Finite Group Databasebeta

← #44487402

Subgroups — #44487402

Order 30592.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#21439916C211yesmaximal, characteristic
2#29504656C211yesmaximal, characteristic, complemented
3#29504656C211yesmaximal, characteristic, complemented
4#29504654C211yesmaximal, characteristic, complemented
5#29504654C211yesmaximal, characteristic, complemented
6#29504645C211yesmaximal, characteristic
7#29504644C211yesmaximal, characteristic
8#29504639C211yesmaximal, characteristic, complemented
9#29504636C211yesmaximal, characteristic, complemented
10#29504633C211yesmaximal, characteristic
11C4×Dic239⋊C4C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic239C211yesmaximal, characteristic
13#29504618C211yesmaximal, characteristic, complemented
14C2×C239⋊#95C211yesmaximal, characteristic
15C4×C22⋊C4×C239C211yesmaximal, characteristic
16C23×Dic239C2211yescharacteristic
17C23×Dic239C2211yescharacteristic
18#21439916C418yescomplemented
19C2×Dic239⋊C4C2211yescharacteristic
20C2×Dic239⋊C4C2211yescharacteristic
21C2×Dic239⋊C4C2211yescharacteristic
22C2×Dic239⋊C4C2211yescharacteristic
23C2×Dic239⋊C4C2211yescharacteristic
24C2×Dic239⋊C4C2211yescharacteristic
25C2×C22⋊Dic239C2211yescharacteristic
26C2×C22⋊Dic239C2211yescharacteristic
27C2×C4×Dic239C2211yescharacteristic
28C2×C4×Dic239C2211yescharacteristic
29C2×C4×Dic239C2211yescharacteristic
30C2×C4×Dic239C2211yescharacteristic
31C239⋊#95C2211yescharacteristic
32C239⋊#95C2211yescharacteristic
33C239⋊#95C2211yescharacteristic
34C239⋊#95C2211yescharacteristic
35C239⋊#95C2211yescharacteristic
36C239⋊#95C2211yescharacteristic
37C239⋊#95C2211yescharacteristic
38C239⋊#95C2211yescharacteristic
39C239⋊#95C2211yescharacteristic
40C239⋊#95C2211yescharacteristic
41C239⋊#95C2211yescharacteristic
42C239⋊#95C2211yescharacteristic
43C239⋊#95C2211yescharacteristic
44C239⋊#95C2211yescharacteristic
45C23×C4×C239C2211yescharacteristic
46C2×C22⋊C4×C239C2211yescharacteristic
47C2×C22⋊C4×C239C2211yescharacteristic
48C2×C42×C239C2211yescharacteristic
49C2×C42×C239C2211yescharacteristic
50#95×C239C2211yescharacteristic
51#95×C239C2211yescharacteristic
52C22×Dic239—22no—
53C22×Dic239—22no—
54C22×Dic239—22no—
55C22×Dic239—22no—
56C22×Dic239—41no—
57C22×Dic239C2×C414yes—
58C22×Dic239C2311yescharacteristic
59C22×Dic239C2311yescharacteristic
60C22×Dic239C2311yescharacteristic
61C22×Dic239C2311yescharacteristic
62C22×Dic239C2311yescharacteristic
63C22×Dic239C2311yescharacteristic
64C22×Dic239C2311yescharacteristic
65C22×Dic239C2311yescharacteristic
66Dic239⋊C4—22no—
67Dic239⋊C4—22no—
68Dic239⋊C4—41no—
69Dic239⋊C4—41no—
70Dic239⋊C4C2×C414yescomplemented
71Dic239⋊C4C2×C414yescomplemented
72C22⋊Dic239—22no—
73C22⋊Dic239C2×C414yes—
74C4×Dic239—22no—
75C4×Dic239—41no—
76C4×Dic239C2×C414yescomplemented
77C4×Dic239C2×C414yescomplemented
78C24×C239C2311yescharacteristic
79C22×C4×C239—22no—
80C22×C4×C239—22no—
81C22×C4×C239—41no—
82C22×C4×C239C2311yescharacteristic
83C22×C4×C239C2311yescharacteristic
84C22×C4×C239C2311yescharacteristic
85C22×C4×C239C2311yescharacteristic
86C22×C4×C239C2311yescharacteristic
87C22×C4×C239C2311yescharacteristic
88C22⋊C4×C239—22no—
89C22⋊C4×C239C2×C414yes—
90C42×C239—22no—
91C42×C239—41no—
92C2×Dic239—28no—
93C2×Dic239—21no—
94C2×Dic239—21no—
95C2×Dic239—21no—
96C2×Dic239—21no—
97C2×Dic239—21no—
98C2×Dic239—21no—
99C2×Dic239—21no—
100C2×Dic239—21no—
101C2×Dic239—21no—
102C2×Dic239—21no—
103C2×Dic239—21no—
104C2×Dic239—21no—
105C2×Dic239—21no—
106C2×Dic239—21no—
107C2×Dic239—21no—
108C2×Dic239—21no—
109C2×Dic239—21no—
110C2×Dic239—21no—
111C2×Dic239—21no—
112C2×Dic239—44no—
113C2×Dic239C22×C412yes—
114C2×Dic239C22×C412yes—
115C2×Dic239C22×C412yes—
116C2×Dic239C22×C412yes—
117C2×Dic239#4112yes—
118C23×C239—21no—
119C23×C239—21no—
120C23×C239—21no—
121C23×C239—21no—
122C23×C239C22×C412yes—
123C23×C239#4112yes—
124C23×C239#4112yes—
125C23×C239C2411yescharacteristic
126C2×C4×C239—21no—
127C2×C4×C239—21no—
128C2×C4×C239—21no—
129C2×C4×C239—21no—
130C2×C4×C239—21no—
131C2×C4×C239—21no—
132C2×C4×C239—21no—
133C2×C4×C239—21no—
134C2×C4×C239—21no—
135C2×C4×C239—21no—
136C2×C4×C239—21no—
137C2×C4×C239—21no—
138C2×C4×C239—21no—
139C2×C4×C239—21no—
140C2×C4×C239—21no—
141C2×C4×C239—44no—
142C2×C4×C239C22×C412yes—
143C2×C4×C239C22×C412yes—
144C2×C4×C239#4112yes—
145C22×C239—22no—
146C22×C239—22no—
147C22×C239—22no—
148C22×C239—22no—
149C22×C239—22no—
150C22×C239—22no—
151C22×C239#11714yes—
152C22×C239C23×C411yescharacteristic
153C22×C239C2×#4111yescharacteristic
154C22×C239C2×#4111yescharacteristic
155C22×C239C2×#4111yescharacteristic
156C22×C239C2×#4111yescharacteristic
157C22×C239H(5,𝔽2)11yescharacteristic
158C22×C2392−1+411yescharacteristic
159C4×C239—22no—
160C4×C239—22no—
161C4×C239—22no—
162C4×C239—41no—
163C4×C239—41no—
164C4×C239—41no—
165Dic239—22no—
166Dic239—22no—
167Dic239—22no—
168Dic239—22no—
169Dic239—22no—
170Dic239—41no—
171Dic239—41no—
172Dic239—41no—
173C2×C239—24no—
174C2×C239C2×#11711yescharacteristic
175C2×C239C4×#4111yescharacteristic
176C2×C239#52011yescharacteristic
177C2×C239#55111yescharacteristic
178C2×C239#55111yescharacteristic
179C2×C239#55211yescharacteristic
180C2×C239#55211yescharacteristic
181C239#234311yescharacteristic, complemented, Sylow
182#2343—2391nomaximal, Sylow
183C2×#117—2391no—
184#387—2391no—
185#387—2391no—
186#387—2391no—
187#387—2391no—
188#385—2391no—
189#385—2391no—
190#383—2391no—
191#382—2391no—
192#381—2391no—
193C4×C4⋊C4—2391no—
194C4×C22⋊C4—2391no—
195C4×C22⋊C4D23911yescharacteristic
196#376—2391no—
197C2×#95—2391no—
198C23×C4—2391no—
199C23×C4—2391no—
200C23×C4C2×D23911yescharacteristic
201#117—2398no—
202C2×C4⋊C4—2391no—
203C2×C4⋊C4—2391no—
204C2×C4⋊C4—2391no—
205C2×C4⋊C4—2391no—
206C2×C4⋊C4—2391no—
207C2×C4⋊C4—2391no—
208C2×C22⋊C4—2391no—
209C2×C22⋊C4—2391no—
210C2×C22⋊C4C2×D23911yescharacteristic
211C2×C22⋊C4C2×D23911yescharacteristic
212C2×C42—2391no—
213C2×C42—2391no—
214C2×C42—2391no—
215C2×C42—2391no—
216C2×C42C2×D23911yescharacteristic
217C2×C42C2×D23911yescharacteristic
218#95—2391no—
219#95—2391no—
220#95—2391no—
221#95—2391no—
222#95—2391no—
223#95—2391no—
224#95—2391no—
225#95—2391no—
226#95—2391no—
227#95—2391no—
228#95—2391no—
229#95—2391no—
230#95—2391no—
231#95—2391no—
232#95C2×D23911yescharacteristic
233#95C2×D23911yescharacteristic
234C24C22×D23911yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—9561no—
239C22×C4—4782no—
240C22×C4—4782no—
241C22×C4—4782no—
242C22×C4—4782no—
243C22×C4—2391no—
244C22×C4—2391no—
245C22×C4—2391no—
246C22×C4—2394no—
247C22×C4—2391no—
248C22×C4—2391no—
249C22×C4—2391no—
250C22×C4—2391no—
251C22×C4—2391no—
252C22×C4C22×D23911yescharacteristic
253C22×C4C22×D23911yescharacteristic
254C22×C4C22×D23911yescharacteristic
255C22×C4C22×D23911yescharacteristic
256C22×C4C22×D23911yescharacteristic
257C22×C4C22×D23911yescharacteristic
258C4⋊C4—9561no—
259C4⋊C4—9561no—
260C4⋊C4—4782no—
261C4⋊C4—4782no—
262C4⋊C4—2394no—
263C4⋊C4—2394no—
264C22⋊C4—22no—
265C22⋊C4—4782no—
266C22⋊C4—2394no—
267C22⋊C4C4×D23914yes—
268C42—22no—
269C42—41no—
270C42—9561no—
271C42—4782no—
272C42—2394no—
273C42—2394no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D23912yes—
279C23#1683854812yes—
280C23#1683854812yes—
281C23C23×D23911yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—9564no—
299C2×C4—4781no—
300C2×C4—4781no—
301C2×C4—4781no—
302C2×C4—4781no—
303C2×C4—4781no—
304C2×C4—4781no—
305C2×C4—4781no—
306C2×C4—4781no—
307C2×C4—4781no—
308C2×C4—4781no—
309C2×C4—4781no—
310C2×C4—4781no—
311C2×C4—4781no—
312C2×C4—4781no—
313C2×C4—4781no—
314C2×C4—4781no—
315C2×C4—4781no—
316C2×C4—4781no—
317C2×C4—4781no—
318C2×C4—4788no—
319C2×C4—2392no—
320C2×C4—2392no—
321C2×C4—2392no—
322C2×C4—2392no—
323C2×C4—2392no—
324C2×C4C2×C4×D23912yes—
325C2×C4C2×C4×D23912yes—
326C2×C4#1683854712yes—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22#2143991714yes—
334C22C22×C4×D23911yescharacteristic
335C22#41×D23911yescharacteristic
336C22C2×#1683854711yescharacteristic
337C22C2×#1683854811yescharacteristic
338C22C2×#1683854811yescharacteristic
339C22#2144000511yescharacteristic
340C22#2144000711yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—9561no—
348C4—9561no—
349C4—9561no—
350C4—4782no—
351C4—4782no—
352C4—4782no—
353C4—4782no—
354C4—4782no—
355C2—24no—
356C2C2×#2143991711yescharacteristic
357C2C4×#1683854711yescharacteristic
358C2#2950525111yescharacteristic
359C2#2950544811yescharacteristic
360C2#2950545111yescharacteristic
361C2#2950545511yescharacteristic
362C2#2950546211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D239
commutatorC22×C239C23×C4
FittingC4×C22⋊C4×C239C2
socle, cosocleC23×C239C24
prime radicalC24×C239C23
hypercenterC4×C22⋊C4D239
hypocenterC239#2343
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #29504656C2semidirect
3. #29504656C2semidirect
4. #29504654C2semidirect
5. #29504654C2semidirect
8. #29504639C2semidirect
9. #29504636C2semidirect
11. C4×Dic239⋊C4C2semidirect
13. #29504618C2semidirect
18. #21439916C4semidirect
70. Dic239⋊C4C2×C4semidirect
71. Dic239⋊C4C2×C4semidirect
76. C4×Dic239C2×C4semidirect
77. C4×Dic239C2×C4semidirect
181. C239#2343semidirect

Maximal subgroups

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

subgroupindex
C2×#214399162
#295046562
#295046542
#295046452
#295046442
#295046392
#295046362
#295046332
C4×Dic239⋊C42
C22⋊C4×Dic2392
#295046182
C2×C239⋊#952
C4×C22⋊C4×C2392
#2343239