Finite Group Databasebeta

← #34058308

Subgroups — #34058308

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#23750347C213yesmaximal, complemented
3#23750316C213yesmaximal, complemented
4#23750310C213yesmaximal, complemented
5Q8×C22⋊D149C211yesmaximal, characteristic, complemented
6#23750287C211yesmaximal, characteristic, complemented
7#23750286C211yesmaximal, characteristic, complemented
8#23750276C213yesmaximal, complemented
9#23750179C213yesmaximal, complemented
10#23750162C213yesmaximal, complemented
11#23750159C213yesmaximal, complemented
12C4×#14558737C211yesmaximal, characteristic, complemented
13C4×#14558736C213yesmaximal, complemented
14C4×Q8×D149C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×#14558736C2213yes—
18C2×Q8×D149C2211yescharacteristic
19#18011910C2213yes—
20#18011909C2213yes—
21#18011906C2216yes—
22#18011905C2216yes—
23#18011903C2213yes—
24#18011902C2213yes—
25#18011901C2216yes—
26#18011900C2213yes—
27#18011899C2213yes—
28#18011896C2216yes—
29#18011894C2216yes—
30#18011891C2213yes—
31#18011890C2213yes—
32#18011889C2216yes—
33#18011887C2213yes—
34#18011887C2213yes—
35#18011886C2213yes—
36#18011885C2213yes—
37Q8×Dic149C2211yescharacteristic
38Q8×Dic149C2211yescharacteristic
39C4×Dic298C2213yes—
40C4×Dic298C2213yes—
41C4×Dic298C2213yes—
42#18011876C2213yes—
43C4×D596C2213yes—
44C4×D596C2213yes—
45C4×D596C2213yes—
46#18011873C2213yes—
47C4×C22⋊D149C2213yes—
48C4×C22⋊D149C2213yes—
49C4×C22⋊D149C2213yes—
50C4×C22⋊D149C2211yescharacteristic
51#18011870C2213yes—
52#18011869C2213yes—
53#18011868C2216yes—
54#18011868C2213yes—
55#18011868C2213yes—
56C4⋊C4×D149C2213yes—
57C42×D149C2213yes—
58C42×D149C2213yes—
59C22×Q8×C149C2211yescharacteristic
60C4×Q8×C149C2213yes—
61C4×Q8×C149C2211yescharacteristic
62C4×Q8×C149C2213yes—
63C4×Q8×C149C2211yescharacteristic
64C2×C4⋊C4×C149C2213yes—
65C2×C42×C149C2213yes—
66#14558737—24no—
67#14558736—212no—
68C2×Dic298C2313yes—
69C2×Dic298C2313yes—
70Q8×D149—24no—
71C2×D596C2313yes—
72C2×D596C2313yes—
73C2×C22⋊D149C2313yes—
74C2×C22⋊D149C2311yescharacteristic
75C2×C4×D149C2313yes—
76C2×C4×D149C2313yes—
77C2×C4×D149C2313yes—
78C2×C4×D149C2313yes—
79C2×C4×D149C2311yescharacteristic
80C2×C4×D149C2313yes—
81C4⋊Dic149C2313yes—
82C4⋊Dic149C2313yes—
83C4⋊Dic149C2313yes—
84C4⋊Dic149C2313yes—
85#14558715C2313yes—
86#14558715C2316yes—
87#14558715C2316yes—
88#14558715C2313yes—
89#14558715C2316yes—
90#14558715C2311yescharacteristic
91#14558715C2313yes—
92C22⋊Dic149C2311yescharacteristic
93C22⋊Dic149C2313yes—
94#14558713C2313yes—
95#14558713C2316yes—
96#14558713C2316yes—
97#14558713C2313yes—
98#14558713C2316yes—
99#14558713C2311yescharacteristic
100#14558713C2313yes—
101C4×Dic149C2313yes—
102C4×Dic149C2313yes—
103C4×Dic149C2311yescharacteristic
104C4×Dic149C2313yes—
105C4×Dic149C2313yes—
106C4×Dic149C2313yes—
107C2×Q8×C149—24no—
108C2×Q8×C149C2313yes—
109C2×Q8×C149C2311yescharacteristic
110C22×C4×C149C2313yes—
111C22×C4×C149C2311yescharacteristic
112C22×C4×C149C2313yes—
113C4⋊C4×C149C2316yes—
114C4⋊C4×C149C2313yes—
115C4⋊C4×C149C2313yes—
116C42×C149C2313yes—
117C42×C149C2313yes—
118C42×C149C2313yes—
119C42×C149C2313yes—
120C22×D149C2413yes—
121C22×D149C2411yescharacteristic
122Dic298—26no—
123Dic298—26no—
124D596—26no—
125D596—26no—
126C22⋊D149—26no—
127C22⋊D149#4114yescomplemented
128C2×Dic149C2411yescharacteristic
129C2×Dic149C2413yes—
130C2×Dic149C2413yes—
131C2×Dic149C2411yescharacteristic
132C2×Dic149C2413yes—
133C2×Dic149C2411yescharacteristic
134C4×D149—26no—
135C4×D149—26no—
136C4×D149—26no—
137C4×D149—22no—
138C4×D149—26no—
139C4×D149—26no—
140C23×C149C2411yescharacteristic
141Q8×C149—26no—
142Q8×C149#4114yescomplemented
143C2×C4×C149—26no—
144C2×C4×C149—26no—
145C2×C4×C149—22no—
146C2×C4×C149C2413yes—
147C2×C4×C149C2413yes—
148C2×C4×C149C2411yescharacteristic
149C2×C4×C149C2413yes—
150C2×C4×C149C2413yes—
151C2×C4×C149C2411yescharacteristic
152C22×C149—21no—
153C22×C149—21no—
154C22×C149C2×#4112yes—
155C22×C149C2511yescharacteristic
156C2×D149—21no—
157C2×D149—21no—
158C2×D149—23no—
159C2×D149—23no—
160C2×D149—23no—
161C2×D149C2×#4112yes—
162C4×C149—23no—
163C4×C149—23no—
164C4×C149—23no—
165C4×C149—21no—
166C4×C149C2×#4116yes—
167C4×C1492−1+412yes—
168Dic149—21no—
169Dic149—23no—
170Dic149—23no—
171Dic149—21no—
172Dic149—23no—
173Dic149C2×#4112yes—
174C2×C149—22no—
175C2×C149C22×#4111yescharacteristic
176C2×C149C22×#4111yescharacteristic
177C2×C149C2×2−1+411yescharacteristic
178D149—22no—
179D149—43no—
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#548—1491no—
189#547—1493no—
190#533—1493no—
191#529—1493no—
192#529—1493no—
193C4×#41—1491no—
194C4×#41—1493no—
195C2×C4×Q8—1491no—
196C2×C4×Q8D14911yescharacteristic, complemented
197C2×#41—1491no—
198C2×#41—1493no—
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#124—1493no—
210#123—1496no—
211#123—1496no—
212C22⋊Q8—1493no—
213C22⋊Q8—1493no—
214C22⋊Q8—1493no—
215C22⋊Q8—1496no—
216C22⋊Q8—1493no—
217#121—1493no—
218#121—1493no—
219C4×Q8—1491no—
220C4×Q8—1491no—
221C4×Q8—1493no—
222C4×Q8—1493no—
223C4×Q8—1493no—
224C4×Q8—1493no—
225C4×Q8C2×D14911yescharacteristic
226C4×Q8C2×D14913yes—
227C4×Q8C2×D14911yescharacteristic
228C4×Q8C2×D14913yes—
229C4×D4—1491no—
230C4×D4—1493no—
231C4×D4—1493no—
232C4×D4—1493no—
233C4×D4—1493no—
234C4×D4—1493no—
235C4×D4—1493no—
236C4×D4—1493no—
237#117—1493no—
238#117—1493no—
239#117—1493no—
240#117—1493no—
241#117—1496no—
242C2×C4⋊C4—1493no—
243C2×C4⋊C4C2×D14913yes—
244C2×C42—1493no—
245C2×C42—1493no—
246C2×C42C2×D14913yes—
247#41—29812no—
248#41—2984no—
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×D4—1493no—
258C2×D4—1493no—
259C22×C4—1493no—
260C22×C4—1493no—
261C22×C4—1493no—
262C22×C4—1493no—
263C22×C4—1491no—
264C22×C4—1493no—
265C22×C4C22×D14913yes—
266C22×C4C22×D14913yes—
267C22×C4C22×D14911yescharacteristic
268C4⋊C4—1493no—
269C4⋊C4—1496no—
270C4⋊C4—1491no—
271C4⋊C4—1493no—
272C4⋊C4—1493no—
273C4⋊C4—1493no—
274C4⋊C4—1493no—
275C4⋊C4—1493no—
276C4⋊C4—1493no—
277C4⋊C4—1496no—
278C4⋊C4—1496no—
279C4⋊C4C22×D14913yes—
280C4⋊C4C22×D14913yes—
281C4⋊C4C22×D14916yes—
282C22⋊C4—1491no—
283C22⋊C4—1493no—
284C22⋊C4—1491no—
285C22⋊C4—1493no—
286C22⋊C4—1496no—
287C22⋊C4—1493no—
288C22⋊C4—1493no—
289C22⋊C4—1496no—
290C22⋊C4—1496no—
291C42—1493no—
292C42—1493no—
293C42—1493no—
294C42—1493no—
295C42—1491no—
296C42—1493no—
297C42C22×D14913yes—
298C42C22×D14913yes—
299C42C22×D14913yes—
300C42C22×D14913yes—
301C23—1493no—
302C23—1491no—
303C23C23×D14911yescharacteristic
304Q8—26no—
305Q8—2986no—
306Q8—2986no—
307Q8#1455873614yescomplemented
308D4—2986no—
309D4—2986no—
310D4—2986no—
311D4—1494no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—22no—
315C2×C4—2986no—
316C2×C4—2986no—
317C2×C4—2986no—
318C2×C4—2986no—
319C2×C4—2982no—
320C2×C4—2986no—
321C2×C4—1491no—
322C2×C4—1491no—
323C2×C4—1493no—
324C2×C4—1493no—
325C2×C4—1493no—
326C2×C4—1491no—
327C2×C4C23×D14913yes—
328C2×C4C23×D14911yescharacteristic
329C2×C4C23×D14913yes—
330C2×C4C23×D14913yes—
331C2×C4C23×D14913yes—
332C2×C4C23×D14911yescharacteristic
333C22—21no—
334C22—21no—
335C22—2981no—
336C22—2981no—
337C22—2983no—
338C22—2983no—
339C22—2983no—
340C22—1492no—
341C22#41×D14912yes—
342C22C24×D14911yescharacteristic
343C4—23no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—2983no—
348C4—2981no—
349C4—2983no—
350C4—2981no—
351C4—2983no—
352C4—1492no—
353C4C2×#1455873616yes—
354C4#1801195912yes—
355C2—22no—
356C2—2982no—
357C2—5963no—
358C2C2×#41×D14911yescharacteristic
359C2C22×#1455873611yescharacteristic
360C2C2×#1801195911yescharacteristic
subgroupG/H
centerC2×C4C23×D149
commutator, socle, cosocleC22×C149C25
FrattiniC22C24×D149
FittingC2×C4×Q8×C149C2
hypercenterC2×C4×Q8D149
hypocenterC149#3482
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #23750351C2semidirect
2. #23750347C2semidirect
3. #23750316C2semidirect
4. #23750310C2semidirect
5. Q8×C22⋊D149C2semidirect
6. #23750287C2semidirect
7. #23750286C2semidirect
8. #23750276C2semidirect
9. #23750179C2semidirect
10. #23750162C2semidirect
11. #23750159C2semidirect
12. C4×#14558737C2semidirect
13. C4×#14558736C2semidirect
14. C4×Q8×D149C2semidirect
15. C2×C4×Q8×C149C2semidirect
127. C22⋊D149#41semidirect
142. Q8×C149#41semidirect
180. C149#3482semidirect
196. C2×C4×Q8D149semidirect
307. Q8#14558736semidirect

Maximal subgroups

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

subgroupindex
#237503512
#237503472
#237503162
#237503102
Q8×C22⋊D1492
#237502872
#237502862
#237502762
#237501792
#237501622
#237501592
C4×#145587372
C4×#145587362
C4×Q8×D1492
C2×C4×Q8×C1492
#3482149