Finite Group Databasebeta

← #44487534

Subgroups — #44487534

Order 30592.

652 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#21439941C211yesmaximal, characteristic
2C2×#21439916C211yesmaximal, characteristic, complemented
3#29504667C211yesmaximal, characteristic, complemented
4#29504666C211yesmaximal, characteristic, complemented
5#29504660C211yesmaximal, characteristic, complemented
6#29504656C211yesmaximal, characteristic, complemented
7#29504653C211yesmaximal, characteristic, complemented
8#29504652C211yesmaximal, characteristic
9#29504645C211yesmaximal, characteristic, complemented
10#29504645C211yesmaximal, characteristic, complemented
11C4×Dic239⋊C4C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic239C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic239C211yesmaximal, characteristic, complemented
14C2×C239⋊#95C211yesmaximal, characteristic, complemented
15C2×C4×D4×C239C211yesmaximal, characteristic
16C23×Dic239C2211yescharacteristic
17C23×Dic239C2211yescharacteristic
18#21439941C418yescomplemented
19#21439916—24no—
20C2×Dic239⋊C4C2211yescharacteristic
21C2×Dic239⋊C4C2211yescharacteristic
22C2×Dic239⋊C4C2211yescharacteristic
23C2×Dic239⋊C4C2211yescharacteristic
24C2×Dic239⋊C4C2211yescharacteristic
25C2×Dic239⋊C4C2211yescharacteristic
26C2×C22⋊Dic239C2211yescharacteristic
27C2×C22⋊Dic239C2211yescharacteristic
28C2×C22⋊Dic239C2211yescharacteristic
29C2×C22⋊Dic239C2211yescharacteristic
30C2×C22⋊Dic239C2211yescharacteristic
31C2×C22⋊Dic239C2211yescharacteristic
32C2×C4×Dic239C2211yescharacteristic
33C2×C4×Dic239C2211yescharacteristic
34C2×C4×Dic239C2211yescharacteristic
35C2×C4×Dic239C2211yescharacteristic
36C239⋊#95C2211yescharacteristic
37C239⋊#95C2211yescharacteristic
38C239⋊#95C2211yescharacteristic
39C239⋊#95C2211yescharacteristic
40C239⋊#95C2211yescharacteristic
41C239⋊#95C2211yescharacteristic
42C239⋊#95C2211yescharacteristic
43C239⋊#95C2211yescharacteristic
44C239⋊#95C2211yescharacteristic
45C239⋊#95C2211yescharacteristic
46C22×D4×C239C2211yescharacteristic
47C23×C4×C239C2211yescharacteristic
48C23×C4×C239C2211yescharacteristic
49C4×D4×C239—24no—
50C2×C4⋊C4×C239C2211yescharacteristic
51C2×C22⋊C4×C239C2211yescharacteristic
52C2×C22⋊C4×C239C2211yescharacteristic
53C2×C42×C239C2211yescharacteristic
54C22×Dic239—22no—
55C22×Dic239—22no—
56C22×Dic239—22no—
57C22×Dic239—22no—
58C22×Dic239—41no—
59C22×Dic239C2×C414yescomplemented
60C22×Dic239C2311yescharacteristic
61C22×Dic239C2311yescharacteristic
62C22×Dic239C2311yescharacteristic
63C22×Dic239C2311yescharacteristic
64C22×Dic239C2311yescharacteristic
65C22×Dic239C2311yescharacteristic
66C22×Dic239C2311yescharacteristic
67C22×Dic239C2311yescharacteristic
68Dic239⋊C4—22no—
69Dic239⋊C4—22no—
70Dic239⋊C4—22no—
71Dic239⋊C4—41no—
72Dic239⋊C4C2×C414yescomplemented
73Dic239⋊C4C2×C414yescomplemented
74C22⋊Dic239—22no—
75C22⋊Dic239—22no—
76C22⋊Dic239—41no—
77C22⋊Dic239C2×C414yescomplemented
78C22⋊Dic239C2×C414yescomplemented
79C22⋊Dic239C2×C414yescomplemented
80C4×Dic239—22no—
81C4×Dic239—22no—
82C4×Dic239—22no—
83C4×Dic239—41no—
84C24×C239C2311yescharacteristic
85C24×C239C2311yescharacteristic
86C2×D4×C239—22no—
87C2×D4×C239—22no—
88C2×D4×C239C2×C414yes—
89C22×C4×C239—22no—
90C22×C4×C239—22no—
91C22×C4×C239—22no—
92C22×C4×C239—22no—
93C22×C4×C239—41no—
94C22×C4×C239—41no—
95C22×C4×C239C2311yescharacteristic
96C22×C4×C239C2311yescharacteristic
97C22×C4×C239C2311yescharacteristic
98C22×C4×C239C2311yescharacteristic
99C22×C4×C239C2311yescharacteristic
100C4⋊C4×C239—22no—
101C22⋊C4×C239—22no—
102C22⋊C4×C239D414yes—
103C42×C239—22no—
104C2×Dic239—28no—
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—21no—
119C2×Dic239—21no—
120C2×Dic239—21no—
121C2×Dic239—21no—
122C2×Dic239—21no—
123C2×Dic239—21no—
124C2×Dic239—44no—
125C2×Dic239C22×C412yes—
126C2×Dic239C22×C412yes—
127C2×Dic239C22×C412yes—
128C2×Dic239C22×C412yes—
129C2×Dic239#4112yes—
130C23×C239—21no—
131C23×C239—21no—
132C23×C239—21no—
133C23×C239—21no—
134C23×C239—21no—
135C23×C239—21no—
136C23×C239—21no—
137C23×C239—21no—
138C23×C239—21no—
139C23×C239—21no—
140C23×C239C22×C412yes—
141C23×C239C22×C412yes—
142C23×C239C2×D412yes—
143C23×C239#4112yes—
144C23×C239C2411yescharacteristic
145D4×C239—28no—
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—21no—
156C2×C4×C239—21no—
157C2×C4×C239—44no—
158C2×C4×C239—44no—
159C2×C4×C239C22×C412yes—
160C2×C4×C239C2×D412yes—
161C2×C4×C239C2×D412yes—
162C2×C4×C239#4112yes—
163C22×C239—22no—
164C22×C239—22no—
165C22×C239—22no—
166C22×C239—22no—
167C22×C239—22no—
168C22×C239—22no—
169C22×C239—22no—
170C22×C239—22no—
171C22×C239—22no—
172C22×C239—41no—
173C22×C239—41no—
174C22×C239—41no—
175C22×C239—41no—
176C22×C239C4×D414yes—
177C22×C239C23×C411yescharacteristic
178C22×C239C22×D411yescharacteristic
179C22×C239C2×#4111yescharacteristic
180C22×C239C2×#4111yescharacteristic
181C22×C239C2×#4111yescharacteristic
182C22×C239H(5,𝔽2)11yescharacteristic
183C22×C2392−1+411yescharacteristic
184C4×C239—22no—
185C4×C239—22no—
186C4×C239—22no—
187C4×C239—22no—
188C4×C239—41no—
189Dic239—22no—
190Dic239—22no—
191Dic239—22no—
192Dic239—22no—
193Dic239—22no—
194Dic239—41no—
195Dic239—41no—
196Dic239—41no—
197C2×C239—24no—
198C2×C239—42no—
199C2×C239C2×C4×D411yescharacteristic
200C2×C239C4×#4111yescharacteristic
201C2×C239#52011yescharacteristic
202C2×C239#54611yescharacteristic
203C2×C239#54711yescharacteristic
204C2×C239#55111yescharacteristic
205C2×C239#55211yescharacteristic
206C239#235911yescharacteristic, complemented, Sylow
207#2359—2391nomaximal, Sylow
208C2×#123—2391no—
209C2×C4×D4D23911yescharacteristic
210C2×#117—2391no—
211#389—2391no—
212#388—2391no—
213#388—2391no—
214#387—2391no—
215#387—2391no—
216#386—2391no—
217#385—2391no—
218#385—2391no—
219C4×C4⋊C4—2391no—
220C4×C22⋊C4—2391no—
221C4×C22⋊C4—2391no—
222C2×#95—2391no—
223C22×D4C2×D23911yescharacteristic
224C23×C4—2391no—
225C23×C4—2391no—
226C23×C4C2×D23911yescharacteristic
227C23×C4C2×D23911yescharacteristic
228#123—2398no—
229C4×D4—24no—
230#117—4784no—
231C2×C4⋊C4—2391no—
232C2×C4⋊C4—2391no—
233C2×C4⋊C4—2391no—
234C2×C4⋊C4—2391no—
235C2×C4⋊C4—2391no—
236C2×C4⋊C4—2391no—
237C2×C4⋊C4C2×D23911yescharacteristic
238C2×C22⋊C4—2391no—
239C2×C22⋊C4—2391no—
240C2×C22⋊C4—2391no—
241C2×C22⋊C4—2391no—
242C2×C22⋊C4—2391no—
243C2×C22⋊C4—2391no—
244C2×C22⋊C4C2×D23911yescharacteristic
245C2×C22⋊C4C2×D23911yescharacteristic
246C2×C42—2391no—
247C2×C42—2391no—
248C2×C42—2391no—
249C2×C42—2391no—
250C2×C42C2×D23911yescharacteristic
251#95—2391no—
252#95—2391no—
253#95—2391no—
254#95—2391no—
255#95—2391no—
256#95—2391no—
257#95—2391no—
258#95—2391no—
259#95—2391no—
260#95—2391no—
261C24C22×D23911yescharacteristic
262C24C22×D23911yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4C4×D23914yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—9561no—
273C22×C4—4782no—
274C22×C4—4782no—
275C22×C4—4782no—
276C22×C4—4782no—
277C22×C4—2391no—
278C22×C4—2391no—
279C22×C4—2391no—
280C22×C4—2394no—
281C22×C4—2391no—
282C22×C4—2391no—
283C22×C4—2391no—
284C22×C4—2391no—
285C22×C4—2391no—
286C22×C4C22×D23911yescharacteristic
287C22×C4C22×D23911yescharacteristic
288C22×C4C22×D23911yescharacteristic
289C22×C4C22×D23911yescharacteristic
290C22×C4C22×D23911yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—9561no—
293C4⋊C4—4782no—
294C4⋊C4—4782no—
295C4⋊C4—4782no—
296C4⋊C4—2394no—
297C4⋊C4—2394no—
298C22⋊C4—22no—
299C22⋊C4—9561no—
300C22⋊C4—4782no—
301C22⋊C4—4782no—
302C22⋊C4—2394no—
303C22⋊C4—2394no—
304C22⋊C4—2394no—
305C22⋊C4C22⋊D23914yes—
306C42—22no—
307C42—9561no—
308C42—4782no—
309C42—4782no—
310C42—4782no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C4×D23912yes—
322C23C2×C4×D23912yes—
323C23C2×C22⋊D23912yes—
324C23#1683854812yes—
325C23C23×D23911yescharacteristic
326D4—28no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—9564no—
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—4781no—
354C2×C4—4781no—
355C2×C4—4781no—
356C2×C4—4781no—
357C2×C4—4781no—
358C2×C4—4788no—
359C2×C4—4781no—
360C2×C4—4781no—
361C2×C4—2392no—
362C2×C4—2392no—
363C2×C4—2392no—
364C2×C4—2392no—
365C2×C4—2392no—
366C2×C4C2×C4×D23912yes—
367C2×C4C2×C22⋊D23912yes—
368C2×C4C2×C22⋊D23912yes—
369C2×C4#1683854712yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22C4×C22⋊D23914yes—
384C22C22×C4×D23911yescharacteristic
385C22C22×C22⋊D23911yescharacteristic
386C22#41×D23911yescharacteristic
387C22C2×#1683854711yescharacteristic
388C22C2×#1683854811yescharacteristic
389C22#2144000511yescharacteristic
390C22#2144000711yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—9561no—
397C4—9561no—
398C4—9561no—
399C4—4782no—
400C4—4782no—
401C4—4782no—
402C4—4782no—
403C4—4782no—
404C2—24no—
405C2—42no—
406C2C2×C4×C22⋊D23911yescharacteristic
407C2C4×#1683854711yescharacteristic
408C2#2950525111yescharacteristic
409C2#2950541611yescharacteristic
410C2#2950542111yescharacteristic
411C2#2950545111yescharacteristic
412C2#2950546211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D239
commutatorC22×C239C23×C4
FittingC2×C4×D4×C239C2
socle, cosocleC23×C239C24
prime radicalC22×D4×C239C22
hypercenterC2×C4×D4D239
hypocenterC239#2359
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#21439916C2semidirect
3. #29504667C2semidirect
4. #29504666C2semidirect
5. #29504660C2semidirect
6. #29504656C2semidirect
7. #29504653C2semidirect
9. #29504645C2semidirect
10. #29504645C2semidirect
11. C4×Dic239⋊C4C2semidirect
12. C22⋊C4×Dic239C2semidirect
13. C4×C22⋊Dic239C2semidirect
14. C2×C239⋊#95C2semidirect
18. #21439941C4semidirect
59. C22×Dic239C2×C4semidirect
72. Dic239⋊C4C2×C4semidirect
73. Dic239⋊C4C2×C4semidirect
77. C22⋊Dic239C2×C4semidirect
78. C22⋊Dic239C2×C4semidirect
79. C22⋊Dic239C2×C4semidirect
206. C239#2359semidirect

Maximal subgroups

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

subgroupindex
C2×#214399412
C2×#214399162
#295046672
#295046662
#295046602
#295046562
#295046532
#295046522
#295046452
C4×Dic239⋊C42
C22⋊C4×Dic2392
C4×C22⋊Dic2392
C2×C239⋊#952
C2×C4×D4×C2392
#2359239