Finite Group Databasebeta

← #30747609

Subgroups — #30747609

Order 17152.

782 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 159 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#22560205C211yesmaximal, characteristic, complemented
2#22558894C212yesmaximal, complemented
3#22558634C212yesmaximal
4#22558533C211yesmaximal, characteristic, complemented
5C2×#554×C67C211yesmaximal, characteristic
6#16863449C2212yes—
7#16863420C2212yes—
8C2×#14523962—21no—
9#16863165—21no—
10#16863142—24no—
11D4⋊(C67⋊C8)—24no—
12#16863122—22no—
13#554×C67—24no—
14#554×C67—24no—
15C2×C4⋊Q8×C67C2211yescharacteristic
16C2×#125×C67C412yes—
17C2×C22⋊Q8×C67—21no—
18C2×C22⋊Q8×C67—22no—
19C2×C4×Q8×C67C2211yescharacteristic
20C2×C4×D4×C67C412yes—
21C2×C4×D4×C67C2211yescharacteristic
22C22×C4⋊C4×C67—21no—
23C2×#11761612—22no—
24C4⋊(C67⋊C8)—22no—
25C4⋊(C67⋊C8)—22no—
26C4⋊(C67⋊C8)—22no—
27#14523962—42no—
28C22⋊(C67⋊C8)—22no—
29C4×C67⋊C8—22no—
30C22×Q8×C67—21no—
31C22×Q8×C67D411yescharacteristic
32C22×D4×C67C2×C411yescharacteristic
33C23×C4×C67—21no—
34C23×C4×C67—22no—
35C4⋊Q8×C67—22no—
36C4⋊Q8×C67D414yes—
37#125×C67—24no—
38#125×C67—24no—
39C22⋊Q8×C67—28no—
40C22⋊Q8×C67—216no—
41C4×Q8×C67—22no—
42C4×Q8×C67—22no—
43C4×D4×C67—24no—
44C4×D4×C67—24no—
45C4×D4×C67—22no—
46C4×D4×C67D414yes—
47C2×C4⋊C4×C67—28no—
48C2×C4⋊C4×C67—22no—
49C2×C4⋊C4×C67—22no—
50C2×C4⋊C4×C67—21no—
51C2×C4⋊C4×C67C2×C412yes—
52C2×C4⋊C4×C67C2×C411yescharacteristic
53C2×C4⋊C4×C67D411yescharacteristic
54C2×C4⋊C4×C67D412yes—
55C2×C22⋊C4×C67—22no—
56C2×C22⋊C4×C67—21no—
57C2×C42×C67C2×C412yes—
58C2×C42×C67C2311yescharacteristic
59#11761612—42no—
60#11761612—42no—
61C2×C67⋊C8—42no—
62C2×C67⋊C8—42no—
63C24×C67—21no—
64C2×Q8×C67—22no—
65C2×Q8×C67—24no—
66C2×Q8×C67—22no—
67C2×Q8×C67—21no—
68C2×Q8×C67—21no—
69C2×Q8×C67—82no—
70C2×D4×C67—24no—
71C2×D4×C67—21no—
72C2×D4×C67C22⋊C412yes—
73C22×C4×C67—21no—
74C22×C4×C67—22no—
75C22×C4×C67—28no—
76C22×C4×C67—24no—
77C22×C4×C67—42no—
78C22×C4×C67—41no—
79C22×C4×C67—41no—
80C22×C4×C67—84no—
81C22×C4×C67C22⋊C412yes—
82C22×C4×C67C22⋊C412yes—
83C22×C4×C67C22×C411yescharacteristic
84C22×C4×C67C2×D411yescharacteristic
85C22×C4×C67C2×D411yescharacteristic
86C4⋊C4×C67—28no—
87C4⋊C4×C67—28no—
88C4⋊C4×C67—21no—
89C4⋊C4×C67—22no—
90C4⋊C4×C67—22no—
91C4⋊C4×C67—24no—
92C4⋊C4×C67—24no—
93C4⋊C4×C67C22⋊C414yes—
94C4⋊C4×C67C22⋊C412yes—
95C22⋊C4×C67—28no—
96C22⋊C4×C67—24no—
97C42×C67—22no—
98C42×C67—22no—
99C42×C67C2×D412yes—
100C42×C67C2×D412yes—
101C23×C67—24no—
102C23×C67—22no—
103C23×C67—42no—
104C23×C67—41no—
105C23×C67—41no—
106C23×C67C2×C22⋊C411yescharacteristic
107Q8×C67—42no—
108Q8×C67—42no—
109Q8×C67—84no—
110D4×C67—24no—
111D4×C67—22no—
112D4×C67#10114yescomplemented
113C2×C4×C67—22no—
114C2×C4×C67—22no—
115C2×C4×C67—22no—
116C2×C4×C67—22no—
117C2×C4×C67—24no—
118C2×C4×C67—22no—
119C2×C4×C67—22no—
120C2×C4×C67—41no—
121C2×C4×C67—44no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—44no—
125C2×C4×C67—44no—
126C2×C4×C67—42no—
127C2×C4×C67—88no—
128C2×C4×C67C2×C22⋊C411yescharacteristic
129C2×C4×C67C2×C22⋊C411yescharacteristic
130C2×C4×C67#12012yes—
131C2×C4×C67#12012yes—
132C2×C4×C67#13512yes—
133C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C67#13711yescharacteristic
135C67⋊C8—82no—
136C67⋊C8—82no—
137C22×C67—21no—
138C22×C67—21no—
139C22×C67—24no—
140C22×C67—44no—
141C22×C67—44no—
142C22×C67—42no—
143C22×C67—42no—
144C22×C67#37911yescharacteristic
145C22×C67#41611yescharacteristic
146C22×C67#41711yescharacteristic
147C4×C67—22no—
148C4×C67—21no—
149C4×C67—44no—
150C4×C67—42no—
151C4×C67—42no—
152C4×C67—42no—
153C4×C67C2×#10112yes—
154C4×C67#44912yes—
155C4×C67#45212yes—
156#8398—671nomaximal, Sylow
157C2×C67—21no—
158C2×C67—21no—
159C2×C67—44no—
160C2×C67#178411yescharacteristic
161C2×C67#178811yescharacteristic
162C2×C67#179511yescharacteristic
163C2×#554D6711yescharacteristic
164#1801—671no—
165#1549—672no—
166#1486—672no—
167#1461—671no—
168C67#839811yescharacteristic, complemented, Sylow
169#554—24no—
170#554—24no—
171C2×C4⋊Q8C2×D6711yescharacteristic
172C2×#125Dic6712yes—
173C2×C22⋊Q8—21no—
174C2×C22⋊Q8—22no—
175C2×C4×Q8C2×D6711yescharacteristic
176C2×C4×D4Dic6712yes—
177C2×C4×D4C2×D6711yescharacteristic
178C22×C4⋊C4—21no—
179#431—672no—
180#423—672no—
181C2×#101—1341no—
182#343—1341no—
183#332—1344no—
184D4⋊C8—1344no—
185#324—1342no—
186C22×Q8—21no—
187C22×Q8C22⋊D6711yescharacteristic
188C22×D4C2×Dic6711yescharacteristic
189C23×C4—21no—
190C23×C4—22no—
191C2×M16—1342no—
192C4⋊Q8—22no—
193C4⋊Q8C22⋊D6714yes—
194#125—24no—
195#125—24no—
196C22⋊Q8—28no—
197C22⋊Q8—216no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—22no—
203C4×D4C22⋊D6714yes—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×Dic6712yes—
209C2×C4⋊C4C2×Dic6711yescharacteristic
210C2×C4⋊C4C22⋊D6711yescharacteristic
211C2×C4⋊C4C22⋊D6712yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C42C2×Dic6712yes—
215C2×C42C22×D6711yescharacteristic
216C4⋊C8—1342no—
217C4⋊C8—1342no—
218C4⋊C8—1342no—
219#101—2682no—
220C22⋊C8—1342no—
221C4×C8—1342no—
222C24—21no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—82no—
229C2×D4—24no—
230C2×D4—21no—
231C2×D4C22⋊Dic6712yes—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—28no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—84no—
240C22×C4C22⋊Dic6712yes—
241C22×C4C22⋊Dic6712yes—
242C22×C4C22×Dic6711yescharacteristic
243C22×C4C2×C22⋊D6711yescharacteristic
244C22×C4C2×C22⋊D6711yescharacteristic
245M16—2682no—
246M16—2682no—
247C2×C8—2682no—
248C2×C8—2682no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4C22⋊Dic6714yes—
257C4⋊C4C22⋊Dic6712yes—
258C22⋊C4—28no—
259C22⋊C4—24no—
260C42—22no—
261C42—22no—
262C42C2×C22⋊D6712yes—
263C42C2×C22⋊D6712yes—
264C23—24no—
265C23—22no—
266C23—42no—
267C23—41no—
268C23—41no—
269C23C2×C22⋊Dic6711yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—84no—
273D4—24no—
274D4—22no—
275D4#1452396214yescomplemented
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—88no—
291C2×C4C2×C22⋊Dic6711yescharacteristic
292C2×C4C2×C22⋊Dic6711yescharacteristic
293C2×C4#1452401212yes—
294C2×C4#1452401212yes—
295C2×C4#1452406312yes—
296C2×C4#1452406811yescharacteristic
297C2×C4#1452407311yescharacteristic
298C8—5362no—
299C8—5362no—
300C22—21no—
301C22—21no—
302C22—24no—
303C22—44no—
304C22—44no—
305C22—42no—
306C22—42no—
307C22#1686325211yescharacteristic
308C22#1686339211yescharacteristic
309C22#1686339711yescharacteristic
310C4—21no—
311C4—22no—
312C4—42no—
313C4—42no—
314C4—44no—
315C4—42no—
316C4C2×#1452396212yes—
317C4#1686354112yes—
318C4#1686355812yes—
319C2—21no—
320C2—21no—
321C2—44no—
322C2#2256013311yescharacteristic
323C2#2256014911yescharacteristic
324C2#2256017911yescharacteristic
subgroupG/H
centerC22#16863252
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC2×#554×C67C2
socleC22×C67#379
prime radicalC22×D4×C67C2×C4
cosocleC2×C42×C67C23
hypercenterC2×#554D67
hypocenterC67#8398
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #22560205C2semidirect
2. #22558894C2semidirect
4. #22558533C2semidirect
112. D4×C67#101semidirect
168. C67#8398semidirect
275. D4#14523962semidirect

Maximal subgroups

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

subgroupindex
#225602052
#225588942
#225586342
#225585332
C2×#554×C672
#839867