Finite Group Databasebeta

← #10692434

Subgroups — #10692434

Order 768.

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

normal subgroups only

This group has 420 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×#3216C211yesmaximal, characteristic
2#83280C211yesmaximal, characteristic
3#71824C211yesmaximal, characteristic, complemented
4#71823C211yesmaximal, characteristic, complemented
5#71807C211yesmaximal, characteristic
6#67756C211yesmaximal, characteristic, complemented
7#67755C211yesmaximal, characteristic, complemented
8#9373—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C3×#497C412yes—
12C3×#489C412yes—
13C3×#462—21no—
14C3×#460—21no—
15C3×#452—22no—
16C3×#432C2211yescharacteristic
17#5234—22no—
18#5012—21no—
19#5011—22no—
20D4×C3⋊C8—21no—
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23#4925C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4546—21no—
26#4544—21no—
27#4543—22no—
28#4535—21no—
29#4535—21no—
30#3216D311yescharacteristic
31#2981—31no—
32#1551—31no—
33#1537—31no—
34#1481—31no—
35#1290—31no—
36#1287—31no—
37C2×C3×#41D412yes—
38C2×C3×Q16—22no—
39C2×C3×SD16—22no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×#127D411yescharacteristic
44C3×#125—21no—
45C3×#124C2×C412yes—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×#117—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#208—22no—
57C22×C3⋊C8—22no—
58C22×C3⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×#103—22no—
61C3×#102—22no—
62C3×C22⋊C8—21no—
63C3×#97C2×C412yes—
64#830—22no—
65#830—22no—
66#830—21no—
67#830—21no—
68#830—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74#803—21no—
75#803D412yes—
76C4×C3⋊C8D411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78#533C2×D311yescharacteristic
79#532C2×D311yescharacteristic
80#497Dic312yes—
81#489Dic312yes—
82#462—21no—
83#460—21no—
84#452—22no—
85#436—61no—
86#435—62no—
87C8×D4—61no—
88#432—31no—
89#432—31no—
90#432C2×D311yescharacteristic
91#431—31no—
92C2×C4⋊C8—31no—
93#408—62no—
94#333—61no—
95#331—61no—
96#330—62no—
97Q8⋊C8—61no—
98Q8⋊C8—61no—
99C3×#41—82no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8C22⋊C412yes—
103C2×C3×D4—22no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4C22⋊C412yes—
107C22×C3×C4—22no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C3×Q16—82no—
112C3×SD16—82no—
113C2×C3×C8—22no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—41no—
125C3×C42—41no—
126C3×C42C22×C411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130#208—82no—
131C2×C3⋊C8—21no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C2×C3⋊C8—82no—
137C2×C3⋊C8—81no—
138C2×#41#6712yes—
139C2×Q16—22no—
140C2×SD16—22no—
141C2×M16—62no—
142C22×C8—62no—
143C22×C8—61no—
144C4⋊Q8C2×Dic311yescharacteristic
145C4⋊Q8C2×Dic311yescharacteristic
146C4⋊Q8#6711yescharacteristic
147#127#6711yescharacteristic
148#125—21no—
149#124C2×Dic312yes—
150C22⋊Q8—21no—
151C22⋊Q8—21no—
152#121—21no—
153#121—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×D4—21no—
157C4×D4—21no—
158#117—21no—
159C2×C42C22×D311yescharacteristic
160C4⋊C8—21no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166#103—22no—
167#102—22no—
168C22⋊C8—21no—
169C22⋊C8—62no—
170C22⋊C8—62no—
171C22⋊C8—61no—
172C22⋊C8—61no—
173C22⋊C8—61no—
174#97—32no—
175#97—61no—
176#97C2×Dic312yes—
177C4×C8—31no—
178C4×C8—31no—
179C23×C3—22no—
180C23×C3#12311yescharacteristic
181C3×Q8—42no—
182C3×Q8—41no—
183C3×Q8—41no—
184C3×Q8—42no—
185C3×D4—42no—
186C3×D4—41no—
187C3×D4—41no—
188C3×D4—81no—
189C3×D4—82no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—22no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4C2×C22⋊C411yescharacteristic
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4C4×D411yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C3×C8—42no—
206C3⋊C8—41no—
207C3⋊C8—42no—
208C3⋊C8—41no—
209C3⋊C8—81no—
210C3⋊C8—81no—
211#41—82no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8#21712yes—
215C2×D4—22no—
216C2×D4—21no—
217C2×D4—21no—
218C2×D4#21712yes—
219C22×C4—22no—
220C22×C4#23711yescharacteristic
221C22×C4#23711yescharacteristic
222C22×C4C2×#6711yescharacteristic
223Q16—82no—
224SD16—82no—
225M16—242no—
226C2×C8—22no—
227C2×C8—61no—
228C2×C8—61no—
229C2×C8—62no—
230C2×C8—121no—
231C2×C8—121no—
232C2×C8—242no—
233C2×C8—241no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—41no—
245C42—41no—
246C42D3×D411yescharacteristic
247C42D3×D411yescharacteristic
248C42C22×Dic311yescharacteristic
249C42C2×#6711yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3#38611yescharacteristic
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—42no—
263C3×C4—81no—
264C3×C4—81no—
265C3×C4#47111yescharacteristic
266C3×C4#47311yescharacteristic
267C23—22no—
268C23#93511yescharacteristic
269Q8—42no—
270Q8—41no—
271Q8—41no—
272Q8—42no—
273D4—42no—
274D4—41no—
275D4—41no—
276D4—81no—
277D4—82no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4D4×Dic311yescharacteristic
288C2×C4D4×Dic311yescharacteristic
289C2×C4#93711yescharacteristic
290C2×C4#93911yescharacteristic
291C2×C4#93911yescharacteristic
292C2×C4C2×#21711yescharacteristic
293C8—42no—
294C8—121no—
295C8—122no—
296C8—121no—
297C8—241no—
298C8—241no—
299C2×C3—41no—
300C2×C3—82no—
301C2×C3#188111yescharacteristic
302C2×C3#190811yescharacteristic
303C2×C3#191111yescharacteristic
304C22—21no—
305C22—21no—
306C22—21no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22#522011yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—81no—
318C4—81no—
319C4#520311yescharacteristic
320C4#520411yescharacteristic
321C3#937311yescharacteristic, complemented, Sylow
322C2—41no—
323C2—82no—
324C2#7185311yescharacteristic
325C2#7210311yescharacteristic
326C2#7211611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3216C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3216D3
hypocenterC3#9373
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3216

Complements

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

NH ≅ G/Ntype
3. #71824C2semidirect
4. #71823C2semidirect
6. #67756C2semidirect
7. #67755C2semidirect
321. C3#9373semidirect

Maximal subgroups

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

subgroupindex
C3×#32162
#832802
#718242
#718232
#718072
#677562
#677552
#93733