Finite Group Databasebeta

← #10691943

Subgroups — #10691943

Order 768.

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

normal subgroups only

This group has 488 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#79286C211yesmaximal, characteristic
2#79059C211yesmaximal, characteristic
3#70793C211yesmaximal, characteristic, complemented
4#70480C211yesmaximal, characteristic
5#70429C211yesmaximal, characteristic, complemented
6C3×#1301C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9303—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5131—21no—
15#5121—21no—
16C2×#821—21no—
17#4936C2211yescharacteristic
18#4925C2211yescharacteristic
19#4847—22no—
20#4846C412yescomplemented
21#4842C412yescomplemented
22#4834—22no—
23D3×C22⋊C8—22no—
24C8×D12—22no—
25#4551—21no—
26#4484—22no—
27#4467—22no—
28#3077—31no—
29#2972—31no—
30#1804—31no—
31#1518—31no—
32#1481—31no—
33#1301D311yescharacteristic, complemented
34#1286—31no—
35C22×D12D411yescharacteristic
36C22×C4×D3—21no—
37C2×C3×M16—21no—
38C22×C3×C8—21no—
39C2×C3×C42C2311yescharacteristic
40C2×(C2×D3)⋊C4—21no—
41#926—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3D411yescharacteristic
44#924—21no—
45#924—21no—
46C2×D3×C8—24no—
47C4×D12—21no—
48C4×D12—21no—
49C4×D12C2×C412yes—
50C4×D12D412yes—
51#870—21no—
52#869C2×C411yescharacteristic
53#869C2×C411yescharacteristic
54C4×Dic6—21no—
55C4×Dic6—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8C2×C412yes—
61C3×C22⋊C8D412yes—
62C3×C4×C8—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65#830—21no—
66#821—22no—
67#821—22no—
68#821—42no—
69(C2×D3)⋊C8—24no—
70#818—22no—
71#817—22no—
72C4⋊(C3⋊C8)—21no—
73C4×C3⋊C8D412yes—
74#533—31no—
75C2×C4×D4—31no—
76#478—62no—
77#476—32no—
78#474—32no—
79C8×D4—62no—
80C8×D4—62no—
81#431—31no—
82#431C2×D311yescharacteristic
83#428—61no—
84#423C2×D311yescharacteristic
85C2×C4⋊C8C2×D311yescharacteristic
86#418—61no—
87C2×#102—61no—
88C2×C22⋊C8—62no—
89#382—31no—
90#340—61no—
91#328—62no—
92Q8⋊C8—62no—
93C23×D3—21no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—41no—
99C2×D12—21no—
100C2×D12—21no—
101C2×D12—21no—
102C2×D12C22⋊C412yes—
103C2×D12SD1612yes—
104C2×D12Q1612yes—
105C2×C4×D3—22no—
106C2×C4×D3—22no—
107C2×C4×D3—82no—
108C2×Dic6—21no—
109C2×Dic6—21no—
110C3×M16—41no—
111C3×M16—41no—
112C2×C3×C8—22no—
113C2×C3×C8—22no—
114C2×C3×C8—41no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C3×C42C22×C411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42#4111yescharacteristic
121#217—41no—
122(C2×D3)⋊C4—22no—
123(C2×D3)⋊C4—22no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131C4⋊Dic3C22⋊C412yes—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134C4×Dic3—41no—
135C2×C3⋊C8—22no—
136D3×C8—84no—
137C22×D4—31no—
138C23×C4—61no—
139C2×M16—21no—
140C22×C8—21no—
141C22×C8—64no—
142C4⋊Q8—31no—
143C4⋊Q8—31no—
144#125—61no—
145C22⋊Q8—61no—
146C22⋊Q8—61no—
147C4×Q8—61no—
148C4×Q8—61no—
149C4×D4—32no—
150C4×D4—32no—
151C4×D4—61no—
152C4×D4—61no—
153#117—61no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—61no—
156C2×C22⋊C4—61no—
157C2×C42C22×D311yescharacteristic
158#107—62no—
159#106—62no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—21no—
164C4⋊C8—61no—
165C4⋊C8C4×D312yes—
166#102—62no—
167#102—62no—
168#102—122no—
169C22⋊C8—64no—
170C22⋊C8—61no—
171C22⋊C8D1212yes—
172C4×C8—21no—
173C4×C8—32no—
174#95—61no—
175#95—61no—
176C23×C3#12111yescharacteristic
177C22×D3—22no—
178C22×D3—22no—
179C22×D3—22no—
180C22×D3—81no—
181C22×D3—81no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
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—41no—
196C2×Dic3—41no—
197C2×Dic3—81no—
198C2×Dic3—81no—
199C2×Dic3—81no—
200C2×Dic3—81no—
201D12—22no—
202D12—22no—
203D12—22no—
204D12#10314yes—
205C4×D3—44no—
206C4×D3—82no—
207Dic6—42no—
208Dic6—81no—
209C3×C8—42no—
210C3×C8—42no—
211C3×C8—81no—
212C3⋊C8—42no—
213C24—61no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×D4—32no—
217C2×D4—32no—
218C2×D4—32no—
219C2×D4—61no—
220C2×D4—61no—
221C2×D4—61no—
222C22×C4—61no—
223C22×C4—62no—
224C22×C4—62no—
225C22×C4—121no—
226C22×C4—242no—
227C22×C4C2×D1211yescharacteristic
228C22×C4D3×D411yescharacteristic
229C22×C4#23911yescharacteristic
230M16—41no—
231M16—41no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—62no—
238C2×C8—244no—
239C4⋊C4—32no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C22⋊C4—62no—
250C22⋊C4—62no—
251C22⋊C4—121no—
252C42—121no—
253C42C2×C4×D311yescharacteristic
254C42#23511yescharacteristic
255C42D3×D411yescharacteristic
256C42D3×D411yescharacteristic
257C22×C3—21no—
258C22×C3—21no—
259C22×C3#38611yescharacteristic
260C22×C3#46511yescharacteristic
261C22×C3#46711yescharacteristic
262C2×D3—24no—
263C2×D3—42no—
264C2×D3—42no—
265C2×D3—82no—
266C2×D3—82no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4C2×#10311yescharacteristic
272C3×C4#41611yescharacteristic
273C3×C4#46511yescharacteristic
274C3×C4#46711yescharacteristic
275Dic3—42no—
276Dic3—81no—
277Dic3—81no—
278Dic3—161no—
279C23—62no—
280C23—62no—
281C23—62no—
282C23—241no—
283C23—241no—
284C23#89511yescharacteristic
285Q8—122no—
286Q8—241no—
287D4—34no—
288D4—62no—
289D4—62no—
290D4—62no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—62no—
294C2×C4—124no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—242no—
298C2×C4—241no—
299C2×C4—241no—
300C2×C4—241no—
301C2×C4—241no—
302C2×C4C4×D1211yescharacteristic
303C2×C4D3×C22⋊C411yescharacteristic
304C2×C4#88211yescharacteristic
305C2×C4#88411yescharacteristic
306C2×C4#89311yescharacteristic
307C2×C4#89411yescharacteristic
308C2×C4#91211yescharacteristic
309C2×C4D3×SD1611yescharacteristic
310C2×C4#91611yescharacteristic
311C2×C4D3×Q1611yescharacteristic
312C8—42no—
313C8—42no—
314C8—81no—
315C8—122no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3#188111yescharacteristic
319C2×C3#189211yescharacteristic
320C2×C3#190411yescharacteristic
321D3—44no—
322C22—21no—
323C22—21no—
324C22—64no—
325C22—122no—
326C22—122no—
327C22—242no—
328C22—242no—
329C22#466511yescharacteristic
330C22#486411yescharacteristic
331C22#488711yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—122no—
337C4—241no—
338C4—241no—
339C4—481no—
340C4#477211yescharacteristic
341C4#477811yescharacteristic
342C4D3×#10311yescharacteristic
343C4#481311yescharacteristic
344C3#930311yescharacteristic, complemented, Sylow
345C2—21no—
346C2—21no—
347C2—124no—
348C2#6888211yescharacteristic
349C2#6894711yescharacteristic
350C2#6902711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#9303
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
3. #70793C2semidirect
5. #70429C2semidirect
6. C3×#1301C2semidirect
7. #67713C2semidirect
20. #4846C4semidirect
21. #4842C4semidirect
33. #1301D3semidirect
344. C3#9303semidirect

Maximal subgroups

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

subgroupindex
#792862
#790592
#707932
#704802
#704292
C3×#13012
#677132
#93033