Finite Group Databasebeta

← #34058307

Subgroups — #34058307

Order 19072.

1094 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#23750351C211yesmaximal, characteristic, complemented
2#23750344C213yesmaximal, complemented
3#23750315C213yesmaximal, complemented
4#23750309C213yesmaximal, complemented
5Q8×C22⋊D149C211yesmaximal, characteristic, complemented
6#23750282C211yesmaximal, characteristic, complemented
7#23750275C213yesmaximal, complemented
8#23750177C213yesmaximal, complemented
9#23750156C213yesmaximal, complemented
10#23750154C213yesmaximal, complemented
11#41×Dic149C211yesmaximal, characteristic, complemented
12C4×#14558735C213yesmaximal, complemented
13C2×Q8×Dic149C211yesmaximal, characteristic, complemented
14C4×Q8×D149C211yesmaximal, characteristic, complemented
15#548×C149C211yesmaximal, characteristic, complemented
16C2×#14558735C2213yes—
17C2×Q8×D149C2211yescharacteristic
18#18011910C2213yes—
19#18011909C2213yes—
20#18011904C2216yes—
21#18011904C2216yes—
22#18011903C2213yes—
23#18011902C2213yes—
24#18011901C2216yes—
25#18011898C2213yes—
26#18011893C2216yes—
27#18011892C2216yes—
28#18011890C2213yes—
29#18011890C2213yes—
30#18011888C2216yes—
31#18011887C2213yes—
32#18011882C2213yes—
33Q8×Dic149C2213yes—
34Q8×Dic149C2211yescharacteristic
35Q8×Dic149C2211yescharacteristic
36Q8×Dic149C2213yes—
37Q8×Dic149C2211yescharacteristic
38C4×Dic298C2213yes—
39C4×Dic298C2213yes—
40#18011876C2213yes—
41#18011876C2213yes—
42D4×Dic149C2213yes—
43D4×Dic149C2213yes—
44D4×Dic149C2213yes—
45C4×C22⋊D149C2213yes—
46C4×C22⋊D149C2213yes—
47#18011871C2213yes—
48#18011871C2211yescharacteristic
49#18011870C2213yes—
50#18011869C2213yes—
51#18011867C2216yes—
52#18011867C2213yes—
53#18011867C2213yes—
54C2×C4⋊Dic149C2213yes—
55C4⋊C4×D149C2213yes—
56C2×C4×Dic149C2213yes—
57C2×C4×Dic149C2213yes—
58C42×D149C2213yes—
59C2×#41×C149C2211yescharacteristic
60C22×Q8×C149C2211yescharacteristic
61#124×C149C2213yes—
62C22⋊Q8×C149C2213yes—
63#121×C149C2213yes—
64C4×Q8×C149C2211yescharacteristic
65C4×D4×C149C2213yes—
66#14558735—212no—
67C2×Dic298C2313yes—
68C2×Dic298C2313yes—
69Q8×D149—24no—
70C2×C22⋊D149C2313yes—
71C2×C22⋊D149C2311yescharacteristic
72C22×Dic149C2313yes—
73C22×Dic149C2313yes—
74C22×Dic149C2311yescharacteristic
75C22×Dic149C2313yes—
76C2×C4×D149C2313yes—
77C2×C4×D149C2313yes—
78C2×C4×D149C2311yescharacteristic
79C4⋊Dic149C2313yes—
80C4⋊Dic149C2316yes—
81C4⋊Dic149C2313yes—
82C4⋊Dic149C2313yes—
83C4⋊Dic149C2313yes—
84#14558715C2316yes—
85#14558715C2313yes—
86#14558715C2316yes—
87#14558715C2313yes—
88#14558715C2316yes—
89#14558715C2311yescharacteristic
90#14558715C2313yes—
91C22⋊Dic149C2313yes—
92C22⋊Dic149C2316yes—
93C22⋊Dic149C2316yes—
94C22⋊Dic149C2313yes—
95#14558713C2311yescharacteristic
96#14558713C2313yes—
97C4×Dic149C2313yes—
98C4×Dic149C2313yes—
99C4×Dic149C2313yes—
100C4×Dic149C2313yes—
101C4×Dic149C2313yes—
102C4×Dic149C2311yescharacteristic
103C4×Dic149C2313yes—
104C4×Dic149C2313yes—
105C4×Dic149C2313yes—
106#41×C149—24no—
107C2×Q8×C149—24no—
108C2×Q8×C149C2313yes—
109C2×Q8×C149C2311yescharacteristic
110C2×D4×C149C2313yes—
111C2×D4×C149C2313yes—
112C22×C4×C149C2313yes—
113C22×C4×C149C2313yes—
114C4⋊C4×C149C2313yes—
115C4⋊C4×C149C2313yes—
116C22⋊C4×C149C2313yes—
117C22⋊C4×C149C2316yes—
118C22⋊C4×C149C2311yescharacteristic
119C42×C149C2313yes—
120C22×D149C2411yescharacteristic
121Dic298—26no—
122Dic298—26no—
123C22⋊D149—26no—
124C22⋊D149#4114yescomplemented
125C2×Dic149—26no—
126C2×Dic149—26no—
127C2×Dic149—22no—
128C2×Dic149—26no—
129C2×Dic149C2413yes—
130C2×Dic149C2413yes—
131C2×Dic149C2413yes—
132C2×Dic149C2411yescharacteristic
133C2×Dic149C2413yes—
134C2×Dic149C2411yescharacteristic
135C2×Dic149C2411yescharacteristic
136C4×D149—26no—
137C4×D149—26no—
138C4×D149—22no—
139C23×C149C2413yes—
140C23×C149C2411yescharacteristic
141Q8×C149—26no—
142Q8×C149#4114yescomplemented
143D4×C149—26no—
144D4×C149—26no—
145C2×C4×C149—26no—
146C2×C4×C149—26no—
147C2×C4×C149C2413yes—
148C2×C4×C149C2411yescharacteristic
149C2×C4×C149C2413yes—
150C2×C4×C149C2411yescharacteristic
151C2×C4×C149C2413yes—
152C22×C149—21no—
153C22×C149—21no—
154C22×C149—23no—
155C22×C149—23no—
156C22×C149—23no—
157C22×C149C2×#4112yes—
158C22×C149C2511yescharacteristic
159C2×D149—21no—
160C2×D149—21no—
161C2×D149C2×#4112yes—
162C4×C149—21no—
163C4×C149—23no—
164C4×C149—23no—
165C4×C149—21no—
166C4×C149C2×#4116yes—
167Dic149—23no—
168Dic149—23no—
169Dic149—23no—
170Dic149—21no—
171Dic149—23no—
172Dic149C2×#4112yes—
173Dic1492−1+412yes—
174C2×C149—22no—
175C2×C149—43no—
176C2×C149C22×#4111yescharacteristic
177C2×C149C22×#4111yescharacteristic
178C2×C149C2×2−1+411yescharacteristic
179D149—22no—
180C149#348211yescharacteristic, complemented, Sylow
181#3482—1491nomaximal, Sylow
182#557—1491no—
183#556—1493no—
184#552—1493no—
185#552—1493no—
186D4×Q8—1491no—
187#548—1491no—
188#548D14911yescharacteristic, complemented
189#547—1493no—
190#533—1493no—
191#529—1493no—
192#529—1493no—
193C4×#41—1491no—
194C4×#41—1493no—
195C2×C4×Q8—1491no—
196C2×C4×Q8—1491no—
197C2×#41—1493no—
198C2×#41C2×D14911yescharacteristic
199C22×Q8—1491no—
200C22×Q8C2×D14911yescharacteristic
201C4⋊Q8—1493no—
202C4⋊Q8—1493no—
203#126—1496no—
204#126—1496no—
205#125—1493no—
206#125—1496no—
207#125—1493no—
208#124—1493no—
209#124C2×D14913yes—
210#123—1496no—
211#123—1496no—
212C22⋊Q8—1493no—
213C22⋊Q8—1493no—
214C22⋊Q8—1493no—
215C22⋊Q8—1496no—
216C22⋊Q8C2×D14913yes—
217#121—1493no—
218#121C2×D14913yes—
219C4×Q8—1491no—
220C4×Q8—1493no—
221C4×Q8—1491no—
222C4×Q8—1493no—
223C4×Q8—1491no—
224C4×Q8—1493no—
225C4×Q8—1493no—
226C4×Q8—1493no—
227C4×Q8—1493no—
228C4×Q8C2×D14911yescharacteristic
229C4×D4—1493no—
230C4×D4—1493no—
231C4×D4—1493no—
232C4×D4—1493no—
233C4×D4—1493no—
234C4×D4—1491no—
235C4×D4—1493no—
236C4×D4C2×D14913yes—
237#117—1493no—
238#117—1493no—
239#117—1493no—
240#117—1493no—
241#117—1496no—
242C2×C4⋊C4—1493no—
243C2×C4⋊C4—1493no—
244C2×C42—1493no—
245C2×C42—1493no—
246C2×C42—1493no—
247#41—24no—
248#41—29812no—
249C2×Q8—24no—
250C2×Q8—2984no—
251C2×Q8—1493no—
252C2×Q8—1493no—
253C2×Q8C22×D14913yes—
254C2×Q8C22×D14911yescharacteristic
255C2×D4—1493no—
256C2×D4—1491no—
257C2×D4C22×D14913yes—
258C2×D4C22×D14913yes—
259C22×C4—1491no—
260C22×C4—1493no—
261C22×C4—1493no—
262C22×C4—1493no—
263C22×C4—1491no—
264C22×C4—1493no—
265C22×C4—1493no—
266C22×C4C22×D14913yes—
267C22×C4C22×D14913yes—
268C4⋊C4—1493no—
269C4⋊C4—1493no—
270C4⋊C4—1496no—
271C4⋊C4—1496no—
272C4⋊C4—1491no—
273C4⋊C4—1493no—
274C4⋊C4—1493no—
275C4⋊C4—1493no—
276C4⋊C4—1493no—
277C4⋊C4—1493no—
278C4⋊C4—1496no—
279C4⋊C4—1496no—
280C4⋊C4C22×D14913yes—
281C4⋊C4C22×D14913yes—
282C22⋊C4—1493no—
283C22⋊C4—1496no—
284C22⋊C4—1493no—
285C22⋊C4—1496no—
286C22⋊C4—1491no—
287C22⋊C4—1493no—
288C22⋊C4C22×D14911yescharacteristic
289C22⋊C4C22×D14913yes—
290C22⋊C4C22×D14916yes—
291C42—1493no—
292C42—1491no—
293C42—1493no—
294C42—1493no—
295C42—1493no—
296C42—1493no—
297C42—1493no—
298C42—1493no—
299C42—1493no—
300C42C22×D14913yes—
301C23—1491no—
302C23C23×D14913yes—
303C23C23×D14911yescharacteristic
304Q8—26no—
305Q8—2986no—
306Q8—2986no—
307Q8#1455873514yescomplemented
308D4—26no—
309D4—26no—
310D4—2986no—
311D4—1494no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—2986no—
315C2×C4—2982no—
316C2×C4—2986no—
317C2×C4—2982no—
318C2×C4—2986no—
319C2×C4—2986no—
320C2×C4—2986no—
321C2×C4—1491no—
322C2×C4—1493no—
323C2×C4—1493no—
324C2×C4—1493no—
325C2×C4—1493no—
326C2×C4—1491no—
327C2×C4—1491no—
328C2×C4C23×D14911yescharacteristic
329C2×C4C23×D14911yescharacteristic
330C2×C4C23×D14913yes—
331C2×C4C23×D14913yes—
332C2×C4C23×D14913yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—2981no—
339C22—2981no—
340C22—1492no—
341C22#41×D14912yes—
342C22C24×D14911yescharacteristic
343C4—23no—
344C4—21no—
345C4—23no—
346C4—21no—
347C4—2983no—
348C4—2983no—
349C4—2983no—
350C4—2981no—
351C4—2983no—
352C4—1492no—
353C4—1492no—
354C4C2×#1455873516yes—
355C2—22no—
356C2—43no—
357C2—2982no—
358C2C2×#41×D14911yescharacteristic
359C2C22×#1455873511yescharacteristic
360C22−1+4×D14911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#548×C149C2
hypercenter#548D149
hypocenterC149#3482
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23750351C2semidirect
2. #23750344C2semidirect
3. #23750315C2semidirect
4. #23750309C2semidirect
5. Q8×C22⋊D149C2semidirect
6. #23750282C2semidirect
7. #23750275C2semidirect
8. #23750177C2semidirect
9. #23750156C2semidirect
10. #23750154C2semidirect
11. #41×Dic149C2semidirect
12. C4×#14558735C2semidirect
13. C2×Q8×Dic149C2semidirect
14. C4×Q8×D149C2semidirect
15. #548×C149C2semidirect
124. C22⋊D149#41semidirect
142. Q8×C149#41semidirect
180. C149#3482semidirect
188. #548D149semidirect
307. Q8#14558735semidirect

Maximal subgroups

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

subgroupindex
#237503512
#237503442
#237503152
#237503092
Q8×C22⋊D1492
#237502822
#237502752
#237501772
#237501562
#237501542
#41×Dic1492
C4×#145587352
C2×Q8×Dic1492
C4×Q8×D1492
#548×C1492
#3482149