Finite Group Databasebeta

← #10697836

Subgroups — #10697836

Order 768.

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

normal subgroups only

This group has 626 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
1C3×#2868C211yesmaximal, characteristic, complemented
2#83216C211yesmaximal, characteristic
3#83213C211yesmaximal, characteristic
4#71775C211yesmaximal, characteristic, complemented
5#71774C211yesmaximal, characteristic, complemented
6#69395C211yesmaximal, characteristic
7#67611C211yesmaximal, characteristic, complemented
8#10283—31nomaximal, Sylow
9C2×C3×C4⋊Q8—22no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12#5477C2211yescharacteristic
13C3×#384—22no—
14C3×#384—22no—
15C3×C43C2211yescharacteristic
16#5227—22no—
17#5088—21no—
18#5085—21no—
19#5084—22no—
20#5083—22no—
21#5080—21no—
22#5079—21no—
23#4968—22no—
24C2×#805—22no—
25#4925C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4547—22no—
29#4546—22no—
30#3227—31no—
31#3206—31no—
32#2868D311yescharacteristic, complemented
33#1962—31no—
34#1549—31no—
35#1548—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—
48C3×C4⋊Q8—48no—
49C2×C3×C4⋊C4—28no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D412yes—
54C2×C3×C42—21no—
55C2×C3×C42—21no—
56C2×C3×C42D412yes—
57C2×C3×C42C2311yescharacteristic
58C2×#215—24no—
59#930—21no—
60#930—21no—
61C2×#208—21no—
62#875—21no—
63#874D411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869D411yescharacteristic
67#835—24no—
68#830—21no—
69#830—21no—
70#809—24no—
71#808—24no—
72#805—44no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)D412yes—
76C4⋊(C3⋊C8)D412yes—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79#532—31no—
80C2×C4⋊Q8—22no—
81C2×C4⋊Q8C2×D311yescharacteristic
82C2×C4⋊Q8C2×D311yescharacteristic
83#498—61no—
84#492—61no—
85#487—61no—
86#486—61no—
87#481—62no—
88#476—62no—
89#461—62no—
90#448—62no—
91#431—31no—
92#431—31no—
93#423—31no—
94C2×#104—62no—
95#384—22no—
96#384—22no—
97C43C2×D311yescharacteristic
98#333—62no—
99#332—62no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—44no—
103C2×C3×Q8—44no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
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—416no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C42—24no—
130C3×C42—24no—
131C3×C42—22no—
132C3×C42SD1614yescomplemented
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137#215—84no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142#208—42no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—41no—
146C2×#41—61no—
147C22×Q8—21no—
148C22×Q8#6712yes—
149C2×SD16—64no—
150C2×M16—61no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—31no—
160C4⋊Q8—48no—
161#127—31no—
162#124—61no—
163#121—61no—
164#121—61no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4#6712yes—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42#6712yes—
175C2×C42C22×D311yescharacteristic
176#106—64no—
177C4⋊C8—32no—
178C4⋊C8—32no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181#104—124no—
182#103—64no—
183#102—64no—
184C22⋊C8—61no—
185C22⋊C8—61no—
186C4×C8—61no—
187C4×C8—61no—
188C23×C3#12011yescharacteristic
189C22×D3—41no—
190C3×Q8—44no—
191C3×Q8—82no—
192C3×Q8—82no—
193C3×Q8—82no—
194C2×C3×C4—24no—
195C2×C3×C4—24no—
196C2×C3×C4—24no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—44no—
200C2×C3×C4—44no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12711yescharacteristic
209C2×C3×C4C2×SD1612yes—
210C2×C3×C4#13712yes—
211#67—82no—
212C2×Dic3—41no—
213D12—42no—
214D12—42no—
215D12—42no—
216C4×D3—81no—
217C4×D3—81no—
218Dic6—42no—
219C3⋊C8—42no—
220C3⋊C8—42no—
221C3⋊C8—81no—
222#41—122no—
223#41—122no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—44no—
227C2×Q8—44no—
228C2×Q8—41no—
229C2×Q8—41no—
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—416no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C42—24no—
261C42—24no—
262C42—22no—
263C42#21514yescomplemented
264C42D3×D411yescharacteristic
265C42D3×D411yescharacteristic
266C42D3×D411yescharacteristic
267C42#23711yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#39211yescharacteristic
271C22×C3#44812yes—
272C2×D3—81no—
273C2×D3—81no—
274C2×D3—81no—
275C3×C4—28no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—82no—
279C3×C4—82no—
280C3×C4#45312yes—
281C3×C4#45612yes—
282C3×C4#46112yes—
283C3×C4#49211yescharacteristic
284C3×C4#49711yescharacteristic
285Dic3—81no—
286C23—121no—
287C23#95311yescharacteristic
288Q8—44no—
289Q8—82no—
290Q8—82no—
291Q8—82no—
292Q8—122no—
293D4—122no—
294D4—122no—
295D4—122no—
296D4—242no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
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×C4C2×#21512yes—
316C2×C4#94212yes—
317C8—122no—
318C8—122no—
319C8—241no—
320C2×C3—22no—
321C2×C3#200211yescharacteristic
322C2×C3#200511yescharacteristic
323C2×C3#201311yescharacteristic
324D3—161no—
325C22—21no—
326C22—21no—
327C22—241no—
328C22—241no—
329C22—241no—
330C22#522311yescharacteristic
331C22#522712yes—
332C4—28no—
333C4—21no—
334C4—21no—
335C4—82no—
336C4—82no—
337C4—241no—
338C4#507712yes—
339C4#508312yes—
340C4#508511yescharacteristic
341C4#508611yescharacteristic
342C4#509512yes—
343C3#1028311yescharacteristic, complemented, Sylow
344C2—22no—
345C2—481no—
346C2#7176911yescharacteristic
347C2#7179111yescharacteristic
348C2#7179511yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2868C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2868D3
hypocenterC3#10283
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
1. C3×#2868C2semidirect
4. #71775C2semidirect
5. #71774C2semidirect
7. #67611C2semidirect
32. #2868D3semidirect
132. C3×C42SD16semidirect
263. C42#215semidirect
343. C3#10283semidirect

Maximal subgroups

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

subgroupindex
C3×#28682
#832162
#832132
#717752
#717742
#693952
#676112
#102833