Finite Group Databasebeta

← #30750502

Subgroups — #30750502

Order 17152.

510 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 131 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#22569294C211yesmaximal, characteristic
2#22558604C212yesmaximal
3#22558533C211yesmaximal, characteristic
4#22558042C212yesmaximal
5C4×C4⋊Q8×C67C211yesmaximal, characteristic
6#16863527—24no—
7#16863453—22no—
8#16863449C2212yes—
9#16863449C2212yes—
10Q8⋊(C67⋊C8)—24no—
11#16863122—22no—
12C2×C4⋊Q8×C67C411yescharacteristic
13C2×C4×Q8×C67C2211yescharacteristic
14C2×C4×Q8×C67C2211yescharacteristic
15#391×C67C411yescharacteristic
16#391×C67C411yescharacteristic
17#389×C67—22no—
18#384×C67C411yescharacteristic
19C4×C4⋊C4×C67—22no—
20C43×C67C2211yescharacteristic
21C4⋊(C67⋊C8)—24no—
22C4⋊(C67⋊C8)—22no—
23C4⋊(C67⋊C8)D414yes—
24C22⋊(C67⋊C8)—22no—
25C22⋊(C67⋊C8)—22no—
26#14523954—24no—
27C4×C67⋊C8—22no—
28C4×C67⋊C8—22no—
29C22×Q8×C67C2×C411yescharacteristic
30C22×Q8×C67C2×C411yescharacteristic
31C4⋊Q8×C67—22no—
32C4⋊Q8×C67—22no—
33C4×Q8×C67—22no—
34C4×Q8×C67—22no—
35C4×Q8×C67—22no—
36C4×Q8×C67—22no—
37C2×C4⋊C4×C67—22no—
38C2×C4⋊C4×C67—22no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67C2×C411yescharacteristic
42C2×C4⋊C4×C67D412yes—
43C2×C42×C67—21no—
44C2×C42×C67—21no—
45C2×C42×C67—21no—
46C2×C42×C67C2×C411yescharacteristic
47C2×C42×C67C2×C411yescharacteristic
48C2×C42×C67D412yes—
49C2×C42×C67C2311yescharacteristic
50#95×C67—22no—
51#95×C67—22no—
52C2×C67⋊C8—24no—
53C2×C67⋊C8—42no—
54C2×Q8×C67—22no—
55C2×Q8×C67—21no—
56C2×Q8×C67—21no—
57C2×Q8×C67—21no—
58C2×Q8×C67—22no—
59C2×Q8×C67—22no—
60C2×Q8×C67—22no—
61C2×Q8×C67M1612yes—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67C22⋊C412yes—
67C22×C4×C67C22⋊C412yes—
68C22×C4×C67C22×C411yescharacteristic
69C22×C4×C67C2×D411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71C4⋊C4×C67—28no—
72C4⋊C4×C67—22no—
73C4⋊C4×C67—24no—
74C4⋊C4×C67—22no—
75C4⋊C4×C67—44no—
76C4⋊C4×C67—42no—
77C42×C67—24no—
78C42×C67—22no—
79C42×C67—22no—
80C42×C67—21no—
81C42×C67—22no—
82C42×C67—42no—
83C42×C67—41no—
84C42×C67—41no—
85C42×C67M1612yes—
86C42×C67C2×D412yes—
87C42×C67#4112yes—
88C23×C67C2×C22⋊C411yescharacteristic
89Q8×C67—24no—
90Q8×C67—42no—
91Q8×C67—42no—
92Q8×C67—42no—
93C2×C4×C67—22no—
94C2×C4×C67—22no—
95C2×C4×C67—22no—
96C2×C4×C67—22no—
97C2×C4×C67—22no—
98C2×C4×C67—22no—
99C2×C4×C67—22no—
100C2×C4×C67—22no—
101C2×C4×C67—22no—
102C2×C4×C67—22no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
107C2×C4×C67—41no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—41no—
111C2×C4×C67C4×D412yes—
112C2×C4×C67#12011yescharacteristic
113C2×C4×C67#12112yes—
114C2×C4×C67#12311yescharacteristic
115C2×C4×C67C2×M1611yescharacteristic
116C2×C4×C67C2×M1611yescharacteristic
117C2×C4×C67#13112yes—
118C67⋊C8—44no—
119C67⋊C8—82no—
120C22×C67—21no—
121C22×C67—21no—
122C22×C67#38611yescharacteristic
123C22×C67#40711yescharacteristic
124C22×C67#40811yescharacteristic
125C4×C67—24no—
126C4×C67—22no—
127C4×C67—21no—
128C4×C67—42no—
129C4×C67—42no—
130C4×C67—42no—
131C4×C67—42no—
132C4×C67—81no—
133C4×C67#43512yes—
134C4×C67#43612yes—
135C4×C67#45612yes—
136#8962—671nomaximal, Sylow
137C2×C67—21no—
138C2×C67—21no—
139C2×C67#187011yescharacteristic
140C2×C67#188411yescharacteristic
141C2×C67#188611yescharacteristic
142#2983—671no—
143C4×C4⋊Q8D6711yescharacteristic
144#1479—672no—
145#1461—671no—
146#1276—672no—
147C67#896211yescharacteristic, complemented, Sylow
148C2×C4⋊Q8Dic6711yescharacteristic
149C2×C4×Q8C2×D6711yescharacteristic
150C2×C4×Q8C2×D6711yescharacteristic
151#446—1344no—
152#432—1342no—
153#431—672no—
154#431—672no—
155#391Dic6711yescharacteristic
156#391Dic6711yescharacteristic
157#389—22no—
158#384Dic6711yescharacteristic
159C4×C4⋊C4—22no—
160C43C2×D6711yescharacteristic
161Q8⋊C8—1344no—
162#324—1342no—
163C22×Q8C2×Dic6711yescharacteristic
164C22×Q8C2×Dic6711yescharacteristic
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C2×Dic6711yescharacteristic
175C2×C4⋊C4C2×Dic6711yescharacteristic
176C2×C4⋊C4C22⋊D6712yes—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42C2×Dic6711yescharacteristic
181C2×C42C2×Dic6711yescharacteristic
182C2×C42C22⋊D6712yes—
183C2×C42C22×D6711yescharacteristic
184C4⋊C8—674no—
185C4⋊C8—1344no—
186C4⋊C8—1342no—
187C22⋊C8—1342no—
188C22⋊C8—1342no—
189#97—1344no—
190C4×C8—1342no—
191C4×C8—1342no—
192#95—22no—
193#95—22no—
194C2×Q8—22no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8#1176161212yes—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4C22⋊Dic6712yes—
207C22×C4C22⋊Dic6712yes—
208C22×C4C22×Dic6711yescharacteristic
209C22×C4C2×C22⋊D6711yescharacteristic
210C22×C4C2×C22⋊D6711yescharacteristic
211C2×C8—2682no—
212C2×C8—1344no—
213C4⋊C4—28no—
214C4⋊C4—22no—
215C4⋊C4—24no—
216C4⋊C4—22no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C42—24no—
220C42—22no—
221C42—22no—
222C42—21no—
223C42—22no—
224C42—42no—
225C42—41no—
226C42—41no—
227C42#1176161212yes—
228C42D4×D6712yes—
229C42#1176162712yes—
230C23C2×C22⋊Dic6711yescharacteristic
231Q8—24no—
232Q8—42no—
233Q8—42no—
234Q8—42no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4D4×Dic6712yes—
254C2×C4#1452401111yescharacteristic
255C2×C4#1452401512yes—
256C2×C4#1452402411yescharacteristic
257C2×C4C2×#1176161211yescharacteristic
258C2×C4C2×#1176161211yescharacteristic
259C2×C4#1452405012yes—
260C8—2684no—
261C8—5362no—
262C22—21no—
263C22—21no—
264C22#1686327611yescharacteristic
265C22#1686336011yescharacteristic
266C22#1686336311yescharacteristic
267C4—24no—
268C4—22no—
269C4—21no—
270C4—42no—
271C4—42no—
272C4—42no—
273C4—42no—
274C4—81no—
275C4#1686346812yes—
276C4#1686347412yes—
277C4#1686357812yes—
278C2—21no—
279C2—21no—
280C2#2256050711yescharacteristic
281C2#2256059311yescharacteristic
282C2#2256060311yescharacteristic
subgroupG/H
centerC2×C4#14524011
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC4×C4⋊Q8×C67C2
socleC22×C67#386
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C42×C67C23
hypercenterC4×C4⋊Q8D67
hypocenterC67#8962
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
147. C67#8962semidirect

Maximal subgroups

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

subgroupindex
#225692942
#225586042
#225585332
#225580422
C4×C4⋊Q8×C672
#896267