Finite Group Databasebeta

← #31995488

Subgroups — #31995488

Order 18176.

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×#17986037C211yesmaximal, characteristic
2#22596758C212yesmaximal
3C2×#17985954C211yesmaximal, characteristic
4#22588227C214yesmaximal, complemented
5#22588222C214yesmaximal, complemented
6#22588051C212yesmaximal
7C22×#108×C71C211yesmaximal, characteristic
8C22×#11765849C2211yescharacteristic
9C2×#14556087C2212yes—
10C2×#14556087C2211yescharacteristic
11#17986037C412yescomplemented
12#17986037C412yescomplemented
13#17986037C2212yes—
14#17986037C2212yes—
15C2×#14556062C412yescomplemented
16C2×#14556060C412yescomplemented
17#17985957C2218yes—
18#17985954—24no—
19#17985954C418yescomplemented
20M16×Dic71—24no—
21C2×#14556038C2212yes—
22C2×C8×Dic71C2212yes—
23#17985760C2218yes—
24C22×M16×C71C2212yes—
25C23×C8×C71C2211yescharacteristic
26C2×#108×C71C418yescomplemented
27C2×#108×C71C2212yes—
28C2×#108×C71C2212yes—
29C2×#11765849C2×C414yes—
30C2×#11765849D412yes—
31C2×#11765849D412yes—
32C2×#11765849Q814yes—
33C22×C71⋊C8C2312yes—
34#14556087—22no—
35#14556087—22no—
36#14556087—22no—
37#14556087C2×C414yes—
38#14556087C2×C414yes—
39#14556087C2314yes—
40C2×C4⋊Dic71—21no—
41C2×C4⋊Dic71C2×C414yes—
42C2×C22⋊Dic71—21no—
43C2×C22⋊Dic71—22no—
44C2×C4×Dic71—22no—
45C2×C4×Dic71C2312yes—
46#14556062C2×C414yes—
47#14556062C2×C414yes—
48#14556060C2×C414yes—
49#14556060C2×C414yes—
50#14556038—28no—
51#14556038C2×C418yescomplemented
52C8×Dic71—28no—
53C8×Dic71C2×C418yescomplemented
54C23×C4×C71C2311yescharacteristic
55C2×M16×C71—22no—
56C2×M16×C71—24no—
57C2×M16×C71C2×C418yes—
58C2×M16×C71C2314yes—
59C22×C8×C71—22no—
60C22×C8×C71C2×C414yes—
61C22×C8×C71C2×C411yescharacteristic
62C22×C8×C71C2×C411yescharacteristic
63C22×C8×C71C2×C411yescharacteristic
64C22×C8×C71C2×C411yescharacteristic
65C22×C8×C71C2×C414yes—
66C22×C8×C71C2311yescharacteristic
67C22×C8×C71C2311yescharacteristic
68#108×C71C2×C418yescomplemented
69#108×C71C2×C418yescomplemented
70C22×Dic71—22no—
71C22×Dic71—24no—
72#11765849C4⋊C418yes—
73#11765849C4⋊C418yes—
74C2×C71⋊C8C22×C414yes—
75C2×C71⋊C8C2×D414yes—
76C2×C71⋊C8C2×Q814yes—
77C4⋊Dic71—24no—
78C4⋊Dic71—24no—
79C4⋊Dic71C22×C418yes—
80C22⋊Dic71—24no—
81C22⋊Dic71—24no—
82C22⋊Dic71—24no—
83C4×Dic71—24no—
84C4×Dic71—24no—
85C4×Dic71—22no—
86C4×Dic71C22×C414yes—
87C24×C71#4111yescharacteristic
88C22×C4×C71C22×C412yes—
89C22×C4×C71C22×C411yescharacteristic
90C22×C4×C71C22×C411yescharacteristic
91C22×C4×C71C22×C412yes—
92C22×C4×C71C2×D411yescharacteristic
93C22×C4×C71C2×D411yescharacteristic
94C22×C4×C71C2×Q812yes—
95C22×C4×C71#4112yes—
96C22×C4×C71#4111yescharacteristic
97C22×C4×C71C2411yescharacteristic
98M16×C71—28no—
99M16×C71—24no—
100M16×C71C22×C418yes—
101C2×C8×C71—28no—
102C2×C8×C71—24no—
103C2×C8×C71C4214yes—
104C2×C8×C71C4214yes—
105C2×C8×C71C4214yes—
106C2×C8×C71C4214yes—
107C2×C8×C71C22×C412yes—
108C2×C8×C71C22×C412yes—
109C2×C8×C71C22×C411yescharacteristic
110C2×C8×C71C22×C411yescharacteristic
111C2×C8×C71C22×C411yescharacteristic
112C2×C8×C71C22×C411yescharacteristic
113C2×C8×C71C22×C412yes—
114C2×C8×C71C22×C412yes—
115C23×C71—42no—
116C23×C71#11712yes—
117C23×C71C4×D411yescharacteristic
118C23×C71C4×D411yescharacteristic
119C23×C71C4×Q812yes—
120C23×C71C2×#4111yescharacteristic
121C2×C4×C71C2×C4212yes—
122C2×C4×C71C2×C4212yes—
123C2×C4×C71C2×C4⋊C412yes—
124C2×C4×C71C2×C4⋊C412yes—
125C2×C4×C71#11712yes—
126C2×C4×C71C4×D411yescharacteristic
127C2×C4×C71C4×D411yescharacteristic
128C2×C4×C71C4×D412yes—
129C2×C4×C71C4×D412yes—
130C2×C4×C71C4×Q812yes—
131C2×C4×C71C4×Q812yes—
132C2×C4×C71C4×Q812yes—
133C2×C4×C71C23×C411yescharacteristic
134C2×C4×C71C23×C411yescharacteristic
135C2×C4×C71C23×C411yescharacteristic
136C2×C4×C71C22×D411yescharacteristic
137C2×C4×C71C22×Q811yescharacteristic
138C2×C4×C71C2×#4111yescharacteristic
139C2×Dic71—24no—
140C2×Dic71—28no—
141C2×Dic71—44no—
142C2×Dic71—84no—
143C8×C71—28no—
144C8×C71C2×C4212yes—
145C8×C71C2×C4212yes—
146C8×C71C2×C4212yes—
147C8×C71C2×C4212yes—
148C71⋊C8C2×C4⋊C418yes—
149C22×C71—22no—
150C22×C71—22no—
151C22×C71—22no—
152C22×C71—42no—
153C22×C71—42no—
154C22×C71C4×C4⋊C412yes—
155C22×C71C4×C4⋊C412yes—
156C22×C71C2×#11711yescharacteristic
157C22×C71C2×C4×D411yescharacteristic
158C22×C71C2×C4×Q811yescharacteristic
159C4×C71C4×C4⋊C412yes—
160C4×C71C4×C4⋊C412yes—
161C4×C71C22×C4211yescharacteristic
162C4×C71C22×C4⋊C411yescharacteristic
163C4×C71C2×C4×D411yescharacteristic
164C4×C71C2×C4×Q811yescharacteristic
165Dic71—44no—
166Dic71—48no—
167#20446—711nomaximal, Sylow
168C2×C71—21no—
169C2×C71—21no—
170C2×C71—22no—
171C2×C71—22no—
172C2×C71C2×C4×C4⋊C411yescharacteristic
173C2×C71#295712yes—
174C22×#108D7111yescharacteristic
175C2×#427—711no—
176#2873—712no—
177C2×#405—711no—
178#1773—714no—
179#1772—714no—
180#1733—712no—
181C71#2044611yescharacteristic, complemented, Sylow
182C22×M16—711no—
183C22×M16C2×D7112yes—
184C23×C8C2×D7111yescharacteristic
185C2×#117—711no—
186C2×#117—712no—
187C2×#108Dic7118yescomplemented
188C2×#108C2×D7112yes—
189C2×#108C2×D7112yes—
190#427—712no—
191#427—712no—
192#427—712no—
193#427—712no—
194C2×#107—712no—
195C2×#106—712no—
196#405—718no—
197#405—718no—
198#405—1424no—
199C4×M16—1424no—
200C2×#97—712no—
201C2×C4×C8—712no—
202#339—718no—
203C23×C4C22×D7111yescharacteristic
204C2×M16—22no—
205C2×M16—24no—
206C2×M16—714no—
207C2×M16—712no—
208C2×M16—712no—
209C2×M16—714no—
210C2×M16C2×Dic7118yes—
211C2×M16C22×D7114yes—
212C22×C8—22no—
213C22×C8—712no—
214C22×C8C4×D7114yes—
215C22×C8C4×D7111yescharacteristic
216C22×C8C4×D7111yescharacteristic
217C22×C8C4×D7111yescharacteristic
218C22×C8C4×D7111yescharacteristic
219C22×C8C2×Dic7114yes—
220C22×C8C22×D7111yescharacteristic
221C22×C8C22×D7111yescharacteristic
222#117—714no—
223#117—714no—
224#117—714no—
225#117—1422no—
226#117—1422no—
227#117—1422no—
228C2×C4⋊C4—714no—
229C2×C4⋊C4—1421no—
230C2×C22⋊C4—1422no—
231C2×C22⋊C4—1421no—
232C2×C42—712no—
233C2×C42—1422no—
234#108C2×Dic7118yescomplemented
235#108C2×Dic7118yescomplemented
236#107—714no—
237#107—714no—
238#106—714no—
239#106—714no—
240#97—718no—
241#97—1428no—
242C4×C8—718no—
243C4×C8—1428no—
244C24#1176586611yescharacteristic
245C22×C4—1422no—
246C22×C4—1424no—
247C22×C4C2×C4×D7112yes—
248C22×C4C2×C4×D7111yescharacteristic
249C22×C4C2×C4×D7111yescharacteristic
250C22×C4C22×Dic7112yes—
251C22×C4D4×D7111yescharacteristic
252C22×C4D4×D7111yescharacteristic
253C22×C4Q8×D7112yes—
254C22×C4#1176586412yes—
255C22×C4#1176586611yescharacteristic
256C22×C4C23×D7111yescharacteristic
257M16—24no—
258M16—28no—
259M16—718no—
260M16—718no—
261M16C22×Dic7118yes—
262C2×C8—24no—
263C2×C8—28no—
264C2×C8—714no—
265C2×C8—714no—
266C2×C8—714no—
267C2×C8C4×Dic7114yes—
268C2×C8C4×Dic7114yes—
269C2×C8C4×Dic7114yes—
270C2×C8C4×Dic7114yes—
271C2×C8C2×C4×D7112yes—
272C2×C8C2×C4×D7112yes—
273C2×C8C2×C4×D7111yescharacteristic
274C2×C8C2×C4×D7111yescharacteristic
275C2×C8C2×C4×D7111yescharacteristic
276C2×C8C2×C4×D7111yescharacteristic
277C2×C8C22×Dic7112yes—
278C2×C8C22×Dic7112yes—
279C4⋊C4—718no—
280C4⋊C4—1424no—
281C4⋊C4—1424no—
282C22⋊C4—1424no—
283C22⋊C4—1424no—
284C22⋊C4—1424no—
285C42—714no—
286C42—1424no—
287C42—1422no—
288C42—1424no—
289C23—42no—
290C23#1455608612yes—
291C23#1455609011yescharacteristic
292C23#1455609011yescharacteristic
293C23Q8×Dic7112yes—
294C23C2×#1176586611yescharacteristic
295C2×C4—2844no—
296C2×C4—5684no—
297C2×C4—1424no—
298C2×C4—1428no—
299C2×C4C2×C4×Dic7112yes—
300C2×C4C2×C4×Dic7112yes—
301C2×C4C4⋊C4×D7112yes—
302C2×C4C4⋊C4×D7112yes—
303C2×C4#1455608612yes—
304C2×C4#1455609011yescharacteristic
305C2×C4#1455609011yescharacteristic
306C2×C4D4×Dic7112yes—
307C2×C4D4×Dic7112yes—
308C2×C4#1455609312yes—
309C2×C4#1455609312yes—
310C2×C4Q8×Dic7112yes—
311C2×C4C22×C4×D7111yescharacteristic
312C2×C4C22×C4×D7111yescharacteristic
313C2×C4C23×Dic7111yescharacteristic
314C2×C4C2×D4×D7111yescharacteristic
315C2×C4C2×Q8×D7111yescharacteristic
316C2×C4C2×#1176586411yescharacteristic
317C8—28no—
318C8—718no—
319C8C2×C4×Dic7112yes—
320C8C2×C4×Dic7112yes—
321C8C2×C4×Dic7112yes—
322C8C2×C4×Dic7112yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22C4⋊C4×Dic7112yes—
329C22C4⋊C4×Dic7112yes—
330C22C2×#1455608611yescharacteristic
331C22C2×#1455609011yescharacteristic
332C22C2×Q8×Dic7111yescharacteristic
333C4—2844no—
334C4—2848no—
335C4C4⋊C4×Dic7112yes—
336C4C4⋊C4×Dic7112yes—
337C4C22×C4×Dic7111yescharacteristic
338C4C2×C4⋊C4×D7111yescharacteristic
339C4C2×D4×Dic7111yescharacteristic
340C4C2×#1455609311yescharacteristic
341C2—21no—
342C2—22no—
343C2—22no—
344C2—21no—
345C2C2×C4⋊C4×Dic7111yescharacteristic
346C2#2259725612yes—
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC4×C71C22×C42
FrattiniC22×C4C23×D71
FittingC22×#108×C71C2
socleC22×C71C2×C4×D4
prime radicalC24×C71#41
cosocleC22×C4×C71C24
hypercenterC22×#108D71
hypocenterC71#20446
LCSG ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
4. #22588227C2semidirect
5. #22588222C2semidirect
11. #17986037C4semidirect
12. #17986037C4semidirect
15. C2×#14556062C4semidirect
16. C2×#14556060C4semidirect
19. #17985954C4semidirect
26. C2×#108×C71C4semidirect
51. #14556038C2×C4semidirect
53. C8×Dic71C2×C4semidirect
68. #108×C71C2×C4semidirect
69. #108×C71C2×C4semidirect
181. C71#20446semidirect
187. C2×#108Dic71semidirect
234. #108C2×Dic71semidirect
235. #108C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860372
#225967582
C2×#179859542
#225882272
#225882222
#225880512
C22×#108×C712
#2044671