Finite Group Databasebeta

← #37583200

Subgroups — #37583200

Order 24448.

374 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133282C211yesmaximal, characteristic, complemented
2#27133267C211yesmaximal, characteristic, complemented
3#27133072C211yesmaximal, characteristic, complemented
4#27133017C211yesmaximal, characteristic, complemented
5#27133009C211yesmaximal, characteristic, complemented
6#27132913C211yesmaximal, characteristic, complemented
7#27132906C211yesmaximal, characteristic, complemented
8#27132896C211yesmaximal, characteristic, complemented
9#27132889C211yesmaximal, characteristic, complemented
10#27132775C211yesmaximal, characteristic, complemented
11#27132764C211yesmaximal, characteristic, complemented
12SD16×Dic191C211yesmaximal, characteristic, complemented
13D8×Dic191C211yesmaximal, characteristic, complemented
14#27132620C211yesmaximal, characteristic, complemented
15#496×C191C211yesmaximal, characteristic, complemented
16C2×#15690270C2211yescharacteristic
17C2×C8×D191C2211yescharacteristic
18#20259322C2211yescharacteristic
19#20259315—21no—
20#20259314C2211yescharacteristic
21#20259308—21no—
22#20259306—21no—
23#20259303—21no—
24#20259303—21no—
25#20259299—21no—
26#20259298C2211yescharacteristic
27#20259294—21no—
28#20259294—21no—
29#20259293C2211yescharacteristic
30#20259293C2211yescharacteristic
31#20259293C2211yescharacteristic
32Q8×Dic191C2211yescharacteristic
33D4×Dic191C2211yescharacteristic
34D4×Dic191C2211yescharacteristic
35D4×Dic191C2211yescharacteristic
36#20259279C2211yescharacteristic
37#20259256C2211yescharacteristic
38#20259256C2211yescharacteristic
39#20259254C2211yescharacteristic
40#20259254C2211yescharacteristic
41#20259246C2211yescharacteristic
42#20259246C2211yescharacteristic
43#20259243C2211yescharacteristic
44#20259243C2211yescharacteristic
45#20259243C2211yescharacteristic
46#20259243C2211yescharacteristic
47#20259243C2211yescharacteristic
48#20259243C2211yescharacteristic
49#20259232C2211yescharacteristic
50C4×C191⋊C8C2211yescharacteristic
51C8×Dic191C2211yescharacteristic
52C2×SD16×C191C2211yescharacteristic
53C2×SD16×C191C2211yescharacteristic
54C2×D8×C191C2211yescharacteristic
55C2×D8×C191C2211yescharacteristic
56#127×C191C2211yescharacteristic
57#124×C191C2211yescharacteristic
58#97×C191C2211yescharacteristic
59C2×Dic382—21no—
60C2×C22⋊D191—21no—
61C2×C22⋊D191—21no—
62C2×C22⋊D191—21no—
63C22×Dic191—21no—
64C22×Dic191—21no—
65C22×Dic191—21no—
66C2×C4×D191C2311yescharacteristic
67#15690270—22no—
68C2×C191⋊C8C2311yescharacteristic
69C2×C191⋊C8C2311yescharacteristic
70C8×D191—22no—
71C4⋊Dic191—21no—
72C4⋊Dic191—21no—
73C4⋊Dic191C2311yescharacteristic
74C4⋊Dic191C2311yescharacteristic
75C4⋊Dic191C2311yescharacteristic
76C4⋊Dic191C2311yescharacteristic
77Dic191⋊C4—21no—
78Dic191⋊C4—21no—
79Dic191⋊C4—21no—
80Dic191⋊C4D411yescharacteristic
81Dic191⋊C4D411yescharacteristic
82C22⋊Dic191—21no—
83C22⋊Dic191—21no—
84C22⋊Dic191—21no—
85C22⋊Dic191—21no—
86C22⋊Dic191—21no—
87C22⋊Dic191—21no—
88(C2×D191)⋊C4—21no—
89(C2×D191)⋊C4D411yescharacteristic
90(C2×D191)⋊C4D411yescharacteristic
91C4×Dic191—21no—
92C4×Dic191C2311yescharacteristic
93C2×Q8×C191C2311yescharacteristic
94C2×D4×C191—21no—
95C2×D4×C191—21no—
96C2×D4×C191C2311yescharacteristic
97C2×D4×C191C2311yescharacteristic
98C2×D4×C191C2311yescharacteristic
99SD16×C191—22no—
100SD16×C191—22no—
101D8×C191—22no—
102D8×C191—22no—
103C2×C8×C191C2311yescharacteristic
104C2×C8×C191C2311yescharacteristic
105C22⋊C4×C191—21no—
106C22⋊C4×C191—21no—
107C42×C191C2311yescharacteristic
108C22×D191C2×D411yescharacteristic
109Dic382—81no—
110C22⋊D191—81no—
111C22⋊D191—81no—
112C22⋊D191—81no—
113C2×Dic191—21no—
114C2×Dic191—21no—
115C2×Dic191—21no—
116C2×Dic191—21no—
117C2×Dic191—81no—
118C2×Dic191—81no—
119C2×Dic191—81no—
120C2×Dic191C2×D411yescharacteristic
121C2×Dic191C2×D411yescharacteristic
122C2×Dic191C2×D411yescharacteristic
123C4×D191—21no—
124C4×D191—21no—
125C191⋊C8—21no—
126C191⋊C8—21no—
127C23×C191—21no—
128C23×C191—21no—
129C23×C191—21no—
130Q8×C191—21no—
131Q8×C191—21no—
132D4×C191—21no—
133D4×C191—21no—
134D4×C191—21no—
135D4×C191—21no—
136D4×C191—21no—
137D4×C191—21no—
138D4×C191—81no—
139D4×C191—81no—
140C2×C4×C191—21no—
141C2×C4×C191C2×D411yescharacteristic
142C2×C4×C191C2×D411yescharacteristic
143C2×C4×C191C2411yescharacteristic
144C8×C191#4112yes—
145C8×C191#4112yes—
146C22×C191—41no—
147C22×C191—41no—
148C22×C191—41no—
149C22×C191—41no—
150C22×C191—41no—
151C22×C191—41no—
152C22×C191—41no—
153C22×C191—41no—
154C22×C191—41no—
155C22×C191C22×D411yescharacteristic
156C2×D191—21no—
157C2×D191—21no—
158C2×D191—21no—
159C4×C191—21no—
160C4×C191—21no—
161C4×C191—41no—
162C4×C191C2×#4111yescharacteristic
163C4×C191C2×#4111yescharacteristic
164Dic191—21no—
165Dic191—21no—
166Dic191—21no—
167Dic191—41no—
168Dic191—41no—
169Dic191—41no—
170Dic191—41no—
171C2×C191—81no—
172C2×C191—81no—
173C2×C191—81no—
174C2×C191#53211yescharacteristic
175C2×C191#57611yescharacteristic
176C2×C191#57711yescharacteristic
177D191—41no—
178C191#317911yescharacteristic, complemented, Sylow
179#3179—1911nomaximal, Sylow
180#542—1911no—
181#540—1911no—
182#501—1911no—
183#496D19111yescharacteristic, complemented
184#488—1911no—
185#486—1911no—
186#469—1911no—
187#468—1911no—
188#466—1911no—
189#465—1911no—
190#442—1911no—
191#440—1911no—
192C4×SD16—1911no—
193C4×D8—1911no—
194#405—1911no—
195C2×SD16C2×D19111yescharacteristic
196C2×SD16C2×D19111yescharacteristic
197C2×D8C2×D19111yescharacteristic
198C2×D8C2×D19111yescharacteristic
199C2×M16—1911no—
200C22×C8—1911no—
201C4⋊Q8—1911no—
202#127C2×D19111yescharacteristic
203#126—3821no—
204#125—1911no—
205#124—3821no—
206#124C2×D19111yescharacteristic
207#123—3821no—
208#123—3821no—
209#123—3821no—
210C22⋊Q8—3821no—
211C22⋊Q8—1911no—
212#121—3821no—
213#121—3821no—
214#121—1911no—
215#121—1911no—
216#121—1911no—
217C4×Q8—1911no—
218C4×D4—1911no—
219C4×D4—1911no—
220C4×D4—1911no—
221#117—1911no—
222#107—1911no—
223#107—1911no—
224#106—1911no—
225#106—1911no—
226#103—1911no—
227#103—1911no—
228#102—1911no—
229#102—1911no—
230#102—1911no—
231#102—1911no—
232#102—1911no—
233#102—1911no—
234#97—1911no—
235#97C2×D19111yescharacteristic
236C4×C8—1911no—
237C4×C8—1911no—
238C2×Q8—3821no—
239C2×Q8C22×D19111yescharacteristic
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—3821no—
243C2×D4—3821no—
244C2×D4—3821no—
245C2×D4C22×D19111yescharacteristic
246C2×D4C22×D19111yescharacteristic
247C2×D4C22×D19111yescharacteristic
248C22×C4—3821no—
249C22×C4—3821no—
250C22×C4—3821no—
251C22×C4—1911no—
252SD16—22no—
253SD16—22no—
254D8—22no—
255D8—22no—
256M16—3822no—
257C2×C8—3822no—
258C2×C8—1911no—
259C2×C8—1911no—
260C2×C8C22×D19111yescharacteristic
261C2×C8C22×D19111yescharacteristic
262C4⋊C4—3821no—
263C4⋊C4—3821no—
264C4⋊C4—3821no—
265C4⋊C4—3821no—
266C4⋊C4—3821no—
267C4⋊C4—1911no—
268C4⋊C4—1911no—
269C4⋊C4—1911no—
270C4⋊C4—1911no—
271C4⋊C4—1911no—
272C4⋊C4—1911no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—3821no—
276C22⋊C4—3821no—
277C22⋊C4—3821no—
278C22⋊C4—3821no—
279C22⋊C4—3821no—
280C22⋊C4—3821no—
281C22⋊C4—3821no—
282C22⋊C4—1911no—
283C22⋊C4—1911no—
284C42—3821no—
285C42—1911no—
286C42C22×D19111yescharacteristic
287C23—21no—
288C23—21no—
289C23—21no—
290C23—1911no—
291Q8—21no—
292Q8—21no—
293Q8—15281no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—81no—
301D4—81no—
302D4—15281no—
303D4—15281no—
304D4—15281no—
305C2×C4—21no—
306C2×C4—3821no—
307C2×C4—3821no—
308C2×C4—3821no—
309C2×C4—3821no—
310C2×C4—3821no—
311C2×C4—3821no—
312C2×C4—1911no—
313C2×C4—1911no—
314C2×C4—1911no—
315C2×C4—15281no—
316C2×C4—15281no—
317C2×C4—15281no—
318C2×C4D4×D19111yescharacteristic
319C2×C4D4×D19111yescharacteristic
320C2×C4C23×D19111yescharacteristic
321C8—3821no—
322C8—3821no—
323C8#1569028612yes—
324C8#1569028612yes—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—3821no—
335C22—3821no—
336C22—3821no—
337C22C2×D4×D19111yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—3821no—
342C4—3821no—
343C4—3821no—
344C4—7641no—
345C4—7641no—
346C4—7641no—
347C4—7641no—
348C4C2×#1569028611yescharacteristic
349C4C2×#1569028611yescharacteristic
350C2—81no—
351C2—81no—
352C2—81no—
353C2—7641no—
354C2#2713321111yescharacteristic
355C2#2713346711yescharacteristic
356C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#496×C191C2
socleC22×C191C22×D4
hypercenter#496D191
hypocenterC191#3179
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #27133282C2semidirect
2. #27133267C2semidirect
3. #27133072C2semidirect
4. #27133017C2semidirect
5. #27133009C2semidirect
6. #27132913C2semidirect
7. #27132906C2semidirect
8. #27132896C2semidirect
9. #27132889C2semidirect
10. #27132775C2semidirect
11. #27132764C2semidirect
12. SD16×Dic191C2semidirect
13. D8×Dic191C2semidirect
14. #27132620C2semidirect
15. #496×C191C2semidirect
178. C191#3179semidirect
183. #496D191semidirect

Maximal subgroups

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

subgroupindex
#271332822
#271332672
#271330722
#271330172
#271330092
#271329132
#271329062
#271328962
#271328892
#271327752
#271327642
SD16×Dic1912
D8×Dic1912
#271326202
#496×C1912
#3179191