Finite Group Databasebeta

← #44487174

Subgroups — #44487174

Order 30592.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic239C211yesmaximal, characteristic
2#29504666C211yesmaximal, characteristic, complemented
3#29504660C211yesmaximal, characteristic, complemented
4#29504657C211yesmaximal, characteristic, complemented
5#29504655C211yesmaximal, characteristic, complemented
6#29504652C211yesmaximal, characteristic
7#29504645C211yesmaximal, characteristic, complemented
8#29504644C211yesmaximal, characteristic, complemented
9#29504639C211yesmaximal, characteristic, complemented
10#29504636C211yesmaximal, characteristic, complemented
11#29504633C211yesmaximal, characteristic, complemented
12#29504630C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic239C211yesmaximal, characteristic, complemented
14C2×C239⋊#95C211yesmaximal, characteristic, complemented
15#386×C239C211yesmaximal, characteristic
16C23×Dic239C2211yescharacteristic
17C23×Dic239C2211yescharacteristic
18C23×Dic239C2211yescharacteristic
19D4×Dic239C418yescomplemented
20C2×C4⋊Dic239C2211yescharacteristic
21C2×Dic239⋊C4C2211yescharacteristic
22C2×Dic239⋊C4C2211yescharacteristic
23C2×Dic239⋊C4C2211yescharacteristic
24C2×Dic239⋊C4C2211yescharacteristic
25C2×C22⋊Dic239C2211yescharacteristic
26C2×C22⋊Dic239C2211yescharacteristic
27C2×C22⋊Dic239C2211yescharacteristic
28C2×C22⋊Dic239C2211yescharacteristic
29C2×C22⋊Dic239C2211yescharacteristic
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
45C22×D4×C239C2211yescharacteristic
46C23×C4×C239C2211yescharacteristic
47C2×C22⋊C4×C239C2211yescharacteristic
48C2×C22⋊C4×C239C2211yescharacteristic
49C2×C22⋊C4×C239C2211yescharacteristic
50#95×C239C2211yescharacteristic
51#95×C239C2211yescharacteristic
52C22×Dic239—22no—
53C22×Dic239—41no—
54C22×Dic239—41no—
55C22×Dic239—41no—
56C22×Dic239—41no—
57C22×Dic239—41no—
58C22×Dic239—41no—
59C22×Dic239C2×C414yes—
60C22×Dic239C2×C414yescomplemented
61C22×Dic239C2311yescharacteristic
62C22×Dic239C2311yescharacteristic
63C22×Dic239C2311yescharacteristic
64C22×Dic239C2311yescharacteristic
65C22×Dic239C2311yescharacteristic
66C22×Dic239C2311yescharacteristic
67C22×Dic239C2311yescharacteristic
68C22×Dic239C2311yescharacteristic
69C4⋊Dic239C2×C414yescomplemented
70Dic239⋊C4—41no—
71Dic239⋊C4—41no—
72Dic239⋊C4—41no—
73Dic239⋊C4—41no—
74C22⋊Dic239—22no—
75C22⋊Dic239—41no—
76C22⋊Dic239—41no—
77C22⋊Dic239C2×C414yes—
78C22⋊Dic239C2×C414yescomplemented
79C4×Dic239—41no—
80C4×Dic239—41no—
81C4×Dic239C2×C414yescomplemented
82C24×C239C2311yescharacteristic
83C24×C239C2311yescharacteristic
84C2×D4×C239—22no—
85C2×D4×C239—22no—
86C2×D4×C239C2×C414yes—
87C22×C4×C239—22no—
88C22×C4×C239—41no—
89C22×C4×C239—41no—
90C22×C4×C239C2311yescharacteristic
91C22×C4×C239C2311yescharacteristic
92C22×C4×C239C2311yescharacteristic
93C22×C4×C239C2311yescharacteristic
94C22×C4×C239C2311yescharacteristic
95C22⋊C4×C239—22no—
96C22⋊C4×C239—41no—
97C22⋊C4×C239—41no—
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—21no—
113C2×Dic239—21no—
114C2×Dic239—21no—
115C2×Dic239—21no—
116C2×Dic239—21no—
117C2×Dic239—21no—
118C2×Dic239—44no—
119C2×Dic239—44no—
120C2×Dic239—44no—
121C2×Dic239C22×C412yes—
122C2×Dic239C22×C412yes—
123C2×Dic239C22×C412yes—
124C2×Dic239C22×C412yes—
125C23×C239—21no—
126C23×C239—21no—
127C23×C239—21no—
128C23×C239—21no—
129C23×C239—21no—
130C23×C239—21no—
131C23×C239—21no—
132C23×C239—21no—
133C23×C239—21no—
134C23×C239—21no—
135C23×C239C22×C412yes—
136C23×C239C22×C412yes—
137C23×C239#4112yes—
138C23×C239#4112yes—
139C23×C239C2411yescharacteristic
140D4×C239—28no—
141C2×C4×C239—21no—
142C2×C4×C239—21no—
143C2×C4×C239—21no—
144C2×C4×C239—21no—
145C2×C4×C239—21no—
146C2×C4×C239—21no—
147C2×C4×C239—21no—
148C2×C4×C239—21no—
149C2×C4×C239—21no—
150C2×C4×C239—21no—
151C2×C4×C239—21no—
152C2×C4×C239—21no—
153C2×C4×C239—21no—
154C2×C4×C239—21no—
155C2×C4×C239—44no—
156C2×C4×C239C22×C412yes—
157C22×C239—22no—
158C22×C239—22no—
159C22×C239—22no—
160C22×C239—22no—
161C22×C239—22no—
162C22×C239—22no—
163C22×C239—22no—
164C22×C239—22no—
165C22×C239—22no—
166C22×C239—41no—
167C22×C239—41no—
168C22×C239—41no—
169C22×C239—41no—
170C22×C239#11714yes—
171C22×C239C23×C411yescharacteristic
172C22×C239C2×#4111yescharacteristic
173C22×C239C2×#4111yescharacteristic
174C22×C239H(5,𝔽2)11yescharacteristic
175C22×C239H(5,𝔽2)11yescharacteristic
176C22×C239H(5,𝔽2)11yescharacteristic
177C22×C2392−1+411yescharacteristic
178C4×C239—22no—
179C4×C239—41no—
180C4×C239—41no—
181C4×C239—41no—
182C4×C239—41no—
183Dic239—22no—
184Dic239—22no—
185Dic239—22no—
186Dic239—22no—
187Dic239—41no—
188Dic239—41no—
189Dic239—41no—
190Dic239—41no—
191C2×C239—24no—
192C2×C239—42no—
193C2×C239C2×#11711yescharacteristic
194C2×C239#51811yescharacteristic
195C2×C239#52011yescharacteristic
196C2×C239#53811yescharacteristic
197C2×C239#53811yescharacteristic
198C2×C239#53911yescharacteristic
199C2×C239#53911yescharacteristic
200C239#231811yescharacteristic, complemented, Sylow
201#2318—2391nomaximal, Sylow
202C2×C4×D4—2391no—
203#388—2391no—
204#388—2391no—
205#387—2391no—
206#387—2391no—
207#386—2391no—
208#386D23911yescharacteristic
209#385—2391no—
210#385—2391no—
211#383—2391no—
212#382—2391no—
213#381—2391no—
214#380—2391no—
215C4×C22⋊C4—2391no—
216C2×#95—2391no—
217C22×D4C2×D23911yescharacteristic
218C23×C4—2391no—
219C23×C4—2391no—
220C23×C4—2391no—
221C23×C4C2×D23911yescharacteristic
222C4×D4—2398no—
223C2×C4⋊C4—2391no—
224C2×C4⋊C4—2391no—
225C2×C4⋊C4—2391no—
226C2×C4⋊C4—2391no—
227C2×C4⋊C4—2391no—
228C2×C22⋊C4—2391no—
229C2×C22⋊C4—2391no—
230C2×C22⋊C4—2391no—
231C2×C22⋊C4—2391no—
232C2×C22⋊C4—2391no—
233C2×C22⋊C4C2×D23911yescharacteristic
234C2×C22⋊C4C2×D23911yescharacteristic
235C2×C22⋊C4C2×D23911yescharacteristic
236C2×C42—2391no—
237C2×C42—2391no—
238C2×C42—2391no—
239#95—2391no—
240#95—2391no—
241#95—2391no—
242#95—2391no—
243#95—2391no—
244#95—2391no—
245#95—2391no—
246#95—2391no—
247#95—2391no—
248#95—2391no—
249#95—2391no—
250#95—2391no—
251#95C2×D23911yescharacteristic
252#95C2×D23911yescharacteristic
253C24C22×D23911yescharacteristic
254C24C22×D23911yescharacteristic
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C4×D23914yes—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—9561no—
262C22×C4—9561no—
263C22×C4—9561no—
264C22×C4—9561no—
265C22×C4—9561no—
266C22×C4—9561no—
267C22×C4—4782no—
268C22×C4—2394no—
269C22×C4—2394no—
270C22×C4—2391no—
271C22×C4—2391no—
272C22×C4—2391no—
273C22×C4—2391no—
274C22×C4—2391no—
275C22×C4—2391no—
276C22×C4—2391no—
277C22×C4—2391no—
278C22×C4C22×D23911yescharacteristic
279C22×C4C22×D23911yescharacteristic
280C22×C4C22×D23911yescharacteristic
281C22×C4C22×D23911yescharacteristic
282C22×C4C22×D23911yescharacteristic
283C4⋊C4—9561no—
284C4⋊C4—9561no—
285C4⋊C4—9561no—
286C4⋊C4—9561no—
287C4⋊C4—2394no—
288C22⋊C4—22no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—9561no—
292C22⋊C4—9561no—
293C22⋊C4—4782no—
294C22⋊C4—2394no—
295C22⋊C4—2394no—
296C42—9561no—
297C42—9561no—
298C42—2394no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C2×C4×D23912yes—
310C23C2×C4×D23912yes—
311C23#1683854712yes—
312C23#1683854712yes—
313C23C23×D23911yescharacteristic
314D4—28no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—44no—
330C2×C4—9564no—
331C2×C4—9564no—
332C2×C4—9564no—
333C2×C4—4781no—
334C2×C4—4781no—
335C2×C4—4781no—
336C2×C4—4781no—
337C2×C4—4781no—
338C2×C4—4781no—
339C2×C4—4781no—
340C2×C4—4781no—
341C2×C4—4781no—
342C2×C4—4781no—
343C2×C4—4781no—
344C2×C4—4781no—
345C2×C4—4781no—
346C2×C4—4781no—
347C2×C4—4781no—
348C2×C4—4781no—
349C2×C4—4781no—
350C2×C4—4781no—
351C2×C4—4781no—
352C2×C4—4781no—
353C2×C4—2392no—
354C2×C4—2392no—
355C2×C4—2392no—
356C2×C4—2392no—
357C2×C4C2×C4×D23912yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22#2143991614yes—
372C22C22×C4×D23911yescharacteristic
373C22C2×#1683854711yescharacteristic
374C22C2×#1683854711yescharacteristic
375C22#2144000511yescharacteristic
376C22#2144000511yescharacteristic
377C22#2144000611yescharacteristic
378C22#2144000711yescharacteristic
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—9561no—
385C4—9561no—
386C4—9561no—
387C4—9561no—
388C4—4782no—
389C4—4782no—
390C4—4782no—
391C4—4782no—
392C2—24no—
393C2—42no—
394C2C2×#2143991611yescharacteristic
395C2#2950524411yescharacteristic
396C2#2950525111yescharacteristic
397C2#2950535711yescharacteristic
398C2#2950535911yescharacteristic
399C2#2950536611yescharacteristic
400C2#2950536711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D239
commutatorC22×C239C23×C4
Fitting#386×C239C2
socle, cosocleC23×C239C24
prime radicalC22×D4×C239C22
hypercenter#386D239
hypocenterC239#2318
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #29504666C2semidirect
3. #29504660C2semidirect
4. #29504657C2semidirect
5. #29504655C2semidirect
7. #29504645C2semidirect
8. #29504644C2semidirect
9. #29504639C2semidirect
10. #29504636C2semidirect
11. #29504633C2semidirect
12. #29504630C2semidirect
13. C22⋊C4×Dic239C2semidirect
14. C2×C239⋊#95C2semidirect
19. D4×Dic239C4semidirect
60. C22×Dic239C2×C4semidirect
69. C4⋊Dic239C2×C4semidirect
78. C22⋊Dic239C2×C4semidirect
81. C4×Dic239C2×C4semidirect
200. C239#2318semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic2392
#295046662
#295046602
#295046572
#295046552
#295046522
#295046452
#295046442
#295046392
#295046362
#295046332
#295046302
C22⋊C4×Dic2392
C2×C239⋊#952
#386×C2392
#2318239