Finite Group Databasebeta

← #26214314

Subgroups — #26214314

Order 12032.

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×#15689161C211yesmaximal, characteristic
2#20251806C212yesmaximal
3C2×#15689078C211yesmaximal, characteristic
4#20243275C214yesmaximal, complemented
5#20243270C214yesmaximal, complemented
6#20243099C212yesmaximal
7C22×#108×C47C211yesmaximal, characteristic
8C22×#10625329C2211yescharacteristic
9C2×#13093983C2212yes—
10C2×#13093983C2211yescharacteristic
11#15689161C412yescomplemented
12#15689161C412yescomplemented
13#15689161C2212yes—
14#15689161C2212yes—
15C2×#13093958C412yescomplemented
16C2×#13093956C412yescomplemented
17#15689081C2218yes—
18#15689078—24no—
19#15689078C418yescomplemented
20M16×Dic47—24no—
21C2×#13093934C2212yes—
22C2×C8×Dic47C2212yes—
23#15688884C2218yes—
24C22×M16×C47C2212yes—
25C23×C8×C47C2211yescharacteristic
26C2×#108×C47C418yescomplemented
27C2×#108×C47C2212yes—
28C2×#108×C47C2212yes—
29C2×#10625329C2×C414yes—
30C2×#10625329D412yes—
31C2×#10625329D412yes—
32C2×#10625329Q814yes—
33C22×C47⋊C8C2312yes—
34#13093983—22no—
35#13093983—22no—
36#13093983—22no—
37#13093983C2×C414yes—
38#13093983C2×C414yes—
39#13093983C2314yes—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47C2×C414yes—
42C2×C22⋊Dic47—21no—
43C2×C22⋊Dic47—22no—
44C2×C4×Dic47—22no—
45C2×C4×Dic47C2312yes—
46#13093958C2×C414yes—
47#13093958C2×C414yes—
48#13093956C2×C414yes—
49#13093956C2×C414yes—
50#13093934—28no—
51#13093934C2×C418yescomplemented
52C8×Dic47—28no—
53C8×Dic47C2×C418yescomplemented
54C23×C4×C47C2311yescharacteristic
55C2×M16×C47—22no—
56C2×M16×C47—24no—
57C2×M16×C47C2×C418yes—
58C2×M16×C47C2314yes—
59C22×C8×C47—22no—
60C22×C8×C47C2×C414yes—
61C22×C8×C47C2×C411yescharacteristic
62C22×C8×C47C2×C411yescharacteristic
63C22×C8×C47C2×C411yescharacteristic
64C22×C8×C47C2×C411yescharacteristic
65C22×C8×C47C2×C414yes—
66C22×C8×C47C2311yescharacteristic
67C22×C8×C47C2311yescharacteristic
68#108×C47C2×C418yescomplemented
69#108×C47C2×C418yescomplemented
70C24×C47#4111yescharacteristic
71C22×C4×C47C22×C412yes—
72C22×C4×C47C22×C412yes—
73C22×C4×C47C22×C411yescharacteristic
74C22×C4×C47C22×C411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×Q812yes—
78C22×C4×C47#4111yescharacteristic
79C22×C4×C47#4112yes—
80C22×C4×C47C2411yescharacteristic
81C22×Dic47—22no—
82C22×Dic47—24no—
83M16×C47—28no—
84M16×C47—24no—
85M16×C47C22×C418yes—
86C2×C8×C47—28no—
87C2×C8×C47—24no—
88C2×C8×C47C4214yes—
89C2×C8×C47C4214yes—
90C2×C8×C47C4214yes—
91C2×C8×C47C4214yes—
92C2×C8×C47C22×C412yes—
93C2×C8×C47C22×C412yes—
94C2×C8×C47C22×C411yescharacteristic
95C2×C8×C47C22×C411yescharacteristic
96C2×C8×C47C22×C411yescharacteristic
97C2×C8×C47C22×C411yescharacteristic
98C2×C8×C47C22×C412yes—
99C2×C8×C47C22×C412yes—
100C22⋊Dic47—24no—
101C22⋊Dic47—24no—
102C22⋊Dic47—24no—
103C4⋊Dic47—24no—
104C4⋊Dic47—24no—
105C4⋊Dic47C22×C418yes—
106C4×Dic47—24no—
107C4×Dic47—24no—
108C4×Dic47—22no—
109C4×Dic47C22×C414yes—
110#10625329C4⋊C418yes—
111#10625329C4⋊C418yes—
112C2×C47⋊C8C22×C414yes—
113C2×C47⋊C8C2×D414yes—
114C2×C47⋊C8C2×Q814yes—
115C23×C47—42no—
116C23×C47#11712yes—
117C23×C47C4×D411yescharacteristic
118C23×C47C4×D411yescharacteristic
119C23×C47C4×Q812yes—
120C23×C47C2×#4111yescharacteristic
121C2×C4×C47C2×C4212yes—
122C2×C4×C47C2×C4212yes—
123C2×C4×C47C2×C4⋊C412yes—
124C2×C4×C47C2×C4⋊C412yes—
125C2×C4×C47#11712yes—
126C2×C4×C47C4×D411yescharacteristic
127C2×C4×C47C4×D411yescharacteristic
128C2×C4×C47C4×D412yes—
129C2×C4×C47C4×D412yes—
130C2×C4×C47C4×Q812yes—
131C2×C4×C47C4×Q812yes—
132C2×C4×C47C4×Q812yes—
133C2×C4×C47C23×C411yescharacteristic
134C2×C4×C47C23×C411yescharacteristic
135C2×C4×C47C23×C411yescharacteristic
136C2×C4×C47C22×D411yescharacteristic
137C2×C4×C47C22×Q811yescharacteristic
138C2×C4×C47C2×#4111yescharacteristic
139C2×Dic47—24no—
140C2×Dic47—28no—
141C2×Dic47—44no—
142C2×Dic47—84no—
143C8×C47—28no—
144C8×C47C2×C4212yes—
145C8×C47C2×C4212yes—
146C8×C47C2×C4212yes—
147C8×C47C2×C4212yes—
148C47⋊C8C2×C4⋊C418yes—
149#20446—471nomaximal, Sylow
150C22×C47—22no—
151C22×C47—22no—
152C22×C47—22no—
153C22×C47—42no—
154C22×C47—42no—
155C22×C47C4×C4⋊C412yes—
156C22×C47C4×C4⋊C412yes—
157C22×C47C2×#11711yescharacteristic
158C22×C47C2×C4×D411yescharacteristic
159C22×C47C2×C4×Q811yescharacteristic
160C4×C47C4×C4⋊C412yes—
161C4×C47C4×C4⋊C412yes—
162C4×C47C22×C4211yescharacteristic
163C4×C47C22×C4⋊C411yescharacteristic
164C4×C47C2×C4×D411yescharacteristic
165C4×C47C2×C4×Q811yescharacteristic
166Dic47—44no—
167Dic47—48no—
168C22×#108D4711yescharacteristic
169C2×#427—471no—
170#2873—472no—
171C2×#405—471no—
172#1773—474no—
173#1772—474no—
174#1733—472no—
175C2×C47—21no—
176C2×C47—21no—
177C2×C47—22no—
178C2×C47—22no—
179C2×C47C2×C4×C4⋊C411yescharacteristic
180C2×C47#295712yes—
181C22×M16—471no—
182C22×M16C2×D4712yes—
183C23×C8C2×D4711yescharacteristic
184C2×#117—471no—
185C2×#117—472no—
186C2×#108Dic4718yescomplemented
187C2×#108C2×D4712yes—
188C2×#108C2×D4712yes—
189#427—472no—
190#427—472no—
191#427—472no—
192#427—472no—
193C2×#107—472no—
194C2×#106—472no—
195#405—478no—
196#405—478no—
197#405—944no—
198C4×M16—944no—
199C2×#97—472no—
200C2×C4×C8—472no—
201#339—478no—
202C47#2044611yescharacteristic, complemented, Sylow
203C23×C4C22×D4711yescharacteristic
204C2×M16—22no—
205C2×M16—24no—
206C2×M16—474no—
207C2×M16—472no—
208C2×M16—472no—
209C2×M16—474no—
210C2×M16C2×Dic4718yes—
211C2×M16C22×D4714yes—
212C22×C8—22no—
213C22×C8—472no—
214C22×C8C4×D4714yes—
215C22×C8C4×D4711yescharacteristic
216C22×C8C4×D4711yescharacteristic
217C22×C8C4×D4711yescharacteristic
218C22×C8C4×D4711yescharacteristic
219C22×C8C2×Dic4714yes—
220C22×C8C22×D4711yescharacteristic
221C22×C8C22×D4711yescharacteristic
222#117—474no—
223#117—474no—
224#117—474no—
225#117—942no—
226#117—942no—
227#117—942no—
228C2×C4⋊C4—474no—
229C2×C4⋊C4—941no—
230C2×C22⋊C4—941no—
231C2×C22⋊C4—942no—
232C2×C42—472no—
233C2×C42—942no—
234#108C2×Dic4718yescomplemented
235#108C2×Dic4718yescomplemented
236#107—474no—
237#107—474no—
238#106—474no—
239#106—474no—
240#97—478no—
241#97—948no—
242C4×C8—478no—
243C4×C8—948no—
244C24#1062535411yescharacteristic
245C22×C4—944no—
246C22×C4—942no—
247C22×C4C2×C4×D4712yes—
248C22×C4C2×C4×D4711yescharacteristic
249C22×C4C2×C4×D4711yescharacteristic
250C22×C4D4×D4711yescharacteristic
251C22×C4D4×D4711yescharacteristic
252C22×C4#1062535212yes—
253C22×C4Q8×D4712yes—
254C22×C4#1062535411yescharacteristic
255C22×C4C22×Dic4712yes—
256C22×C4C23×D4711yescharacteristic
257M16—28no—
258M16—24no—
259M16—478no—
260M16—478no—
261M16C22×Dic4718yes—
262C2×C8—28no—
263C2×C8—24no—
264C2×C8—474no—
265C2×C8—474no—
266C2×C8—474no—
267C2×C8C4×Dic4714yes—
268C2×C8C4×Dic4714yes—
269C2×C8C4×Dic4714yes—
270C2×C8C4×Dic4714yes—
271C2×C8C2×C4×D4712yes—
272C2×C8C2×C4×D4712yes—
273C2×C8C2×C4×D4711yescharacteristic
274C2×C8C2×C4×D4711yescharacteristic
275C2×C8C2×C4×D4711yescharacteristic
276C2×C8C2×C4×D4711yescharacteristic
277C2×C8C22×Dic4712yes—
278C2×C8C22×Dic4712yes—
279C4⋊C4—478no—
280C4⋊C4—944no—
281C4⋊C4—944no—
282C22⋊C4—944no—
283C22⋊C4—944no—
284C22⋊C4—944no—
285C42—474no—
286C42—944no—
287C42—944no—
288C42—942no—
289C23—42no—
290C23#1309398212yes—
291C23#1309398611yescharacteristic
292C23#1309398611yescharacteristic
293C23Q8×Dic4712yes—
294C23C2×#1062535411yescharacteristic
295C2×C4—944no—
296C2×C4—948no—
297C2×C4—3764no—
298C2×C4—1884no—
299C2×C4C2×C4×Dic4712yes—
300C2×C4C2×C4×Dic4712yes—
301C2×C4C4⋊C4×D4712yes—
302C2×C4C4⋊C4×D4712yes—
303C2×C4#1309398212yes—
304C2×C4#1309398611yescharacteristic
305C2×C4#1309398611yescharacteristic
306C2×C4D4×Dic4712yes—
307C2×C4D4×Dic4712yes—
308C2×C4#1309398912yes—
309C2×C4#1309398912yes—
310C2×C4Q8×Dic4712yes—
311C2×C4C22×C4×D4711yescharacteristic
312C2×C4C22×C4×D4711yescharacteristic
313C2×C4C23×Dic4711yescharacteristic
314C2×C4C2×D4×D4711yescharacteristic
315C2×C4C2×Q8×D4711yescharacteristic
316C2×C4C2×#1062535211yescharacteristic
317C8—28no—
318C8—478no—
319C8C2×C4×Dic4712yes—
320C8C2×C4×Dic4712yes—
321C8C2×C4×Dic4712yes—
322C8C2×C4×Dic4712yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22C4⋊C4×Dic4712yes—
329C22C4⋊C4×Dic4712yes—
330C22C2×#1309398211yescharacteristic
331C22C2×#1309398611yescharacteristic
332C22C2×Q8×Dic4711yescharacteristic
333C4—1884no—
334C4—1888no—
335C4C4⋊C4×Dic4712yes—
336C4C4⋊C4×Dic4712yes—
337C4C22×C4×Dic4711yescharacteristic
338C4C2×C4⋊C4×D4711yescharacteristic
339C4C2×D4×Dic4711yescharacteristic
340C4C2×#1309398911yescharacteristic
341C2—21no—
342C2—22no—
343C2—22no—
344C2—21no—
345C2C2×C4⋊C4×Dic4711yescharacteristic
346C2#2025230412yes—
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC22×#108×C47C2
socleC22×C47C2×C4×D4
prime radicalC24×C47#41
cosocleC22×C4×C47C24
hypercenterC22×#108D47
hypocenterC47#20446
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
4. #20243275C2semidirect
5. #20243270C2semidirect
11. #15689161C4semidirect
12. #15689161C4semidirect
15. C2×#13093958C4semidirect
16. C2×#13093956C4semidirect
19. #15689078C4semidirect
26. C2×#108×C47C4semidirect
51. #13093934C2×C4semidirect
53. C8×Dic47C2×C4semidirect
68. #108×C47C2×C4semidirect
69. #108×C47C2×C4semidirect
186. C2×#108Dic47semidirect
202. C47#20446semidirect
234. #108C2×Dic47semidirect
235. #108C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156891612
#202518062
C2×#156890782
#202432752
#202432702
#202430992
C22×#108×C472
#2044647