Finite Group Databasebeta

← #34056980

Subgroups — #34056980

Order 19072.

386 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750339C211yesmaximal, characteristic
2#23750288C211yesmaximal, characteristic, complemented
3#23749974C211yesmaximal, characteristic, complemented
4#23749968C211yesmaximal, characteristic, complemented
5#23749963C211yesmaximal, characteristic, complemented
6#23749919C211yesmaximal, characteristic, complemented
7#23749870C211yesmaximal, characteristic, complemented
8#23749828C211yesmaximal, characteristic, complemented
9#23749763C211yesmaximal, characteristic, complemented
10#23749762C211yesmaximal, characteristic, complemented
11#23749716C211yesmaximal, characteristic, complemented
12#23749701C211yesmaximal, characteristic, complemented
13#23749635C211yesmaximal, characteristic, complemented
14#23749632C211yesmaximal, characteristic, complemented
15#468×C149C211yesmaximal, characteristic, complemented
16C22×Dic298C2211yescharacteristic
17C2×Q8×D149C2211yescharacteristic
18C2×#14558724C2211yescharacteristic
19C2×#14558724C2211yescharacteristic
20C2×#14558719C2211yescharacteristic
21#18011910—21no—
22#18011910C2211yescharacteristic
23#18011909C2211yescharacteristic
24#18011898—21no—
25#18011897—21no—
26#18011891C2211yescharacteristic
27#18011889—21no—
28#18011888—21no—
29#18011883—21no—
30#18011882C2211yescharacteristic
31#18011876C2211yescharacteristic
32D4×Dic149C2211yescharacteristic
33C4×C22⋊D149C2211yescharacteristic
34#18011871—21no—
35#18011870C2211yescharacteristic
36#18011869C2211yescharacteristic
37#18011867—21no—
38#18011844C2211yescharacteristic
39#18011842C2211yescharacteristic
40#18011840C2211yescharacteristic
41#18011836C2211yescharacteristic
42#18011836C2211yescharacteristic
43#18011836C2211yescharacteristic
44#18011836C2211yescharacteristic
45#18011835C2211yescharacteristic
46#18011835C2211yescharacteristic
47#18011834C2211yescharacteristic
48#18011834C2211yescharacteristic
49C22⋊#11924746C2211yescharacteristic
50#18011823C2211yescharacteristic
51C8×Dic149C2211yescharacteristic
52C2×SD16×C149C2211yescharacteristic
53C2×M16×C149C2211yescharacteristic
54C22⋊Q8×C149C2211yescharacteristic
55#121×C149C2211yescharacteristic
56#107×C149C2211yescharacteristic
57#103×C149C2211yescharacteristic
58#102×C149C2211yescharacteristic
59C2×Dic298—21no—
60C2×Dic298—21no—
61C2×Dic298—22no—
62C2×Dic298—21no—
63C2×Dic298—41no—
64C2×Dic298C2311yescharacteristic
65C2×Dic298C2311yescharacteristic
66Q8×D149—22no—
67Q8×D149—41no—
68C2×C22⋊D149—21no—
69C2×C22⋊D149D411yescharacteristic
70C22×Dic149—21no—
71C22×Dic149—21no—
72C2×C4×D149—21no—
73C2×C4×D149C2311yescharacteristic
74#14558724—22no—
75#14558724—41no—
76#14558719—41no—
77C2×#11924746C2311yescharacteristic
78C2×#11924746C2311yescharacteristic
79C4⋊Dic149—21no—
80C4⋊Dic149C2311yescharacteristic
81C4⋊Dic149C2311yescharacteristic
82#14558715—21no—
83#14558715—21no—
84#14558715—21no—
85#14558715—21no—
86#14558715D411yescharacteristic
87C22⋊Dic149—21no—
88C22⋊Dic149—21no—
89C22⋊Dic149—21no—
90C22⋊Dic149D411yescharacteristic
91#14558713—21no—
92#14558713—21no—
93#14558713D411yescharacteristic
94C4×Dic149—21no—
95C4×Dic149—21no—
96C4×Dic149—21no—
97C4×Dic149C2311yescharacteristic
98C2×Q8×C149C2311yescharacteristic
99C2×D4×C149—21no—
100C2×D4×C149C2311yescharacteristic
101C22×C4×C149C2311yescharacteristic
102SD16×C149—41no—
103M16×C149—41no—
104C2×C8×C149C2311yescharacteristic
105C2×C8×C149C2311yescharacteristic
106C4⋊C4×C149—21no—
107C4⋊C4×C149C2311yescharacteristic
108C4⋊C4×C149C2311yescharacteristic
109C22⋊C4×C149—21no—
110C22⋊C4×C149—21no—
111C22×D149C2×D411yescharacteristic
112Dic298—21no—
113Dic298—21no—
114Dic298—21no—
115Dic298—42no—
116Dic298—42no—
117Dic298—81no—
118Dic298#4112yes—
119C22⋊D149—22no—
120C22⋊D149—81no—
121C2×Dic149—21no—
122C2×Dic149—21no—
123C2×Dic149—21no—
124C2×Dic149—21no—
125C2×Dic149—42no—
126C2×Dic149—42no—
127C2×Dic149C2×D411yescharacteristic
128C2×Dic149C2×D411yescharacteristic
129C2×Dic149C2×D411yescharacteristic
130C4×D149—21no—
131C4×D149—21no—
132C4×D149—42no—
133#11924746—21no—
134#11924746—21no—
135C23×C149—21no—
136C23×C149C2×D411yescharacteristic
137Q8×C149—21no—
138Q8×C149—21no—
139D4×C149—21no—
140D4×C149—21no—
141D4×C149—81no—
142C2×C4×C149—21no—
143C2×C4×C149—21no—
144C2×C4×C149—21no—
145C2×C4×C149—21no—
146C2×C4×C149—21no—
147C2×C4×C149C2×D411yescharacteristic
148C2×C4×C149C2411yescharacteristic
149C8×C149—21no—
150C8×C149—21no—
151C22×C149—21no—
152C22×C149—21no—
153C22×C149—21no—
154C22×C149—41no—
155C22×C149—41no—
156C22×C149—41no—
157C22×C149C22×D411yescharacteristic
158C2×D149—21no—
159C2×D149—21no—
160C2×D149—21no—
161C4×C149—21no—
162C4×C149—41no—
163C4×C149—41no—
164C4×C149—41no—
165C4×C149C2×#4111yescharacteristic
166C4×C149H(5,𝔽2)11yescharacteristic
167Dic149—21no—
168Dic149—22no—
169Dic149—21no—
170Dic149—41no—
171Dic149—41no—
172Dic149—41no—
173Dic149#13712yes—
174C2×C149—41no—
175C2×C149—81no—
176C2×C149#54611yescharacteristic
177C2×C149C2×#13711yescharacteristic
178C2×C149#57711yescharacteristic
179D149—41no—
180C149#332911yescharacteristic, complemented, Sylow
181#3329—1491nomaximal, Sylow
182#555—1491no—
183D4×Q8—1491no—
184#487—1491no—
185#485—1491no—
186#485—1491no—
187#477—1491no—
188#468—1491no—
189#468D14911yescharacteristic, complemented
190#461—1491no—
191#448—1491no—
192#448—1491no—
193C4×SD16—1491no—
194#436—1491no—
195#419—1491no—
196#419—1491no—
197C22×Q8—1491no—
198C22×Q8—1491no—
199C2×SD16—1491no—
200C2×SD16—1491no—
201C2×SD16C2×D14911yescharacteristic
202C2×M16—1491no—
203C2×M16C2×D14911yescharacteristic
204C4⋊Q8—2981no—
205C4⋊Q8—1491no—
206C4⋊Q8—1491no—
207#124—2981no—
208#124—2981no—
209C22⋊Q8—2981no—
210C22⋊Q8—2981no—
211C22⋊Q8—1491no—
212C22⋊Q8C2×D14911yescharacteristic
213#121—2981no—
214#121—1491no—
215#121C2×D14911yescharacteristic
216C4×Q8—1491no—
217C4×D4—2981no—
218C4×D4—1491no—
219C4×D4—1491no—
220#117—2981no—
221#117—1491no—
222#117—1491no—
223#107—1491no—
224#107C2×D14911yescharacteristic
225#106—1491no—
226C4⋊C8—1491no—
227#103—1491no—
228#103—1491no—
229#103—1491no—
230#103—1491no—
231#103—1491no—
232#103—1491no—
233#103—1491no—
234#103—1491no—
235#103C2×D14911yescharacteristic
236#102C2×D14911yescharacteristic
237C22⋊C8—1491no—
238C22⋊C8—1491no—
239C4×C8—1491no—
240C2×Q8—2981no—
241C2×Q8—2981no—
242C2×Q8—2982no—
243C2×Q8—2981no—
244C2×Q8—2982no—
245C2×Q8—5961no—
246C2×Q8—5961no—
247C2×Q8—1491no—
248C2×Q8—1491no—
249C2×Q8C22×D14911yescharacteristic
250C2×D4—21no—
251C2×D4—2981no—
252C2×D4—1491no—
253C2×D4C22×D14911yescharacteristic
254C22×C4—2981no—
255C22×C4—2981no—
256C22×C4—2981no—
257C22×C4—1491no—
258C22×C4C22×D14911yescharacteristic
259SD16—41no—
260SD16—2982no—
261SD16—5961no—
262M16—41no—
263M16—5961no—
264C2×C8—1491no—
265C2×C8—1491no—
266C2×C8C22×D14911yescharacteristic
267C2×C8C22×D14911yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—2981no—
270C4⋊C4—2981no—
271C4⋊C4—2981no—
272C4⋊C4—2981no—
273C4⋊C4—2981no—
274C4⋊C4—1491no—
275C4⋊C4—1491no—
276C4⋊C4—1491no—
277C4⋊C4C22×D14911yescharacteristic
278C4⋊C4C22×D14911yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—2981no—
282C22⋊C4—2981no—
283C22⋊C4—2981no—
284C22⋊C4—2981no—
285C22⋊C4—2981no—
286C22⋊C4—1491no—
287C22⋊C4—1491no—
288C42—2981no—
289C42—2981no—
290C42—2981no—
291C42—1491no—
292C23—21no—
293C23—1491no—
294C23D4×D14911yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—2981no—
298Q8—2981no—
299Q8—2981no—
300Q8—5962no—
301Q8—5962no—
302Q8—1492no—
303Q8—11921no—
304D4—21no—
305D4—21no—
306D4—81no—
307D4—2982no—
308D4—11921no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—2981no—
315C2×C4—2981no—
316C2×C4—2981no—
317C2×C4—2981no—
318C2×C4—2981no—
319C2×C4—2981no—
320C2×C4—5962no—
321C2×C4—5962no—
322C2×C4—5962no—
323C2×C4—1491no—
324C2×C4—1491no—
325C2×C4—1491no—
326C2×C4D4×D14911yescharacteristic
327C2×C4C23×D14911yescharacteristic
328C8—21no—
329C8—21no—
330C8—2981no—
331C8—2981no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—2981no—
339C22—2981no—
340C22—2981no—
341C22C2×D4×D14911yescharacteristic
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—2981no—
347C4—2981no—
348C4—2982no—
349C4—5961no—
350C4—5961no—
351C4—5961no—
352C4—1492no—
353C4#41×D14911yescharacteristic
354C4#1801195611yescharacteristic
355C2—41no—
356C2—81no—
357C2—5961no—
358C2#2375026411yescharacteristic
359C2#137×D14911yescharacteristic
360C2#2375043511yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#468×C149C2
socleC22×C149C22×D4
prime radical#23750339C2
hypercenter#468D149
hypocenterC149#3329
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #23750288C2semidirect
3. #23749974C2semidirect
4. #23749968C2semidirect
5. #23749963C2semidirect
6. #23749919C2semidirect
7. #23749870C2semidirect
8. #23749828C2semidirect
9. #23749763C2semidirect
10. #23749762C2semidirect
11. #23749716C2semidirect
12. #23749701C2semidirect
13. #23749635C2semidirect
14. #23749632C2semidirect
15. #468×C149C2semidirect
180. C149#3329semidirect
189. #468D149semidirect

Maximal subgroups

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

subgroupindex
#237503392
#237502882
#237499742
#237499682
#237499632
#237499192
#237498702
#237498282
#237497632
#237497622
#237497162
#237497012
#237496352
#237496322
#468×C1492
#3329149