Finite Group Databasebeta

← #44487147

Subgroups — #44487147

Order 30592.

512 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×C4⋊C4×D239C211yesmaximal, characteristic
2#29504659C211yesmaximal, characteristic, complemented
3#29504658C211yesmaximal, characteristic, complemented
4#29504657C211yesmaximal, characteristic, complemented
5#29504656C211yesmaximal, characteristic, complemented
6#29504655C211yesmaximal, characteristic, complemented
7#29504654C211yesmaximal, characteristic, complemented
8#29504653C211yesmaximal, characteristic, complemented
9#29504634C211yesmaximal, characteristic
10#29504634C211yesmaximal, characteristic
11#29504631C211yesmaximal, characteristic
12#29504629C211yesmaximal, characteristic
13C4×(C2×D239)⋊C4C211yesmaximal, characteristic
14#29504619C211yesmaximal, characteristic
15#387×C239C211yesmaximal, characteristic, complemented
16C22×C4×D239C2211yescharacteristic
17C22×C4×D239C2211yescharacteristic
18C22×C4×D239C2211yescharacteristic
19C2×C4⋊Dic239C2211yescharacteristic
20C2×C4⋊Dic239C2211yescharacteristic
21C2×Dic239⋊C4C2211yescharacteristic
22C2×Dic239⋊C4C2211yescharacteristic
23C2×Dic239⋊C4C2211yescharacteristic
24C2×Dic239⋊C4C2211yescharacteristic
25C4⋊C4×D239C418yescomplemented
26C2×(C2×D239)⋊C4C2211yescharacteristic
27C2×(C2×D239)⋊C4C2211yescharacteristic
28C2×(C2×D239)⋊C4C2211yescharacteristic
29C2×(C2×D239)⋊C4C2211yescharacteristic
30C2×C4×Dic239C2211yescharacteristic
31C2×C4×Dic239C2211yescharacteristic
32C2×C4×Dic239C2211yescharacteristic
33C239⋊#95C2211yescharacteristic
34C239⋊#95C2211yescharacteristic
35C239⋊#95C2211yescharacteristic
36C239⋊#95C2211yescharacteristic
37C239⋊#95C2211yescharacteristic
38C239⋊#95C2211yescharacteristic
39C239⋊#95C2211yescharacteristic
40C239⋊#95C2211yescharacteristic
41C239⋊#95C2211yescharacteristic
42C239⋊#95C2211yescharacteristic
43C239⋊#95C2211yescharacteristic
44C239⋊#95C2211yescharacteristic
45C2×C4⋊C4×C239C2211yescharacteristic
46C2×C4⋊C4×C239C2211yescharacteristic
47C2×C42×C239C2211yescharacteristic
48#95×C239C2211yescharacteristic
49#95×C239C2211yescharacteristic
50#95×C239C2211yescharacteristic
51#95×C239C2211yescharacteristic
52C23×D239C2311yescharacteristic
53C22×Dic239C2311yescharacteristic
54C22×Dic239C2311yescharacteristic
55C22×Dic239C2311yescharacteristic
56C22×Dic239C2311yescharacteristic
57C22×Dic239C2311yescharacteristic
58C22×Dic239C2311yescharacteristic
59C22×Dic239C2311yescharacteristic
60C2×C4×D239—22no—
61C2×C4×D239—22no—
62C2×C4×D239—41no—
63C2×C4×D239—41no—
64C2×C4×D239—41no—
65C2×C4×D239—41no—
66C2×C4×D239C2×C414yes—
67C2×C4×D239C2×C414yes—
68C2×C4×D239C2×C414yes—
69C4⋊Dic239—41no—
70C4⋊Dic239C2×C414yes—
71Dic239⋊C4—41no—
72Dic239⋊C4—41no—
73Dic239⋊C4C2×C414yes—
74Dic239⋊C4C2×C414yes—
75(C2×D239)⋊C4—22no—
76(C2×D239)⋊C4—22no—
77(C2×D239)⋊C4—22no—
78(C2×D239)⋊C4—22no—
79C4×Dic239—41no—
80C4×Dic239—41no—
81C4×Dic239—41no—
82C22×C4×C239C2311yescharacteristic
83C22×C4×C239C2311yescharacteristic
84C22×C4×C239C2311yescharacteristic
85C22×C4×C239C2311yescharacteristic
86C22×C4×C239C2311yescharacteristic
87C22×C4×C239C2311yescharacteristic
88C22×C4×C239C2311yescharacteristic
89C4⋊C4×C239—41no—
90C4⋊C4×C239C2×C414yes—
91C42×C239—41no—
92C22×D239—21no—
93C22×D239—21no—
94C22×D239—21no—
95C22×D239—21no—
96C22×D239C22×C412yes—
97C22×D239#4112yes—
98C22×D239#4112yes—
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—21no—
113C2×Dic239—21no—
114C2×Dic239—21no—
115C2×Dic239—21no—
116C2×Dic239—21no—
117C2×Dic239C22×C412yes—
118C2×Dic239C22×C412yes—
119C2×Dic239C22×C412yes—
120C4×D239—28no—
121C4×D239—44no—
122C4×D239—44no—
123C23×C239C2411yescharacteristic
124C2×C4×C239—21no—
125C2×C4×C239—21no—
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—21no—
142C2×C4×C239C22×C412yes—
143C2×C4×C239C22×C412yes—
144C2×C4×C239C22×C412yes—
145C22×C239C23×C411yescharacteristic
146C22×C239C2×#4111yescharacteristic
147C22×C239C2×#4111yescharacteristic
148C22×C239H(5,𝔽2)11yescharacteristic
149C22×C239H(5,𝔽2)11yescharacteristic
150C22×C2392−1+411yescharacteristic
151C22×C2392−1+411yescharacteristic
152C2×D239—22no—
153C2×D239—22no—
154C2×D239—22no—
155C2×D239—22no—
156C2×D239—22no—
157C2×D239—22no—
158C2×D239#11714yes—
159C4×C239—22no—
160C4×C239—22no—
161C4×C239—22no—
162C4×C239—41no—
163C4×C239—41no—
164C4×C239—41no—
165C4×C239—41no—
166Dic239—22no—
167Dic239—22no—
168Dic239—22no—
169Dic239—41no—
170Dic239—41no—
171Dic239—41no—
172Dic239—41no—
173C2×C239C2×#11711yescharacteristic
174C2×C239#51811yescharacteristic
175C2×C239#51911yescharacteristic
176C2×C239#53911yescharacteristic
177C2×C239#53911yescharacteristic
178C2×C239#53911yescharacteristic
179C2×C239#53911yescharacteristic
180D239—24no—
181C239#231311yescharacteristic, complemented, Sylow
182#2313—2391nomaximal, Sylow
183C22×C4⋊C4—2391no—
184#387—2391no—
185#387—2391no—
186#387—2391no—
187#387—2391no—
188#387—2391no—
189#387—2391no—
190#387—2391no—
191#387D23911yescharacteristic, complemented
192#381—2391no—
193#381—2391no—
194#380—2391no—
195#380—2391no—
196C4×C22⋊C4—2391no—
197C4×C22⋊C4—2391no—
198C23×C4—2391no—
199C23×C4—2391no—
200C23×C4—2391no—
201C2×C4⋊C4—2391no—
202C2×C4⋊C4—2391no—
203C2×C4⋊C4—2391no—
204C2×C4⋊C4—2391no—
205C2×C4⋊C4—2391no—
206C2×C4⋊C4—2391no—
207C2×C4⋊C4—2398no—
208C2×C4⋊C4C2×D23911yescharacteristic
209C2×C4⋊C4C2×D23911yescharacteristic
210C2×C22⋊C4—2391no—
211C2×C22⋊C4—2391no—
212C2×C22⋊C4—2391no—
213C2×C22⋊C4—2391no—
214C2×C42—2391no—
215C2×C42—2391no—
216C2×C42—2391no—
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#95C2×D23911yescharacteristic
231#95C2×D23911yescharacteristic
232#95C2×D23911yescharacteristic
233#95C2×D23911yescharacteristic
234C24—2391no—
235C22×C4—9561no—
236C22×C4—9561no—
237C22×C4—9561no—
238C22×C4—9561no—
239C22×C4—4782no—
240C22×C4—4782no—
241C22×C4—2394no—
242C22×C4—2394no—
243C22×C4—2391no—
244C22×C4—2391no—
245C22×C4—2391no—
246C22×C4—2391no—
247C22×C4—2391no—
248C22×C4—2391no—
249C22×C4—2391no—
250C22×C4—2394no—
251C22×C4C22×D23911yescharacteristic
252C22×C4C22×D23911yescharacteristic
253C22×C4C22×D23911yescharacteristic
254C22×C4C22×D23911yescharacteristic
255C22×C4C22×D23911yescharacteristic
256C22×C4C22×D23911yescharacteristic
257C22×C4C22×D23911yescharacteristic
258C4⋊C4—41no—
259C4⋊C4—9561no—
260C4⋊C4—9561no—
261C4⋊C4—9561no—
262C4⋊C4—2394no—
263C4⋊C4—2394no—
264C4⋊C4—2394no—
265C4⋊C4C4×D23914yes—
266C22⋊C4—4782no—
267C22⋊C4—4782no—
268C22⋊C4—4782no—
269C22⋊C4—4782no—
270C42—41no—
271C42—9561no—
272C42—9561no—
273C42—9561no—
274C23—4781no—
275C23—4781no—
276C23—4781no—
277C23—4781no—
278C23—2392no—
279C23—2392no—
280C23—2392no—
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—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—9564no—
301C2×C4—9564no—
302C2×C4—4788no—
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—4781no—
319C2×C4—4781no—
320C2×C4—4781no—
321C2×C4—2392no—
322C2×C4—2392no—
323C2×C4—2392no—
324C2×C4C2×C4×D23912yes—
325C2×C4C2×C4×D23912yes—
326C2×C4C2×C4×D23912yes—
327C22—4782no—
328C22—4782no—
329C22—4782no—
330C22—4782no—
331C22—4782no—
332C22—4782no—
333C22—2394no—
334C22C22×C4×D23911yescharacteristic
335C22#41×D23911yescharacteristic
336C22#41×D23911yescharacteristic
337C22#2144000511yescharacteristic
338C22#2144000611yescharacteristic
339C22#2144000711yescharacteristic
340C22#2144000811yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—9561no—
349C4—9561no—
350C4—9561no—
351C4—9561no—
352C4—4782no—
353C4—4782no—
354C4—4782no—
355C2—4784no—
356C2#117×D23911yescharacteristic
357C2#2950524411yescharacteristic
358C2#2950524811yescharacteristic
359C2#2950536311yescharacteristic
360C2#2950536511yescharacteristic
361C2#2950536811yescharacteristic
362C2#2950537011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D239
commutatorC22×C239C23×C4
Fitting#387×C239C2
socle, cosocleC23×C239C24
prime radicalC23×D239C23
hypercenter#387D239
hypocenterC239#2313
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #29504659C2semidirect
3. #29504658C2semidirect
4. #29504657C2semidirect
5. #29504656C2semidirect
6. #29504655C2semidirect
7. #29504654C2semidirect
8. #29504653C2semidirect
15. #387×C239C2semidirect
25. C4⋊C4×D239C4semidirect
181. C239#2313semidirect
191. #387D239semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D2392
#295046592
#295046582
#295046572
#295046562
#295046552
#295046542
#295046532
#295046342
#295046312
#295046292
C4×(C2×D239)⋊C42
#295046192
#387×C2392
#2313239