Finite Group Databasebeta

← #10697867

Subgroups — #10697867

Order 768.

554 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2862C211yesmaximal, characteristic, complemented
2#83215C211yesmaximal, characteristic
3#83213C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic, complemented
5#71773C211yesmaximal, characteristic, complemented
6#69395C211yesmaximal, characteristic
7#67611C211yesmaximal, characteristic, complemented
8#10288—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#391—22no—
13C3×#391—22no—
14C3×#382—22no—
15C3×C43C2211yescharacteristic
16#5227—22no—
17#5088—21no—
18#5087—22no—
19#5085—21no—
20#5083—22no—
21#5079—21no—
22#5078—21no—
23#4967—22no—
24C2×#805—22no—
25#4925C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4547—22no—
29#4546—22no—
30#3226—31no—
31#3206—31no—
32#2862D311yescharacteristic, complemented
33#1962—31no—
34#1548—31no—
35#1546—31no—
36#1276—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8D412yes—
39C2×#235—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D412yes—
52C2×C3×C42—21no—
53C2×C3×C42—21no—
54C2×C3×C42D412yes—
55C2×C3×C42C2311yescharacteristic
56C2×#215—24no—
57#930—21no—
58#930—21no—
59C2×#208—21no—
60#875—21no—
61#874D411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869D411yescharacteristic
65C3×#95—28no—
66#835—24no—
67#830—21no—
68#830—21no—
69#809—24no—
70#807—24no—
71#805—44no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78#532—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×C4⋊Q8C2×D311yescharacteristic
81#500—61no—
82#492—61no—
83#487—61no—
84#486—61no—
85#480—62no—
86#474—62no—
87#461—62no—
88#448—62no—
89#431—31no—
90#431—31no—
91#423—31no—
92C2×#104—62no—
93#391—22no—
94#391—22no—
95#382—22no—
96C43C2×D311yescharacteristic
97#333—62no—
98#332—62no—
99C2×C3×Q8—22no—
100C2×C3×Q8—22no—
101C2×C3×Q8—44no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C2×#67—41no—
113#235—42no—
114#235—42no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—21no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C42—24no—
129C3×C42—22no—
130C3×C42—22no—
131C3×C42—42no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#215—84no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141#208—42no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C2×#41—61no—
146C22×Q8—21no—
147C22×Q8#6712yes—
148C2×SD16—64no—
149C2×M16—61no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—31no—
159#127—31no—
160#124—61no—
161#121—61no—
162#121—61no—
163C4×D4—61no—
164C4×D4—61no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4#6712yes—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42#6712yes—
172C2×C42C22×D311yescharacteristic
173#107—64no—
174C4⋊C8—32no—
175C4⋊C8—32no—
176C4⋊C8—61no—
177C4⋊C8—61no—
178#104—124no—
179#103—64no—
180#102—64no—
181C22⋊C8—61no—
182C22⋊C8—61no—
183C4×C8—61no—
184C4×C8—61no—
185#95—28no—
186C23×C3#12011yescharacteristic
187C22×D3—41no—
188C3×Q8—44no—
189C3×Q8—82no—
190C3×Q8—82no—
191C3×Q8—81no—
192C3×Q8—81no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—24no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—44no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—44no—
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12711yescharacteristic
208C2×C3×C4#13512yes—
209C2×C3×C4AGL(1,ℤ/8)12yes—
210#67—82no—
211C2×Dic3—41no—
212D12—42no—
213D12—42no—
214D12—42no—
215C4×D3—81no—
216C4×D3—81no—
217Dic6—42no—
218C3⋊C8—42no—
219C3⋊C8—42no—
220C3⋊C8—81no—
221#41—122no—
222#41—122no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—44no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—121no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—121no—
239C22×C4C2×#6711yescharacteristic
240C22×C4C2×#6711yescharacteristic
241C22×C4C2×#6711yescharacteristic
242SD16—244no—
243M16—122no—
244C2×C8—62no—
245C2×C8—62no—
246C2×C8—121no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C22⋊C4—121no—
258C22⋊C4—121no—
259C42—24no—
260C42—22no—
261C42—22no—
262C42—42no—
263C42D3×D411yescharacteristic
264C42D3×D411yescharacteristic
265C42D3×D411yescharacteristic
266C42#23711yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3#39211yescharacteristic
270C22×C3#44912yes—
271C2×D3—81no—
272C2×D3—81no—
273C2×D3—81no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—44no—
277C3×C4—82no—
278C3×C4—82no—
279C3×C4#45512yes—
280C3×C4#45612yes—
281C3×C4#46312yes—
282C3×C4#49511yescharacteristic
283C3×C4#49611yescharacteristic
284Dic3—81no—
285C23—121no—
286C23#95311yescharacteristic
287Q8—44no—
288Q8—82no—
289Q8—82no—
290Q8—81no—
291Q8—81no—
292Q8—122no—
293D4—122no—
294D4—122no—
295D4—122no—
296D4—242no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—121no—
307C2×C4—241no—
308C2×C4—241no—
309C2×C4#93711yescharacteristic
310C2×C4#93711yescharacteristic
311C2×C4#93811yescharacteristic
312C2×C4#93911yescharacteristic
313C2×C4#93911yescharacteristic
314C2×C4#94011yescharacteristic
315C2×C4#94912yes—
316C2×C4#95012yes—
317C8—122no—
318C8—122no—
319C8—241no—
320C2×C3—22no—
321C2×C3#200411yescharacteristic
322C2×C3#200511yescharacteristic
323C2×C3#201211yescharacteristic
324D3—161no—
325C22—21no—
326C22—21no—
327C22—241no—
328C22—241no—
329C22—241no—
330C22#522311yescharacteristic
331C22#523812yes—
332C4—21no—
333C4—21no—
334C4—44no—
335C4—82no—
336C4—82no—
337C4—241no—
338C4#505712yes—
339C4#505811yescharacteristic
340C4#505911yescharacteristic
341C4#506112yes—
342C4#509512yes—
343C3#1028811yescharacteristic, complemented, Sylow
344C2—22no—
345C2—481no—
346C2#7172811yescharacteristic
347C2#7173111yescharacteristic
348C2#7179511yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2862C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2862D3
hypocenterC3#10288
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
1. C3×#2862C2semidirect
4. #71774C2semidirect
5. #71773C2semidirect
7. #67611C2semidirect
32. #2862D3semidirect
343. C3#10288semidirect

Maximal subgroups

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

subgroupindex
C3×#28622
#832152
#832132
#717742
#717732
#693952
#676112
#102883