Finite Group Databasebeta

← #32999506

Subgroups — #32999506

Order 18688.

322 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 75 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#22612188C211yesmaximal, characteristic
2#22611971C211yesmaximal, characteristic
3#22609291C211yesmaximal, characteristic
4#22607980C211yesmaximal, characteristic, complemented
5Q8⋊#11924003C211yesmaximal, characteristic, complemented
6#22607895C211yesmaximal, characteristic, complemented
7C73×#3395C211yesmaximal, characteristic, complemented
8C2×#14556534—21no—
9C2×#14556534—22no—
10#17988964C2211yescharacteristic
11#17988787—21no—
12C4⋊#11924003C2211yescharacteristic
13#17988387—21no—
14C4×#11924003C2211yescharacteristic
15#17988340C2211yescharacteristic
16#17988330—21no—
17#557×C73—21no—
18#494×C73—21no—
19#494×C73C2211yescharacteristic
20#462×C73—22no—
21#462×C73—21no—
22C8×Q8×C73C2211yescharacteristic
23C4×Q16×C73—21no—
24C4×Q16×C73C2211yescharacteristic
25C2×#11924015—41no—
26C2×D584—21no—
27C2×D584—21no—
28#14556588—21no—
29#14556534—42no—
30#14556534—44no—
31C2×#11924003—21no—
32C2×#11924003—21no—
33C4⋊#10596831—21no—
34#14556511—41no—
35C2×Q16×C73—22no—
36C2×Q16×C73—21no—
37C2×Q16×C73D411yescharacteristic
38C2×Q16×C73D412yes—
39C4⋊Q8×C73—22no—
40C4⋊Q8×C73—21no—
41#125×C73—42no—
42#125×C73—41no—
43C4×Q8×C73—22no—
44C4×Q8×C73—21no—
45C4×Q8×C73—41no—
46C4×Q8×C73D411yescharacteristic
47#107×C73D411yescharacteristic
48C4⋊C8×C73—21no—
49C4⋊C8×C73D411yescharacteristic
50#103×C73—22no—
51#103×C73—21no—
52C4×C8×C73—21no—
53C4×C8×C73C2311yescharacteristic
54C2×D292—41no—
55C2×D292—41no—
56C2×D292—41no—
57#11924015—82no—
58D584—42no—
59D584—41no—
60D584—41no—
61C2×#10596831—41no—
62#11924003—22no—
63#11924003—41no—
64C2×Q8×C73—21no—
65C2×Q8×C73—22no—
66C2×Q8×C73D811yescharacteristic
67Q16×C73—21no—
68Q16×C73—21no—
69Q16×C73—42no—
70Q16×C73—41no—
71Q16×C73—41no—
72Q16×C73—82no—
73C2×C8×C73—21no—
74C2×C8×C73C2×D411yescharacteristic
75C2×C8×C73C2×D411yescharacteristic
76C4⋊C4×C73—21no—
77C4⋊C4×C73—21no—
78C4⋊C4×C73—42no—
79C4⋊C4×C73—42no—
80C4⋊C4×C73—42no—
81C4⋊C4×C73—41no—
82C4⋊C4×C73—41no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73D811yescharacteristic
85C42×C73—21no—
86C42×C73—42no—
87C42×C73—41no—
88C42×C73C2×D411yescharacteristic
89C22×D73—81no—
90Q8×C73—21no—
91Q8×C73—41no—
92Q8×C73—42no—
93Q8×C73—42no—
94Q8×C73—41no—
95Q8×C73SD3212yescomplemented
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—41no—
99C2×C4×C73—42no—
100C2×C4×C73—41no—
101C2×C4×C73#12011yescharacteristic
102C2×C4×C73C2×D811yescharacteristic
103C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
104D292—81no—
105D292—81no—
106D292—81no—
107D292—81no—
108D292—161no—
109C8×C73—21no—
110C8×C73—41no—
111C8×C73#13511yescharacteristic
112C8×C73AGL(1,ℤ/8)11yescharacteristic
113#10596831—81no—
114C22×C73#44711yescharacteristic
115C2×D73—161no—
116C2×D73—161no—
117C2×D73—161no—
118C4×C73—22no—
119C4×C73—21no—
120C4×C73—41no—
121C4×C73—81no—
122C4×C73—82no—
123C4×C73—81no—
124C4×C73#44911yescharacteristic
125C4×C73#45311yescharacteristic
126C4×C73C2×SD3211yescharacteristic
127C4×C73#51011yescharacteristic
128#12330—731nomaximal, Sylow
129C2×C73#162111yescharacteristic
130C2×C73#218511yescharacteristic
131C2×C73#219311yescharacteristic
132D73—321no—
133#3395D7311yescharacteristic, complemented
134#2248—731no—
135#2207—731no—
136#1668—731no—
137#1362—731no—
138Q8⋊C16—731no—
139#1332—731no—
140C73#1233011yescharacteristic, complemented, Sylow
141#557—21no—
142C2×SD32—1461no—
143C2×SD32—1462no—
144#494—21no—
145#494C2×D7311yescharacteristic
146#493—731no—
147#462—22no—
148#462—21no—
149#460—1461no—
150C8×Q8C2×D7311yescharacteristic
151C4×Q16—21no—
152C4×Q16C2×D7311yescharacteristic
153C4⋊C16—731no—
154#357—1461no—
155C4×C16—731no—
156#335—731no—
157#331—1461no—
158C2×Q16—22no—
159C2×Q16—21no—
160C2×Q16C22⋊D7311yescharacteristic
161C2×Q16C22⋊D7312yes—
162C2×SD16—2921no—
163C2×D8—1461no—
164C2×D8—1461no—
165C4⋊Q8—22no—
166C4⋊Q8—21no—
167#127—1461no—
168#125—42no—
169#125—41no—
170C4×Q8—22no—
171C4×Q8—21no—
172C4×Q8—41no—
173C4×Q8C22⋊D7311yescharacteristic
174SD32—2922no—
175SD32—2924no—
176C2×C16—1461no—
177C2×C16—1461no—
178#107C22⋊D7311yescharacteristic
179C4⋊C8—21no—
180C4⋊C8—1461no—
181C4⋊C8C22⋊D7311yescharacteristic
182#103—22no—
183#103—21no—
184#102—2921no—
185C4×C8—21no—
186C4×C8C22×D7311yescharacteristic
187C2×Q8—21no—
188C2×Q8—22no—
189C2×Q8#1192401311yescharacteristic
190C2×D4—2921no—
191C2×D4—2921no—
192C2×D4—2921no—
193Q16—21no—
194Q16—21no—
195Q16—42no—
196Q16—41no—
197Q16—41no—
198Q16—82no—
199SD16—5842no—
200D8—2922no—
201D8—2921no—
202D8—2921no—
203C2×C8—21no—
204C2×C8—2921no—
205C2×C8C2×C22⋊D7311yescharacteristic
206C2×C8C2×C22⋊D7311yescharacteristic
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4#1192401311yescharacteristic
216C42—21no—
217C42—42no—
218C42—41no—
219C42C2×C22⋊D7311yescharacteristic
220C16—2921no—
221C16—1462no—
222C23—5841no—
223Q8—21no—
224Q8—41no—
225Q8—42no—
226Q8—42no—
227Q8—41no—
228Q8#1455653412yescomplemented
229D4—5841no—
230D4—5841no—
231D4—5841no—
232D4—5841no—
233D4—11681no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4#1455656011yescharacteristic
240C2×C4C2×#1192401311yescharacteristic
241C2×C4#1455661711yescharacteristic
242C8—21no—
243C8—41no—
244C8—5841no—
245C8#1455661111yescharacteristic
246C8#1455661611yescharacteristic
247C22—11681no—
248C22—11681no—
249C22—11681no—
250C22#1798872111yescharacteristic
251C4—22no—
252C4—21no—
253C4—41no—
254C4—81no—
255C4—82no—
256C4—81no—
257C4#1798873211yescharacteristic
258C4#1798875511yescharacteristic
259C4C2×#1455653411yescharacteristic
260C4#1798902811yescharacteristic
261C2—23361no—
262C2#2260899611yescharacteristic
263C2#2261184811yescharacteristic
264C2#2261189511yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#3395C2
socleC22×C73#447
prime radical#17988964C22
hypercenter#3395D73
hypocenterC73#12330
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3395

Complements

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

NH ≅ G/Ntype
4. #22607980C2semidirect
5. Q8⋊#11924003C2semidirect
6. #22607895C2semidirect
7. C73×#3395C2semidirect
95. Q8×C73SD32semidirect
133. #3395D73semidirect
140. C73#12330semidirect
228. Q8#14556534semidirect

Maximal subgroups

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

subgroupindex
#226121882
#226119712
#226092912
#226079802
Q8⋊#119240032
#226078952
C73×#33952
#1233073