Finite Group Databasebeta

← #30853867

Subgroups — #30853867

Order 17152.

1010 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863434C211yesmaximal, characteristic
2#22568627C212yesmaximal
3C2×#16863351C211yesmaximal, characteristic
4#22560096C214yesmaximal, complemented
5#22560091C214yesmaximal, complemented
6#22559920C212yesmaximal
7C22×#108×C67C211yesmaximal, characteristic
8C22×#11761612C2211yescharacteristic
9C2×#14524002C2212yes—
10C2×#14524002C2211yescharacteristic
11#16863434C412yescomplemented
12#16863434C412yescomplemented
13#16863434C2212yes—
14#16863434C2212yes—
15C2×#14523977C412yescomplemented
16C2×#14523975C412yescomplemented
17#16863354C2218yes—
18#16863351—24no—
19#16863351C418yescomplemented
20M16×Dic67—24no—
21C2×#14523953C2212yes—
22C2×C8×Dic67C2212yes—
23#16863157C2218yes—
24C22×M16×C67C2212yes—
25C23×C8×C67C2211yescharacteristic
26C2×#108×C67C418yescomplemented
27C2×#108×C67C2212yes—
28C2×#108×C67C2212yes—
29C2×#11761612C2×C414yes—
30C2×#11761612D412yes—
31C2×#11761612D412yes—
32C2×#11761612Q814yes—
33C22×C67⋊C8C2312yes—
34#14524002—22no—
35#14524002—22no—
36#14524002—22no—
37#14524002C2×C414yes—
38#14524002C2×C414yes—
39#14524002C2314yes—
40C2×C4⋊Dic67—21no—
41C2×C4⋊Dic67C2×C414yes—
42C2×C22⋊Dic67—21no—
43C2×C22⋊Dic67—22no—
44C2×C4×Dic67—22no—
45C2×C4×Dic67C2312yes—
46#14523977C2×C414yes—
47#14523977C2×C414yes—
48#14523975C2×C414yes—
49#14523975C2×C414yes—
50#14523953—28no—
51#14523953C2×C418yescomplemented
52C8×Dic67—28no—
53C8×Dic67C2×C418yescomplemented
54C23×C4×C67C2311yescharacteristic
55C2×M16×C67—22no—
56C2×M16×C67—24no—
57C2×M16×C67C2×C418yes—
58C2×M16×C67C2314yes—
59C22×C8×C67—22no—
60C22×C8×C67C2×C414yes—
61C22×C8×C67C2×C411yescharacteristic
62C22×C8×C67C2×C411yescharacteristic
63C22×C8×C67C2×C411yescharacteristic
64C22×C8×C67C2×C411yescharacteristic
65C22×C8×C67C2×C414yes—
66C22×C8×C67C2311yescharacteristic
67C22×C8×C67C2311yescharacteristic
68#108×C67C2×C418yescomplemented
69#108×C67C2×C418yescomplemented
70C22×Dic67—22no—
71C22×Dic67—24no—
72#11761612C4⋊C418yes—
73#11761612C4⋊C418yes—
74C2×C67⋊C8C22×C414yes—
75C2×C67⋊C8C2×D414yes—
76C2×C67⋊C8C2×Q814yes—
77C4⋊Dic67—24no—
78C4⋊Dic67—24no—
79C4⋊Dic67C22×C418yes—
80C22⋊Dic67—24no—
81C22⋊Dic67—24no—
82C22⋊Dic67—24no—
83C4×Dic67—24no—
84C4×Dic67—24no—
85C4×Dic67—22no—
86C4×Dic67C22×C414yes—
87C24×C67#4111yescharacteristic
88C22×C4×C67C22×C412yes—
89C22×C4×C67C22×C411yescharacteristic
90C22×C4×C67C22×C411yescharacteristic
91C22×C4×C67C22×C412yes—
92C22×C4×C67C2×D411yescharacteristic
93C22×C4×C67C2×D411yescharacteristic
94C22×C4×C67C2×Q812yes—
95C22×C4×C67#4112yes—
96C22×C4×C67#4111yescharacteristic
97C22×C4×C67C2411yescharacteristic
98M16×C67—28no—
99M16×C67—24no—
100M16×C67C22×C418yes—
101C2×C8×C67—28no—
102C2×C8×C67—24no—
103C2×C8×C67C4214yes—
104C2×C8×C67C4214yes—
105C2×C8×C67C4214yes—
106C2×C8×C67C4214yes—
107C2×C8×C67C22×C412yes—
108C2×C8×C67C22×C412yes—
109C2×C8×C67C22×C411yescharacteristic
110C2×C8×C67C22×C411yescharacteristic
111C2×C8×C67C22×C411yescharacteristic
112C2×C8×C67C22×C411yescharacteristic
113C2×C8×C67C22×C412yes—
114C2×C8×C67C22×C412yes—
115C23×C67—42no—
116C23×C67#11712yes—
117C23×C67C4×D411yescharacteristic
118C23×C67C4×D411yescharacteristic
119C23×C67C4×Q812yes—
120C23×C67C2×#4111yescharacteristic
121C2×C4×C67C2×C4212yes—
122C2×C4×C67C2×C4212yes—
123C2×C4×C67C2×C4⋊C412yes—
124C2×C4×C67C2×C4⋊C412yes—
125C2×C4×C67#11712yes—
126C2×C4×C67C4×D411yescharacteristic
127C2×C4×C67C4×D411yescharacteristic
128C2×C4×C67C4×D412yes—
129C2×C4×C67C4×D412yes—
130C2×C4×C67C4×Q812yes—
131C2×C4×C67C4×Q812yes—
132C2×C4×C67C4×Q812yes—
133C2×C4×C67C23×C411yescharacteristic
134C2×C4×C67C23×C411yescharacteristic
135C2×C4×C67C23×C411yescharacteristic
136C2×C4×C67C22×D411yescharacteristic
137C2×C4×C67C22×Q811yescharacteristic
138C2×C4×C67C2×#4111yescharacteristic
139C2×Dic67—24no—
140C2×Dic67—28no—
141C2×Dic67—44no—
142C2×Dic67—84no—
143C8×C67—28no—
144C8×C67C2×C4212yes—
145C8×C67C2×C4212yes—
146C8×C67C2×C4212yes—
147C8×C67C2×C4212yes—
148C67⋊C8C2×C4⋊C418yes—
149C22×C67—22no—
150C22×C67—22no—
151C22×C67—22no—
152C22×C67—42no—
153C22×C67—42no—
154C22×C67C4×C4⋊C412yes—
155C22×C67C4×C4⋊C412yes—
156C22×C67C2×#11711yescharacteristic
157C22×C67C2×C4×D411yescharacteristic
158C22×C67C2×C4×Q811yescharacteristic
159C4×C67C4×C4⋊C412yes—
160C4×C67C4×C4⋊C412yes—
161C4×C67C22×C4211yescharacteristic
162C4×C67C22×C4⋊C411yescharacteristic
163C4×C67C2×C4×D411yescharacteristic
164C4×C67C2×C4×Q811yescharacteristic
165Dic67—44no—
166Dic67—48no—
167#20446—671nomaximal, Sylow
168C2×C67—21no—
169C2×C67—21no—
170C2×C67—22no—
171C2×C67—22no—
172C2×C67C2×C4×C4⋊C411yescharacteristic
173C2×C67#295712yes—
174C22×#108D6711yescharacteristic
175C2×#427—671no—
176#2873—672no—
177C2×#405—671no—
178#1773—674no—
179#1772—674no—
180#1733—672no—
181C67#2044611yescharacteristic, complemented, Sylow
182C22×M16—671no—
183C22×M16C2×D6712yes—
184C23×C8C2×D6711yescharacteristic
185C2×#117—671no—
186C2×#117—672no—
187C2×#108Dic6718yescomplemented
188C2×#108C2×D6712yes—
189C2×#108C2×D6712yes—
190#427—672no—
191#427—672no—
192#427—672no—
193#427—672no—
194C2×#107—672no—
195C2×#106—672no—
196#405—678no—
197#405—678no—
198#405—1344no—
199C4×M16—1344no—
200C2×#97—672no—
201C2×C4×C8—672no—
202#339—678no—
203C23×C4C22×D6711yescharacteristic
204C2×M16—22no—
205C2×M16—24no—
206C2×M16—674no—
207C2×M16—672no—
208C2×M16—672no—
209C2×M16—674no—
210C2×M16C2×Dic6718yes—
211C2×M16C22×D6714yes—
212C22×C8—22no—
213C22×C8—672no—
214C22×C8C4×D6714yes—
215C22×C8C4×D6711yescharacteristic
216C22×C8C4×D6711yescharacteristic
217C22×C8C4×D6711yescharacteristic
218C22×C8C4×D6711yescharacteristic
219C22×C8C2×Dic6714yes—
220C22×C8C22×D6711yescharacteristic
221C22×C8C22×D6711yescharacteristic
222#117—674no—
223#117—674no—
224#117—674no—
225#117—1342no—
226#117—1342no—
227#117—1342no—
228C2×C4⋊C4—674no—
229C2×C4⋊C4—1341no—
230C2×C22⋊C4—1341no—
231C2×C22⋊C4—1342no—
232C2×C42—672no—
233C2×C42—1342no—
234#108C2×Dic6718yescomplemented
235#108C2×Dic6718yescomplemented
236#107—674no—
237#107—674no—
238#106—674no—
239#106—674no—
240#97—678no—
241#97—1348no—
242C4×C8—678no—
243C4×C8—1348no—
244C24#1176162911yescharacteristic
245C22×C4—1342no—
246C22×C4—1344no—
247C22×C4C2×C4×D6712yes—
248C22×C4C2×C4×D6711yescharacteristic
249C22×C4C2×C4×D6711yescharacteristic
250C22×C4C22×Dic6712yes—
251C22×C4D4×D6711yescharacteristic
252C22×C4D4×D6711yescharacteristic
253C22×C4Q8×D6712yes—
254C22×C4#1176162712yes—
255C22×C4#1176162911yescharacteristic
256C22×C4C23×D6711yescharacteristic
257M16—28no—
258M16—24no—
259M16—678no—
260M16—678no—
261M16C22×Dic6718yes—
262C2×C8—28no—
263C2×C8—24no—
264C2×C8—674no—
265C2×C8—674no—
266C2×C8—674no—
267C2×C8C4×Dic6714yes—
268C2×C8C4×Dic6714yes—
269C2×C8C4×Dic6714yes—
270C2×C8C4×Dic6714yes—
271C2×C8C2×C4×D6712yes—
272C2×C8C2×C4×D6712yes—
273C2×C8C2×C4×D6711yescharacteristic
274C2×C8C2×C4×D6711yescharacteristic
275C2×C8C2×C4×D6711yescharacteristic
276C2×C8C2×C4×D6711yescharacteristic
277C2×C8C22×Dic6712yes—
278C2×C8C22×Dic6712yes—
279C4⋊C4—678no—
280C4⋊C4—1344no—
281C4⋊C4—1344no—
282C22⋊C4—1344no—
283C22⋊C4—1344no—
284C22⋊C4—1344no—
285C42—674no—
286C42—1344no—
287C42—1344no—
288C42—1342no—
289C23—42no—
290C23#1452400112yes—
291C23#1452400511yescharacteristic
292C23#1452400511yescharacteristic
293C23Q8×Dic6712yes—
294C23C2×#1176162911yescharacteristic
295C2×C4—2684no—
296C2×C4—5364no—
297C2×C4—1344no—
298C2×C4—1348no—
299C2×C4C2×C4×Dic6712yes—
300C2×C4C2×C4×Dic6712yes—
301C2×C4C4⋊C4×D6712yes—
302C2×C4C4⋊C4×D6712yes—
303C2×C4#1452400112yes—
304C2×C4#1452400511yescharacteristic
305C2×C4#1452400511yescharacteristic
306C2×C4D4×Dic6712yes—
307C2×C4D4×Dic6712yes—
308C2×C4#1452400812yes—
309C2×C4#1452400812yes—
310C2×C4Q8×Dic6712yes—
311C2×C4C22×C4×D6711yescharacteristic
312C2×C4C22×C4×D6711yescharacteristic
313C2×C4C23×Dic6711yescharacteristic
314C2×C4C2×D4×D6711yescharacteristic
315C2×C4C2×Q8×D6711yescharacteristic
316C2×C4C2×#1176162711yescharacteristic
317C8—28no—
318C8—678no—
319C8C2×C4×Dic6712yes—
320C8C2×C4×Dic6712yes—
321C8C2×C4×Dic6712yes—
322C8C2×C4×Dic6712yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22C4⋊C4×Dic6712yes—
329C22C4⋊C4×Dic6712yes—
330C22C2×#1452400111yescharacteristic
331C22C2×#1452400511yescharacteristic
332C22C2×Q8×Dic6711yescharacteristic
333C4—2684no—
334C4—2688no—
335C4C4⋊C4×Dic6712yes—
336C4C4⋊C4×Dic6712yes—
337C4C22×C4×Dic6711yescharacteristic
338C4C2×C4⋊C4×D6711yescharacteristic
339C4C2×D4×Dic6711yescharacteristic
340C4C2×#1452400811yescharacteristic
341C2—21no—
342C2—22no—
343C2—22no—
344C2—21no—
345C2C2×C4⋊C4×Dic6711yescharacteristic
346C2#2256912512yes—
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC4×C67C22×C42
FrattiniC22×C4C23×D67
FittingC22×#108×C67C2
socleC22×C67C2×C4×D4
prime radicalC24×C67#41
cosocleC22×C4×C67C24
hypercenterC22×#108D67
hypocenterC67#20446
LCSG ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
4. #22560096C2semidirect
5. #22560091C2semidirect
11. #16863434C4semidirect
12. #16863434C4semidirect
15. C2×#14523977C4semidirect
16. C2×#14523975C4semidirect
19. #16863351C4semidirect
26. C2×#108×C67C4semidirect
51. #14523953C2×C4semidirect
53. C8×Dic67C2×C4semidirect
68. #108×C67C2×C4semidirect
69. #108×C67C2×C4semidirect
181. C67#20446semidirect
187. C2×#108Dic67semidirect
234. #108C2×Dic67semidirect
235. #108C2×Dic67semidirect

Maximal subgroups

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

subgroupindex
C2×#168634342
#225686272
C2×#168633512
#225600962
#225600912
#225599202
C22×#108×C672
#2044667