Finite Group Databasebeta

← #30853789

Subgroups — #30853789

Order 17152.

1110 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863423C211yesmaximal, characteristic
2C2×#16863351C211yesmaximal, characteristic
3C22×#14523953C211yesmaximal, characteristic
4#22562693C211yesmaximal, characteristic
5#22560081C214yesmaximal, complemented
6#22560076C214yesmaximal, complemented
7#22559907C211yesmaximal, characteristic
8#22559902C211yesmaximal, characteristic
9C2×#405×C67C211yesmaximal, characteristic
10C22×#11761612C2211yescharacteristic
11C23×C67⋊C8C2211yescharacteristic
12C2×#14524002C2211yescharacteristic
13C22×C4×Dic67C2211yescharacteristic
14#16863423C418yescomplemented
15C2×C4⋊(C67⋊C8)C2214yes—
16#16863351—24no—
17C2×#14523953—24no—
18C2×#14523953C2212yes—
19C2×#14523953C2212yes—
20C2×#14523953C2212yes—
21C2×C8×Dic67C2212yes—
22#16863295C414yes—
23#16863271C414yes—
24C4×C4⋊Dic67C2214yes—
25#16863150C2214yes—
26#16863150C2214yes—
27C22×M16×C67C2211yescharacteristic
28C23×C8×C67C2211yescharacteristic
29C2×#117×C67C2211yescharacteristic
30#405×C67C418yescomplemented
31C2×#97×C67C2212yes—
32C2×C4×C8×C67C2212yes—
33C23×Dic67C2×C412yes—
34C2×#11761612—22no—
35C2×#11761612—22no—
36C2×#11761612C2×C414yes—
37C22×C67⋊C8—22no—
38C22×C67⋊C8—22no—
39C22×C67⋊C8C2×C414yes—
40C22×C67⋊C8C2312yes—
41C22×C67⋊C8C2312yes—
42#14524002—22no—
43#14524002—22no—
44C2×C4⋊Dic67C2×C412yes—
45C2×C4⋊Dic67C2×C418yes—
46C2×C22⋊Dic67C2×C412yes—
47C2×C4×Dic67—22no—
48C2×C4×Dic67—22no—
49C2×C4×Dic67C2×C412yes—
50C2×C4×Dic67C2312yes—
51C2×C4×Dic67C2312yes—
52C4⋊(C67⋊C8)C2×C4116yescomplemented
53#14523953—24no—
54#14523953—24no—
55#14523953—24no—
56C8×Dic67—24no—
57C67⋊#95C2×C418yes—
58C23×C4×C67C2311yescharacteristic
59C2×M16×C67—22no—
60C2×M16×C67—22no—
61C2×M16×C67C2×C414yes—
62C22×C8×C67C2×C414yes—
63C22×C8×C67D414yes—
64C22×C8×C67Q814yes—
65C22×C8×C67C2312yes—
66C22×C8×C67C2311yescharacteristic
67C22×C8×C67C2311yescharacteristic
68#117×C67C2×C414yes—
69#117×C67C2×C414yes—
70C2×C4⋊C4×C67C2×C412yes—
71C2×C22⋊C4×C67C2×C412yes—
72C2×C42×C67C2312yes—
73#97×C67C2×C418yescomplemented
74C4×C8×C67C2×C418yescomplemented
75C22×Dic67—24no—
76C22×Dic67—44no—
77C22×Dic67C22×C414yes—
78C22×Dic67C22×C414yes—
79#11761612—28no—
80C2×C67⋊C8—28no—
81C2×C67⋊C8—22no—
82C2×C67⋊C8—22no—
83C2×C67⋊C8—22no—
84C2×C67⋊C8—22no—
85C2×C67⋊C8C22×C414yes—
86C2×C67⋊C8C22×C414yes—
87C4⋊Dic67—24no—
88C4⋊Dic67—48no—
89C22⋊Dic67—24no—
90C4×Dic67—24no—
91C4×Dic67—22no—
92C4×Dic67—22no—
93C4×Dic67—22no—
94C4×Dic67—22no—
95C24×C67C22×C411yescharacteristic
96C22×C4×C67C22×C414yes—
97C22×C4×C67C22×C411yescharacteristic
98C22×C4×C67C22×C412yes—
99C22×C4×C67C22×C412yes—
100C22×C4×C67C2×D412yes—
101C22×C4×C67C2×Q812yes—
102C22×C4×C67#4112yes—
103C22×C4×C67#4112yes—
104C22×C4×C67C2411yescharacteristic
105M16×C67—28no—
106C2×C8×C67—22no—
107C2×C8×C67—22no—
108C2×C8×C67C4⋊C4116yescomplemented
109C2×C8×C67C22×C414yes—
110C2×C8×C67C22×C412yes—
111C2×C8×C67C22×C412yes—
112C2×C8×C67C2×D412yes—
113C2×C8×C67C2×D412yes—
114C2×C8×C67C2×Q812yes—
115C2×C8×C67C2×Q812yes—
116C4⋊C4×C67C4218yes—
117C22⋊C4×C67C4218yes—
118C42×C67C22×C414yes—
119C42×C67C22×C414yes—
120C23×C67—24no—
121C23×C67C2×C4212yes—
122C23×C67C4×D412yes—
123C23×C67C4×Q812yes—
124C23×C67C23×C411yescharacteristic
125C2×C4×C67—28no—
126C2×C4×C67C2×C4218yes—
127C2×C4×C67C2×C4212yes—
128C2×C4×C67C2×C4⋊C414yes—
129C2×C4×C67#11714yes—
130C2×C4×C67C4×D414yes—
131C2×C4×C67C4×D412yes—
132C2×C4×C67C4×Q814yes—
133C2×C4×C67C4×Q812yes—
134C2×C4×C67C23×C411yescharacteristic
135C2×C4×C67C23×C411yescharacteristic
136C2×C4×C67C22×D411yescharacteristic
137C2×C4×C67C22×Q811yescharacteristic
138C2×C4×C67C2×#4111yescharacteristic
139C2×C4×C67C2×#4111yescharacteristic
140C2×Dic67—24no—
141C2×Dic67—24no—
142C2×Dic67—24no—
143C2×Dic67—28no—
144C2×Dic67—24no—
145C2×Dic67—48no—
146C8×C67—24no—
147C8×C67C2×C4⋊C414yes—
148C8×C67C2×C4⋊C414yes—
149C67⋊C8—24no—
150C67⋊C8—24no—
151C22×C67—24no—
152C22×C67—24no—
153C22×C67—24no—
154C22×C67C4×C4⋊C414yes—
155C22×C67C22×C4211yescharacteristic
156C22×C67C2×C4×D411yescharacteristic
157C22×C67C2×C4×Q811yescharacteristic
158C22×C67#56814yes—
159C4×C67—28no—
160C4×C67C4×C4⋊C414yes—
161C4×C67C22×C4⋊C411yescharacteristic
162C4×C67C2×#11711yescharacteristic
163C4×C67C2×C4×D411yescharacteristic
164C4×C67C2×C4×Q811yescharacteristic
165Dic67—44no—
166Dic67—44no—
167#20436—671nomaximal, Sylow
168C2×C67—24no—
169C2×C67C2×C4×C4⋊C411yescharacteristic
170C2×C67#287512yes—
171C2×C67#293012yes—
172C2×C67#296112yes—
173C2×#424—671no—
174C2×#405—671no—
175C2×#405D6711yescharacteristic
176C22×#97—671no—
177#2296—671no—
178#1771—674no—
179#1770—674no—
180#1729—671no—
181#1728—671no—
182C67#2043611yescharacteristic, complemented, Sylow
183C22×M16—671no—
184C22×M16C2×D6711yescharacteristic
185C23×C8—671no—
186C23×C8C2×D6711yescharacteristic
187C2×#117—671no—
188C2×#117C2×D6711yescharacteristic
189C22×C42—671no—
190#424—678no—
191C2×C4⋊C8—674no—
192#405—1344no—
193#405Dic6718yescomplemented
194C2×#97—672no—
195C2×#97—672no—
196C2×#97—672no—
197C2×#97—1344no—
198C2×#97C2×D6712yes—
199C2×C4×C8—672no—
200C2×C4×C8C2×D6712yes—
201#389—674no—
202#385—674no—
203C4×C4⋊C4—674no—
204#336—674no—
205#336—674no—
206C23×C4—672no—
207C23×C4C22×D6711yescharacteristic
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—674no—
211C2×M16—1342no—
212C2×M16—1342no—
213C2×M16C2×Dic6714yes—
214C22×C8—674no—
215C22×C8—672no—
216C22×C8—672no—
217C22×C8—1342no—
218C22×C8—1342no—
219C22×C8Dic13414yes—
220C22×C8D26814yes—
221C22×C8C2×Dic6714yes—
222C22×C8C22×D6712yes—
223C22×C8C22×D6711yescharacteristic
224C22×C8C22×D6711yescharacteristic
225#117—1342no—
226#117—1342no—
227#117C4×D6714yes—
228#117C2×Dic6714yes—
229C2×C4⋊C4—678no—
230C2×C4⋊C4—672no—
231C2×C4⋊C4C4×D6712yes—
232C2×C22⋊C4—672no—
233C2×C22⋊C4C4×D6712yes—
234C2×C42—672no—
235C2×C42—672no—
236C2×C42—672no—
237C2×C42—1342no—
238C2×C42—1342no—
239C2×C42C22×D6712yes—
240C4⋊C8—6716no—
241#97—1344no—
242#97—1344no—
243#97—1344no—
244#97C2×Dic6718yescomplemented
245C4×C8—1344no—
246C4×C8C2×Dic6718yescomplemented
247#95—678no—
248C24C2×C4×D6711yescharacteristic
249C22×C4—2684no—
250C22×C4—674no—
251C22×C4—674no—
252C22×C4—1344no—
253C22×C4C2×C4×D6714yes—
254C22×C4C2×C4×D6711yescharacteristic
255C22×C4C2×C4×D6712yes—
256C22×C4C22×Dic6712yes—
257C22×C4C2×D26812yes—
258C22×C4C2×Dic13412yes—
259C22×C4#1176162812yes—
260C22×C4#1176162812yes—
261C22×C4C23×D6711yescharacteristic
262M16—28no—
263M16—1348no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—674no—
267C2×C8—674no—
268C2×C8—1348no—
269C2×C8—1342no—
270C2×C8—1342no—
271C2×C8—1342no—
272C2×C8—1342no—
273C2×C8C4⋊Dic67116yescomplemented
274C2×C8C22×Dic6714yes—
275C2×C8C22×Dic6712yes—
276C2×C8C22×Dic6712yes—
277C2×C8C2×D26812yes—
278C2×C8C2×D26812yes—
279C2×C8C2×Dic13412yes—
280C2×C8C2×Dic13412yes—
281C4⋊C4—2688no—
282C4⋊C4—1344no—
283C4⋊C4C4×Dic6718yes—
284C22⋊C4—1344no—
285C22⋊C4C4×Dic6718yes—
286C42—1344no—
287C42—1342no—
288C42—1342no—
289C42—1342no—
290C42—1342no—
291C42C2×C4×D6714yes—
292C42C22×Dic6714yes—
293C23—24no—
294C23C2×C4×Dic6712yes—
295C23C4×D26812yes—
296C23C4×Dic13412yes—
297C23C22×C4×D6711yescharacteristic
298C2×C4—28no—
299C2×C4—2688no—
300C2×C4—1344no—
301C2×C4—1344no—
302C2×C4—1344no—
303C2×C4—1348no—
304C2×C4—1344no—
305C2×C4C2×C4×Dic6718yes—
306C2×C4C2×C4×Dic6712yes—
307C2×C4C2×C4⋊Dic6714yes—
308C2×C4#1452400214yes—
309C2×C4C4×D26812yes—
310C2×C4C4×D26814yes—
311C2×C4C4×Dic13412yes—
312C2×C4C4×Dic13414yes—
313C2×C4C22×C4×D6711yescharacteristic
314C2×C4C23×Dic6711yescharacteristic
315C2×C4C22×D26811yescharacteristic
316C2×C4C22×Dic13411yescharacteristic
317C2×C4C2×#1176162811yescharacteristic
318C2×C4C2×#1176162811yescharacteristic
319C8—24no—
320C8—1344no—
321C8—1344no—
322C8C2×C4⋊Dic6714yes—
323C8C2×C4⋊Dic6714yes—
324C22—24no—
325C22—24no—
326C22—24no—
327C22C4×C4⋊Dic6714yes—
328C22C22×C4×Dic6711yescharacteristic
329C22C2×C4×D26811yescharacteristic
330C22C2×C4×Dic13411yescharacteristic
331C22#1686416114yes—
332C4—28no—
333C4—2684no—
334C4—2684no—
335C4C4×C4⋊Dic6714yes—
336C4C22×C4⋊Dic6711yescharacteristic
337C4C2×#1452400211yescharacteristic
338C4C2×C4×D26811yescharacteristic
339C4C2×C4×Dic13411yescharacteristic
340C2—24no—
341C2C2×C4×C4⋊Dic6711yescharacteristic
342C2#2256863512yes—
343C2#2256891412yes—
344C2#2256914712yes—
subgroupG/H
center, FrattiniC22×C4C23×D67
commutatorC22×C67C22×C42
FittingC2×#405×C67C2
socleC23×C67C23×C4
prime radicalC24×C67C22×C4
cosocleC22×C4×C67C24
hypercenterC2×#405D67
hypocenterC67#20436
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
5. #22560081C2semidirect
6. #22560076C2semidirect
14. #16863423C4semidirect
30. #405×C67C4semidirect
52. C4⋊(C67⋊C8)C2×C4semidirect
73. #97×C67C2×C4semidirect
74. C4×C8×C67C2×C4semidirect
108. C2×C8×C67C4⋊C4semidirect
182. C67#20436semidirect
193. #405Dic67semidirect
244. #97C2×Dic67semidirect
246. C4×C8C2×Dic67semidirect
273. C2×C8C4⋊Dic67semidirect

Maximal subgroups

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

subgroupindex
C2×#168634232
C2×#168633512
C22×#145239532
#225626932
#225600812
#225600762
#225599072
#225599022
C2×#405×C672
#2043667