411yescharacteristic
185C2×C3×C4C4×D411yescharacteristic
186C2×C3×C4C4×D411yescharacteristic
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12311yescharacteristic
190C2×C3×C4C2×SD1611yescharacteristic
191C2×C3×C4C2×Q1611yescharacteristic
192C2×C3×C4#13511yescharacteristic
193C2×C3×C4#13511yescharacteristic
194C2×Dic3—22no—
195C2×Dic3—22no—
196C2×Dic3—22no—
197C2×Dic3—22no—
198C2×Dic3—81no—
199Dic6—22no—
200Dic6—22no—
201Dic6—22no—
202Dic6#10314yes—
203C3×C8—42no—
204C3⋊C8—42no—
205C3⋊C8—42no—
206C3⋊C8—81no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—32no—
210C2×Q8—32no—
211C2×Q8—32no—
212C2×Q8—61no—
213C2×Q8—61no—
214C2×Q8—61no—
215C22×C4—41no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4D3×D411yescharacteristic
219C22×C4#23711yescharacteristic
220C22×C4C2×#6711yescharacteristic
221M16—121no—
222M16—121no—
223C2×C8—22no—
224C2×C8—62no—
225C2×C8—62no—
226C2×C8—121no—
227C2×C8—121no—
228C2×C8—121no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—32no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—62no—
239C4⋊C4—62no—
240C4⋊C4—61no—
241C22⋊C4—41no—
242C42—41no—
243C42—62no—
244C42—62no—
245C42C2×C4×D311yescharacteristic
246C42#23511yescharacteristic
247C42D3×D411yescharacteristic
248C42D3×D411yescharacteristic
249C22×C3—21no—
250C22×C3—21no—
251C22×C3#38611yescharacteristic
252C22×C3#46511yescharacteristic
253C22×C3#46711yescharacteristic
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—81no—
259C3×C4—81no—
260C3×C4—161no—
261C3×C4C2×#10311yescharacteristic
262C3×C4#41611yescharacteristic
263C3×C4#46511yescharacteristic
264C3×C4#46711yescharacteristic
265Dic3—24no—
266Dic3—42no—
267C23#93811yescharacteristic
268Q8—34no—
269Q8—42no—
270Q8—62no—
271Q8—62no—
272Q8—62no—
273Q8—81no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—241no—
286C2×C4D3×C22⋊C411yescharacteristic
287C2×C4#67⋊C411yescharacteristic
288C2×C4#88311yescharacteristic
289C2×C4#88411yescharacteristic
290C2×C4D3×SD1611yescharacteristic
291C2×C4#91611yescharacteristic
292C2×C4D3×Q1611yescharacteristic
293C2×C4#91911yescharacteristic
294C2×C4C4×#6711yescharacteristic
295C2×C4#93711yescharacteristic
296C8—42no—
297C8—122no—
298C8—122no—
299C8—241no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3#188111yescharacteristic
303C2×C3#189211yescharacteristic
304C2×C3#190511yescharacteristic
305C22—21no—
306C22—21no—
307C22#495611yescharacteristic
308C22#517111yescharacteristic
309C22#518811yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—64no—
315C4—81no—
316C4—81no—
317C4—122no—
318C4—161no—
319C4D3×#10311yescharacteristic
320C4#480411yescharacteristic
321C4#480711yescharacteristic
322C4#481311yescharacteristic
323C3#930411yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2#7005211yescharacteristic
327C2#7027011yescharacteristic
328C2#7041611yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1518C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1518D3
hypocenterC3#9304
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
13. #5094C4semidirect
14. #5091C4semidirect
323. C3#9304semidirect

Maximal subgroups

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

subgroupindex
#833672
#792632
C3×#15182
#718242
#695592
#677452
#676312
#93043