Finite Group Databasebeta

← #10697838

Subgroups — #10697838

Order 768.

626 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 107 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#83253C211yesmaximal, characteristic, complemented
2#83216C211yesmaximal, characteristic
3#79312C211yesmaximal, characteristic
4C3×#1549C211yesmaximal, characteristic, complemented
5#70512C211yesmaximal, characteristic
6#70507C211yesmaximal, characteristic, complemented
7#67736C211yesmaximal, characteristic, complemented
8#10283—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#944—22no—
11#5477C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C2×#825—22no—
16#5088—21no—
17#5084—22no—
18#5080—21no—
19#4975—22no—
20#4975—22no—
21#4968—22no—
22C42×Dic3C2211yescharacteristic
23#4925C2211yescharacteristic
24#4841—22no—
25#4738—22no—
26#4693—21no—
27#4681—21no—
28C3×#332—22no—
29#4481—22no—
30#3227—31no—
31#3206—31no—
32#2868—31no—
33#1962—31no—
34#1549D311yescharacteristic, complemented
35#1548—31no—
36#1276—31no—
37C22×C3×Q8—21no—
38C2×#235—21no—
39C22×Dic6D412yes—
40C2×C3×M16—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D412yes—
47C2×C3×C42C2311yescharacteristic
48#944—48no—
49#930—21no—
50#930—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×Dic3⋊C4—28no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3D412yes—
57C2×#204—24no—
58#875—21no—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72#830—21no—
73#825—44no—
74#816—24no—
75#809—24no—
76#808—24no—
77C4⋊(C3⋊C8)D412yes—
78C4×C3⋊C8—21no—
79#532—31no—
80C2×C4⋊Q8—31no—
81C2×C4⋊Q8—62no—
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#431C2×D311yescharacteristic
93#423C2×D311yescharacteristic
94C2×#104—62no—
95#384—62no—
96#384—62no—
97C43—31no—
98#333—62no—
99#332—22no—
100C2×C3×Q8—44no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C22×C3×C4—22no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C2×#67—41no—
108C22×Dic3—22no—
109C22×Dic3—22no—
110#235—42no—
111#235—42no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×C4×D3—41no—
116C2×Dic6—22no—
117C2×Dic6—21no—
118C2×Dic6—22no—
119C2×Dic6—44no—
120C3×M16—42no—
121C2×C3×C8—22no—
122C2×C3×C8—41no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—41no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139Dic3⋊C4—416no—
140C4×Dic3—24no—
141C4×Dic3—24no—
142C4×Dic3—22no—
143C4×Dic3SD1614yescomplemented
144C2×C3⋊C8—22no—
145#204—84no—
146C2×#41—61no—
147C22×Q8—21no—
148C22×Q8—32no—
149C2×SD16—64no—
150C2×M16—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—31no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—128no—
161#127—31no—
162#124—61no—
163#121—61no—
164#121—61no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—68no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4D1212yes—
172C2×C42—32no—
173C2×C42—61no—
174C2×C42—61no—
175C2×C42C22×D311yescharacteristic
176#106—64no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—32no—
180C4⋊C8#6712yes—
181#104—124no—
182#103—64no—
183#102—64no—
184C22⋊C8—21no—
185C22⋊C8—61no—
186C4×C8—21no—
187C4×C8—61no—
188C23×C3#12011yescharacteristic
189C22×D3—41no—
190C3×Q8—82no—
191C3×Q8—82no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—44no—
195C2×C3×C4—42no—
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12711yescharacteristic
202C2×C3×C4C2×SD1612yes—
203C2×C3×C4#13712yes—
204#67—82no—
205C2×Dic3—24no—
206C2×Dic3—24no—
207C2×Dic3—24no—
208C2×Dic3—44no—
209C2×Dic3—41no—
210C2×Dic3—42no—
211D12—42no—
212D12—42no—
213D12—42no—
214C4×D3—81no—
215C4×D3—81no—
216Dic6—44no—
217Dic6—42no—
218Dic6—82no—
219C3×C8—42no—
220C3×C8—81no—
221C3⋊C8—42no—
222#41—122no—
223#41—122no—
224C2×Q8—44no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—61no—
228C2×Q8—62no—
229C2×Q8—62no—
230C2×Q8—124no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—121no—
235C22×C4—22no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4—121no—
239C22×C4C2×D1211yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242SD16—244no—
243M16—42no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—62no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—1216no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C42—34no—
261C42—64no—
262C42—64no—
263C42—62no—
264C42#23511yescharacteristic
265C42D3×D411yescharacteristic
266C42D3×D411yescharacteristic
267C42C2×#6711yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#39211yescharacteristic
271C22×C3#44812yes—
272C2×D3—81no—
273C2×D3—81no—
274C2×D3—81no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—82no—
278C3×C4#45312yes—
279C3×C4#45612yes—
280C3×C4#46112yes—
281C3×C4#49211yescharacteristic
282C3×C4#49711yescharacteristic
283Dic3—28no—
284Dic3—82no—
285Dic3—81no—
286C23—121no—
287C23#88211yescharacteristic
288Q8—82no—
289Q8—82no—
290Q8—124no—
291Q8—122no—
292Q8—242no—
293D4—122no—
294D4—122no—
295D4—122no—
296D4—242no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—64no—
302C2×C4—64no—
303C2×C4—64no—
304C2×C4—124no—
305C2×C4—121no—
306C2×C4—122no—
307C2×C4—241no—
308C2×C4—241no—
309C2×C4#88211yescharacteristic
310C2×C4#88411yescharacteristic
311C2×C4#88411yescharacteristic
312C2×C4D3×SD1612yes—
313C2×C4#91812yes—
314C2×C4#93011yescharacteristic
315C2×C4#93711yescharacteristic
316C2×C4#94011yescharacteristic
317C8—42no—
318C8—81no—
319C8—122no—
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#480612yes—
331C22#496011yescharacteristic
332C4—21no—
333C4—21no—
334C4—68no—
335C4—82no—
336C4—241no—
337C4—242no—
338C4#475212yes—
339C4#481812yes—
340C4#517511yescharacteristic
341C4#519011yescharacteristic
342C4#519812yes—
343C3#1028311yescharacteristic, complemented, Sylow
344C2—22no—
345C2—481no—
346C2#7011911yescharacteristic
347C2#7028211yescharacteristic
348C2#7032411yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1549C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1549D3
hypocenterC3#10283
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
1. #83253C2semidirect
4. C3×#1549C2semidirect
6. #70507C2semidirect
7. #67736C2semidirect
34. #1549D3semidirect
143. C4×Dic3SD16semidirect
343. C3#10283semidirect

Maximal subgroups

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

subgroupindex
#832532
#832162
#793122
C3×#15492
#705122
#705072
#677362
#102833