Finite Group Databasebeta

← #10691945

Subgroups — #10691945

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#83350C211yesmaximal, characteristic
2#79286C211yesmaximal, characteristic
3C3×#1481C211yesmaximal, characteristic, complemented
4#71838C211yesmaximal, characteristic, complemented
5#70480C211yesmaximal, characteristic
6#67738C211yesmaximal, characteristic, complemented
7#67707C211yesmaximal, characteristic, complemented
8#9303—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×#821—21no—
14#5087C412yescomplemented
15#5084C412yescomplemented
16#5066—22no—
17#5001—21no—
18#4995—21no—
19#4936C2211yescharacteristic
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22#4834—22no—
23D3×C22⋊C8—22no—
24C8×D12—22no—
25C3×Q8⋊C8—22no—
26#4551—21no—
27#4538—22no—
28#3077—31no—
29#2972—31no—
30#1804—31no—
31#1518—31no—
32#1481D311yescharacteristic, complemented
33#1301—31no—
34#1286—31no—
35C22×D12D411yescharacteristic
36C22×C4×D3—21no—
37C22×C3×C8—21no—
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×#125—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×#117—21no—
46C2×C3×C42C2311yescharacteristic
47C2×(C2×D3)⋊C4—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D411yescharacteristic
50C2×#208—21no—
51C2×D3×C8—24no—
52C4×D12—21no—
53C4×D12—21no—
54C4×D12C2×C412yes—
55C4×D12D412yes—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C4×C8D412yes—
61C3⋊#95—21no—
62C3⋊#95—21no—
63#830D412yes—
64#821—42no—
65(C2×D3)⋊C8—24no—
66#809—22no—
67#809—22no—
68#808—22no—
69#807—22no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)C2×C412yes—
73C4×C3⋊C8—21no—
74#533C2×D311yescharacteristic
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#423—31no—
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—22no—
93C23×D3—21no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4#4111yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—41no—
101C2×D12—21no—
102C2×D12—21no—
103C2×D12—21no—
104C2×D12C22⋊C412yes—
105C2×D12SD1612yes—
106C2×D12Q1612yes—
107C2×C4×D3—22no—
108C2×C4×D3—22no—
109C2×C4×D3—82no—
110C2×C3×C8—22no—
111C2×C3×C8—41no—
112C2×C3×C8—41no—
113C2×C3×C8—41no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C42—41no—
122C3×C42C22×C411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126(C2×D3)⋊C4—22no—
127(C2×D3)⋊C4—22no—
128C4⋊Dic3—21no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131C4⋊Dic3C22⋊C412yes—
132#208—41no—
133#208—41no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8—22no—
136D3×C8—84no—
137C22×D4—31no—
138C23×C4—61no—
139C2×M16—61no—
140C22×C8—21no—
141C22×C8—64no—
142C4⋊Q8C4×D311yescharacteristic
143C4⋊Q8C4×D311yescharacteristic
144#125—21no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147C4×Q8—21no—
148C4×Q8—21no—
149C4×D4—32no—
150C4×D4—32no—
151C4×D4—61no—
152C4×D4—61no—
153#117—21no—
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—32no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166#102—62no—
167#102—62no—
168#102—122no—
169C22⋊C8—21no—
170C22⋊C8—32no—
171C22⋊C8—64no—
172C4×C8—61no—
173C4×C8#6712yes—
174#95—61no—
175#95—61no—
176C23×C3#12111yescharacteristic
177C22×D3—22no—
178C22×D3—22no—
179C22×D3—22no—
180C22×D3—81no—
181C22×D3—81no—
182C3×Q8—42no—
183C3×Q8—81no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4C2×C22⋊C411yescharacteristic
189C2×C3×C4C4×D411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12311yescharacteristic
194C2×C3×C4C2×SD1611yescharacteristic
195C2×C3×C4C2×Q1611yescharacteristic
196C2×C3×C4#13511yescharacteristic
197C2×C3×C4#13511yescharacteristic
198C2×Dic3—22no—
199C2×Dic3—81no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203D12—22no—
204D12—22no—
205D12—22no—
206D12#10314yes—
207C4×D3—44no—
208C4×D3—82no—
209C3×C8—42no—
210C3×C8—81no—
211C3⋊C8—42no—
212C3⋊C8—42no—
213C24—61no—
214C2×Q8—21no—
215C2×Q8—21no—
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×C4#23511yescharacteristic
228C22×C4D3×D411yescharacteristic
229C22×C4D3×D411yescharacteristic
230M16—121no—
231M16—121no—
232C2×C8—22no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—62no—
237C2×C8—62no—
238C2×C8—244no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—32no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—61no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C22⋊C4—41no—
250C22⋊C4—62no—
251C22⋊C4—62no—
252C42—41no—
253C42C2×C4×D311yescharacteristic
254C42D3×D411yescharacteristic
255C42#23711yescharacteristic
256C42C2×#6711yescharacteristic
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×C4—81no—
272C3×C4—81no—
273C3×C4C2×#10311yescharacteristic
274C3×C4#41611yescharacteristic
275C3×C4#46511yescharacteristic
276C3×C4#46711yescharacteristic
277Dic3—42no—
278Dic3—161no—
279C23—62no—
280C23—62no—
281C23—62no—
282C23—241no—
283C23—241no—
284C23#88411yescharacteristic
285Q8—42no—
286Q8—81no—
287D4—34no—
288D4—62no—
289D4—62no—
290D4—62no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—62no—
296C2×C4—124no—
297C2×C4—242no—
298C2×C4—241no—
299C2×C4—241no—
300C2×C4—241no—
301C2×C4—241no—
302C2×C4D3×C22⋊C411yescharacteristic
303C2×C4#67⋊C411yescharacteristic
304C2×C4#88311yescharacteristic
305C2×C4D3×SD1611yescharacteristic
306C2×C4#91611yescharacteristic
307C2×C4D3×Q1611yescharacteristic
308C2×C4#91911yescharacteristic
309C2×C4C4×#6711yescharacteristic
310C2×C4#93711yescharacteristic
311C2×C4#93811yescharacteristic
312C8—42no—
313C8—81no—
314C8—122no—
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#480711yescharacteristic
330C22#481311yescharacteristic
331C22#495611yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—81no—
337C4—81no—
338C4—122no—
339C4—481no—
340C4D3×#10311yescharacteristic
341C4#480411yescharacteristic
342C4#517111yescharacteristic
343C4#518811yescharacteristic
344C3#930311yescharacteristic, complemented, Sylow
345C2—21no—
346C2—21no—
347C2—124no—
348C2#7001211yescharacteristic
349C2#7027011yescharacteristic
350C2#7040711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1481C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1481D3
hypocenterC3#9303
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
3. C3×#1481C2semidirect
4. #71838C2semidirect
6. #67738C2semidirect
7. #67707C2semidirect
14. #5087C4semidirect
15. #5084C4semidirect
32. #1481D3semidirect
344. C3#9303semidirect

Maximal subgroups

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

subgroupindex
#833502
#792862
C3×#14812
#718382
#704802
#677382
#677072
#93033