Finite Group Databasebeta

← #30853805

Subgroups — #30853805

Order 17152.

1110 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863419C211yesmaximal, characteristic
2C2×#16863351C211yesmaximal, characteristic
3C22×#14523953C211yesmaximal, characteristic
4#22562612C211yesmaximal, characteristic
5#22560085C214yesmaximal, complemented
6#22560083C214yesmaximal, complemented
7#22559907C211yesmaximal, characteristic
8#22559906C211yesmaximal, characteristic
9C2×#405×C67C211yesmaximal, characteristic
10C22×#11761612C2211yescharacteristic
11C23×C67⋊C8C2211yescharacteristic
12C2×#14524002C2211yescharacteristic
13C22×C4×Dic67C2211yescharacteristic
14#16863419C418yescomplemented
15C2×Dic67⋊C8C2214yes—
16#16863351—24no—
17C2×#14523953C418yescomplemented
18C2×#14523953C2212yes—
19C2×#14523953C2212yes—
20C2×#14523953C2212yes—
21C2×C8×Dic67C2212yes—
22#16863265C412yes—
23#16863261C412yes—
24#16863254C414yes—
25C4×Dic67⋊C4C2214yes—
26#16863151C2214yes—
27#16863150C2214yes—
28C22×M16×C67C2211yescharacteristic
29C23×C8×C67C2211yescharacteristic
30C2×#117×C67C2211yescharacteristic
31#405×C67—24no—
32C2×#97×C67C2212yes—
33C2×C4×C8×C67C2212yes—
34C23×Dic67C2×C412yes—
35C2×#11761612—22no—
36C2×#11761612—22no—
37C2×#11761612C2×C414yes—
38C22×C67⋊C8—22no—
39C22×C67⋊C8—22no—
40C22×C67⋊C8C2×C414yes—
41C22×C67⋊C8C2312yes—
42C22×C67⋊C8C2312yes—
43#14524002—22no—
44#14524002—22no—
45C2×C4⋊Dic67C2×C412yes—
46C2×Dic67⋊C4C2×C418yes—
47C2×C22⋊Dic67C2×C412yes—
48C2×C4×Dic67C2×C411yescharacteristic
49C2×C4×Dic67C2×C411yescharacteristic
50C2×C4×Dic67C2×C414yes—
51C2×C4×Dic67C2×C414yes—
52C2×C4×Dic67C2312yes—
53C2×C4×Dic67C2312yes—
54Dic67⋊C8C2×C4116yes—
55#14523953—24no—
56#14523953C2×C418yes—
57#14523953C2×C418yes—
58C8×Dic67—24no—
59C67⋊#95C2×C418yes—
60C23×C4×C67C2311yescharacteristic
61C2×M16×C67—22no—
62C2×M16×C67—22no—
63C2×M16×C67C2×C414yes—
64C22×C8×C67C2×C414yes—
65C22×C8×C67D414yes—
66C22×C8×C67Q814yes—
67C22×C8×C67C2312yes—
68C22×C8×C67C2311yescharacteristic
69C22×C8×C67C2311yescharacteristic
70#117×C67—22no—
71#117×C67—22no—
72C2×C4⋊C4×C67C2×C412yes—
73C2×C22⋊C4×C67C2×C412yes—
74C2×C42×C67C2312yes—
75#97×C67—24no—
76C4×C8×C67—24no—
77C22×Dic67—44no—
78C22×Dic67C4218yes—
79C22×Dic67C22×C414yes—
80C22×Dic67C22×C412yes—
81C22×Dic67C22×C412yes—
82#11761612—28no—
83C2×C67⋊C8—28no—
84C2×C67⋊C8—22no—
85C2×C67⋊C8—22no—
86C2×C67⋊C8—22no—
87C2×C67⋊C8—22no—
88C2×C67⋊C8C22×C414yes—
89C2×C67⋊C8C22×C414yes—
90C4⋊Dic67—24no—
91Dic67⋊C4—48no—
92C22⋊Dic67—24no—
93C4×Dic67—22no—
94C4×Dic67—22no—
95C4×Dic67C4214yes—
96C4×Dic67C4214yes—
97C4×Dic67C22×C414yes—
98C4×Dic67C22×C414yes—
99C24×C67C22×C411yescharacteristic
100C22×C4×C67C22×C414yes—
101C22×C4×C67C22×C411yescharacteristic
102C22×C4×C67C22×C412yes—
103C22×C4×C67C22×C412yes—
104C22×C4×C67C2×D412yes—
105C22×C4×C67C2×Q812yes—
106C22×C4×C67#4112yes—
107C22×C4×C67#4112yes—
108C22×C4×C67C2411yescharacteristic
109M16×C67—28no—
110C2×C8×C67—22no—
111C2×C8×C67—22no—
112C2×C8×C67C4⋊C4116yescomplemented
113C2×C8×C67C22×C414yes—
114C2×C8×C67C22×C412yes—
115C2×C8×C67C22×C412yes—
116C2×C8×C67C2×D412yes—
117C2×C8×C67C2×D412yes—
118C2×C8×C67C2×Q812yes—
119C2×C8×C67C2×Q812yes—
120C4⋊C4×C67—24no—
121C22⋊C4×C67—24no—
122C42×C67—22no—
123C42×C67—22no—
124C23×C67—24no—
125C23×C67C2×C4212yes—
126C23×C67C4×D412yes—
127C23×C67C4×Q812yes—
128C23×C67C23×C411yescharacteristic
129C2×C4×C67—24no—
130C2×C4×C67—28no—
131C2×C4×C67C2×C4212yes—
132C2×C4×C67C2×C4⋊C414yes—
133C2×C4×C67#11714yes—
134C2×C4×C67C4×D414yes—
135C2×C4×C67C4×D412yes—
136C2×C4×C67C4×Q814yes—
137C2×C4×C67C4×Q812yes—
138C2×C4×C67C23×C411yescharacteristic
139C2×C4×C67C23×C411yescharacteristic
140C2×C4×C67C22×D411yescharacteristic
141C2×C4×C67C22×Q811yescharacteristic
142C2×C4×C67C2×#4111yescharacteristic
143C2×C4×C67C2×#4111yescharacteristic
144C2×Dic67—24no—
145C2×Dic67—24no—
146C2×Dic67—24no—
147C2×Dic67—28no—
148C2×Dic67—48no—
149C2×Dic67C2×C4214yes—
150C2×Dic67C2×C4214yes—
151C8×C67—24no—
152C8×C67C2×C4⋊C414yes—
153C8×C67C2×C4⋊C414yes—
154C67⋊C8—24no—
155C67⋊C8—24no—
156C22×C67—24no—
157C22×C67—24no—
158C22×C67—24no—
159C22×C67C4×C4⋊C414yes—
160C22×C67C22×C4211yescharacteristic
161C22×C67C2×C4×D411yescharacteristic
162C22×C67C2×C4×Q811yescharacteristic
163C22×C67#56814yes—
164C4×C67—44no—
165C4×C67C4×C4⋊C414yes—
166C4×C67C22×C4⋊C411yescharacteristic
167C4×C67C2×#11711yescharacteristic
168C4×C67C2×C4×D411yescharacteristic
169C4×C67C2×C4×Q811yescharacteristic
170Dic67—24no—
171Dic67—24no—
172Dic67—44no—
173#20438—671nomaximal, Sylow
174C2×C67—24no—
175C2×C67C2×C4×C4⋊C411yescharacteristic
176C2×C67#287312yes—
177C2×C67#293212yes—
178C2×C67#296112yes—
179C2×#423—671no—
180C2×#405—671no—
181C2×#405D6711yescharacteristic
182C22×#97—671no—
183#2285—671no—
184#1771—674no—
185#1771—674no—
186#1729—671no—
187#1729—671no—
188C67#2043811yescharacteristic, complemented, Sylow
189C22×M16—671no—
190C22×M16C2×D6711yescharacteristic
191C23×C8—671no—
192C23×C8C2×D6711yescharacteristic
193C2×#117—671no—
194C2×#117C2×D6711yescharacteristic
195C22×C42—671no—
196#423—678no—
197C2×C4⋊C8—674no—
198#405—24no—
199#405—1344no—
200C2×#97—678no—
201C2×#97—672no—
202C2×#97—672no—
203C2×#97—672no—
204C2×#97C2×D6712yes—
205C2×C4×C8—672no—
206C2×C4×C8C2×D6712yes—
207#384—672no—
208#382—672no—
209#380—674no—
210C4×C4⋊C4—674no—
211#336—674no—
212#336—674no—
213C23×C4—672no—
214C23×C4C22×D6711yescharacteristic
215C2×M16—22no—
216C2×M16—22no—
217C2×M16—674no—
218C2×M16—1342no—
219C2×M16—1342no—
220C2×M16C4×D6714yes—
221C22×C8—674no—
222C22×C8—672no—
223C22×C8—672no—
224C22×C8—1342no—
225C22×C8—1342no—
226C22×C8Dic13414yes—
227C22×C8C4×D6714yes—
228C22×C8C22⋊D6714yes—
229C22×C8C22×D6712yes—
230C22×C8C22×D6711yescharacteristic
231C22×C8C22×D6711yescharacteristic
232#117—22no—
233#117—22no—
234#117—1342no—
235#117—1342no—
236C2×C4⋊C4—678no—
237C2×C4⋊C4—672no—
238C2×C4⋊C4C4×D6712yes—
239C2×C22⋊C4—672no—
240C2×C22⋊C4C4×D6712yes—
241C2×C42—671no—
242C2×C42—671no—
243C2×C42—674no—
244C2×C42—674no—
245C2×C42—672no—
246C2×C42—672no—
247C2×C42C22×D6712yes—
248C4⋊C8—6716no—
249#97—24no—
250#97—678no—
251#97—678no—
252#97—1344no—
253C4×C8—24no—
254C4×C8—1344no—
255#95—678no—
256C24C2×C4×D6711yescharacteristic
257C22×C4—2684no—
258C22×C4—678no—
259C22×C4—672no—
260C22×C4—672no—
261C22×C4—674no—
262C22×C4C2×C4×D6714yes—
263C22×C4C2×C4×D6711yescharacteristic
264C22×C4C2×C4×D6712yes—
265C22×C4C2×C4×D6712yes—
266C22×C4C2×C22⋊D6712yes—
267C22×C4C2×Dic13412yes—
268C22×C4#1176162812yes—
269C22×C4#1176162812yes—
270C22×C4C23×D6711yescharacteristic
271M16—28no—
272M16—1348no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—674no—
276C2×C8—674no—
277C2×C8—1348no—
278C2×C8—1342no—
279C2×C8—1342no—
280C2×C8—1342no—
281C2×C8—1342no—
282C2×C8Dic67⋊C4116yescomplemented
283C2×C8C2×C4×D6714yes—
284C2×C8C2×C4×D6712yes—
285C2×C8C2×C4×D6712yes—
286C2×C8C2×C22⋊D6712yes—
287C2×C8C2×C22⋊D6712yes—
288C2×C8C2×Dic13412yes—
289C2×C8C2×Dic13412yes—
290C4⋊C4—24no—
291C4⋊C4—2688no—
292C4⋊C4—1344no—
293C22⋊C4—24no—
294C22⋊C4—1344no—
295C42—22no—
296C42—22no—
297C42—674no—
298C42—674no—
299C42—674no—
300C42—674no—
301C42—1342no—
302C42—1342no—
303C23—24no—
304C23C42×D6712yes—
305C23C4×C22⋊D6712yes—
306C23C4×Dic13412yes—
307C23C22×C4×D6711yescharacteristic
308C2×C4—24no—
309C2×C4—28no—
310C2×C4—2688no—
311C2×C4—674no—
312C2×C4—674no—
313C2×C4—1344no—
314C2×C4—1348no—
315C2×C4—1344no—
316C2×C4—1344no—
317C2×C4C42×D6712yes—
318C2×C4C2×Dic67⋊C414yes—
319C2×C4#1452400014yes—
320C2×C4C4×C22⋊D6712yes—
321C2×C4C4×C22⋊D6714yes—
322C2×C4C4×Dic13412yes—
323C2×C4C4×Dic13414yes—
324C2×C4C22×C4×D6711yescharacteristic
325C2×C4C22×C4×D6711yescharacteristic
326C2×C4C22×C22⋊D6711yescharacteristic
327C2×C4C22×Dic13411yescharacteristic
328C2×C4C2×#1176162811yescharacteristic
329C2×C4C2×#1176162811yescharacteristic
330C8—24no—
331C8—1344no—
332C8—1344no—
333C8C2×Dic67⋊C414yes—
334C8C2×Dic67⋊C414yes—
335C22—24no—
336C22—24no—
337C22—24no—
338C22C4×Dic67⋊C414yes—
339C22C2×C42×D6711yescharacteristic
340C22C2×C4×C22⋊D6711yescharacteristic
341C22C2×C4×Dic13411yescharacteristic
342C22#1686416114yes—
343C4—44no—
344C4—2684no—
345C4—1344no—
346C4—1344no—
347C4C4×Dic67⋊C414yes—
348C4C22×Dic67⋊C411yescharacteristic
349C4C2×#1452400011yescharacteristic
350C4C2×C4×C22⋊D6711yescharacteristic
351C4C2×C4×Dic13411yescharacteristic
352C2—24no—
353C2C2×C4×Dic67⋊C411yescharacteristic
354C2#2256862512yes—
355C2#2256892512yes—
356C2#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#20438
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
5. #22560085C2semidirect
6. #22560083C2semidirect
14. #16863419C4semidirect
17. C2×#14523953C4semidirect
112. C2×C8×C67C4⋊C4semidirect
188. C67#20438semidirect
282. C2×C8Dic67⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168634192
C2×#168633512
C22×#145239532
#225626122
#225600852
#225600832
#225599072
#225599062
C2×#405×C672
#2043867