Finite Group Databasebeta

← #34058498

Subgroups — #34058498

Order 19072.

964 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic298C211yesmaximal, characteristic, complemented
2#23750355C213yesmaximal, complemented
3#23750353C213yesmaximal, complemented
4#23750352C213yesmaximal, complemented
5#23750352C213yesmaximal, complemented
6#23750351C211yesmaximal, characteristic, complemented
7#23750351C211yesmaximal, characteristic, complemented
8#23750255C213yesmaximal
9#23750252C213yesmaximal
10#23750181C213yesmaximal
11#23750179C213yesmaximal
12#23750101C211yesmaximal, characteristic
13#23750101C211yesmaximal, characteristic
14C4×Q8×D149C211yesmaximal, characteristic
15#557×C149C211yesmaximal, characteristic, complemented
16C2×Q8×D149C2211yescharacteristic
17#18011911C2213yes—
18#18011910C2213yes—
19#18011910C2213yes—
20#18011910C2213yes—
21#18011910C2213yes—
22#18011910C2213yes—
23#18011910C2216yes—
24#18011909C2213yes—
25#18011909C2216yes—
26#18011903C2213yes—
27#18011903C2213yes—
28#18011902C2213yes—
29#18011902C2213yes—
30#18011901C2213yes—
31#18011901C2213yes—
32#18011901C2216yes—
33#18011901C2216yes—
34#18011891C2213yes—
35#18011889C2211yescharacteristic
36#18011889C2213yes—
37#18011889C2216yes—
38#18011887C2213yes—
39Q8×Dic149C2211yescharacteristic
40Q8×Dic149C2211yescharacteristic
41Q8×Dic149C2211yescharacteristic
42C4×Dic298C2213yes—
43C4×Dic298C2213yes—
44C4×Dic298C2213yes—
45C4×Dic298C2213yes—
46C4×Dic298C2213yes—
47C4×Dic298C2213yes—
48#18011876C2216yes—
49#18011876C2216yes—
50#18011876C2213yes—
51#18011876C2213yes—
52#18011876C2213yes—
53#18011869C2213yes—
54#18011869C2213yes—
55#18011868C2213yes—
56#18011868C2213yes—
57C4⋊C4×D149C2213yes—
58C42×D149C2213yes—
59C4⋊Q8×C149C2213yes—
60#125×C149C2213yes—
61#125×C149C2213yes—
62C4×Q8×C149C2213yes—
63C4×Q8×C149C2211yescharacteristic
64C4×Q8×C149C2211yescharacteristic
65C4×Q8×C149C2211yescharacteristic
66C2×Dic298C2313yes—
67C2×Dic298C2311yescharacteristic
68C2×Dic298C2313yes—
69C2×Dic298C2316yes—
70C2×Dic298C2313yes—
71Q8×D149Q818yescomplemented
72C2×C4×D149C2311yescharacteristic
73C2×C4×D149C2313yes—
74C2×C4×D149C2313yes—
75C4⋊Dic149C2313yes—
76C4⋊Dic149C2313yes—
77C4⋊Dic149C2313yes—
78C4⋊Dic149C2313yes—
79C4⋊Dic149C2313yes—
80C4⋊Dic149C2313yes—
81#14558715C2313yes—
82#14558715C2311yescharacteristic
83#14558715C2313yes—
84#14558715C2313yes—
85#14558715C2313yes—
86#14558715C2316yes—
87#14558715C2316yes—
88#14558715C2313yes—
89#14558715C2313yes—
90#14558715C2313yes—
91#14558715C2316yes—
92#14558715C2313yes—
93#14558715C2311yescharacteristic
94#14558713C2311yescharacteristic
95#14558713C2311yescharacteristic
96#14558713C2313yes—
97#14558713C2313yes—
98C4×Dic149C2313yes—
99C4×Dic149C2316yes—
100C4×Dic149C2313yes—
101C4×Dic149C2313yes—
102C4×Dic149C2313yes—
103C4×Dic149C2316yes—
104C4×Dic149C2313yes—
105C2×Q8×C149C2313yes—
106C2×Q8×C149C2311yescharacteristic
107C4⋊C4×C149C2313yes—
108C4⋊C4×C149C2313yes—
109C4⋊C4×C149C2313yes—
110C4⋊C4×C149C2313yes—
111C4⋊C4×C149C2316yes—
112C4⋊C4×C149C2311yescharacteristic
113C4⋊C4×C149C2313yes—
114C42×C149C2313yes—
115C42×C149C2313yes—
116C42×C149C2313yes—
117C22×D149C2411yescharacteristic
118Dic298—26no—
119Dic298—22no—
120Dic298—46no—
121Dic298—43no—
122Dic298C2×Q8112yes—
123C2×Dic149C2413yes—
124C2×Dic149C2411yescharacteristic
125C2×Dic149C2413yes—
126C2×Dic149C2413yes—
127C2×Dic149C2411yescharacteristic
128C2×Dic149C2413yes—
129C2×Dic149C2411yescharacteristic
130C4×D149—22no—
131C4×D149—43no—
132C4×D149C2×Q8112yes—
133Q8×C149—43no—
134Q8×C149C2×Q814yes—
135C2×C4×C149C2413yes—
136C2×C4×C149C2411yescharacteristic
137C2×C4×C149C2413yes—
138C2×C4×C149C2411yescharacteristic
139C2×C4×C149C2413yes—
140C2×C4×C149C2411yescharacteristic
141C2×C4×C149C2413yes—
142C22×C149C2511yescharacteristic
143C2×D149—21no—
144C2×D149—21no—
145C2×D149C22×Q812yes—
146C4×C149—21no—
147C4×C149—21no—
148C4×C149—23no—
149C4×C149—21no—
150C4×C149—23no—
151C4×C149—23no—
152C4×C149C22×Q816yes—
153Dic149—23no—
154Dic149—21no—
155Dic149—23no—
156Dic149—21no—
157Dic149—23no—
158Dic149C22×Q816yes—
159Dic1492−1+412yes—
160C2×C149C23×Q811yescharacteristic
161C2×C149C2×2−1+411yescharacteristic
162C2×C149#58511yescharacteristic
163D149—22no—
164C149#350211yescharacteristic, complemented, Sylow
165#3502—1491nomaximal, Sylow
166Q82—1493no—
167Q82—1491no—
168#557—1493no—
169#557—1493no—
170#557—1491no—
171#557—1491no—
172#557—1493no—
173#557D14911yescharacteristic, complemented
174#544—1493no—
175#544—1493no—
176#533—1493no—
177#533—1493no—
178#519—1491no—
179#519—1491no—
180C2×C4×Q8—1491no—
181C22×Q8—1491no—
182C4⋊Q8—1493no—
183C4⋊Q8—1496no—
184C4⋊Q8—1493no—
185C4⋊Q8—1493no—
186C4⋊Q8—1493no—
187C4⋊Q8—1496no—
188C4⋊Q8—1493no—
189C4⋊Q8—1493no—
190C4⋊Q8—1493no—
191C4⋊Q8C2×D14913yes—
192#125—1493no—
193#125—1493no—
194#125—1496no—
195#125—1496no—
196#125—1493no—
197#125—1493no—
198#125—1493no—
199#125—1493no—
200#125C2×D14913yes—
201#125C2×D14913yes—
202C22⋊Q8—1493no—
203C22⋊Q8—1493no—
204C22⋊Q8—1491no—
205C22⋊Q8—1496no—
206C22⋊Q8—1493no—
207C4×Q8—1491no—
208C4×Q8—1491no—
209C4×Q8—1491no—
210C4×Q8—1493no—
211C4×Q8—1493no—
212C4×Q8—1493no—
213C4×Q8—1496no—
214C4×Q8—1496no—
215C4×Q8—1493no—
216C4×Q8—1493no—
217C4×Q8—1493no—
218C4×Q8—1493no—
219C4×Q8—1493no—
220C4×Q8—1493no—
221C4×Q8C2×D14911yescharacteristic
222C4×Q8C2×D14911yescharacteristic
223C4×Q8C2×D14911yescharacteristic
224C4×Q8C2×D14913yes—
225#117—1493no—
226#117—1493no—
227#117—1493no—
228#117—1493no—
229C2×C4⋊C4—1493no—
230C2×C42—1493no—
231C2×Q8—1498no—
232C2×Q8—1493no—
233C2×Q8—1491no—
234C2×Q8—1493no—
235C2×Q8—1496no—
236C2×Q8—1493no—
237C2×Q8C22×D14913yes—
238C2×Q8C22×D14911yescharacteristic
239C22×C4—1491no—
240C22×C4—1493no—
241C22×C4—1493no—
242C4⋊C4—1496no—
243C4⋊C4—1493no—
244C4⋊C4—1493no—
245C4⋊C4—1493no—
246C4⋊C4—1493no—
247C4⋊C4—1493no—
248C4⋊C4—1493no—
249C4⋊C4—1493no—
250C4⋊C4—1493no—
251C4⋊C4—1491no—
252C4⋊C4—1493no—
253C4⋊C4—1491no—
254C4⋊C4—1496no—
255C4⋊C4—1493no—
256C4⋊C4—1496no—
257C4⋊C4—1493no—
258C4⋊C4—1493no—
259C4⋊C4—1493no—
260C4⋊C4—1493no—
261C4⋊C4C22×D14913yes—
262C4⋊C4C22×D14913yes—
263C4⋊C4C22×D14913yes—
264C4⋊C4C22×D14913yes—
265C4⋊C4C22×D14916yes—
266C4⋊C4C22×D14911yescharacteristic
267C4⋊C4C22×D14913yes—
268C22⋊C4—1493no—
269C22⋊C4—1491no—
270C22⋊C4—1493no—
271C22⋊C4—1491no—
272C42—1493no—
273C42—1493no—
274C42—1493no—
275C42—1493no—
276C42—1493no—
277C42—1496no—
278C42—1496no—
279C42C22×D14913yes—
280C42C22×D14913yes—
281C42C22×D14913yes—
282C23—1491no—
283Q8—43no—
284Q8—2986no—
285Q8—2982no—
286Q8—5963no—
287Q8—5966no—
288Q8—14912no—
289Q8Q8×D14914yes—
290C2×C4—2982no—
291C2×C4—5963no—
292C2×C4—1491no—
293C2×C4—1493no—
294C2×C4—1491no—
295C2×C4—1493no—
296C2×C4—1493no—
297C2×C4—1493no—
298C2×C4—1491no—
299C2×C4—14912no—
300C2×C4C23×D14911yescharacteristic
301C2×C4C23×D14913yes—
302C2×C4C23×D14913yes—
303C2×C4C23×D14913yes—
304C2×C4C23×D14911yescharacteristic
305C2×C4C23×D14911yescharacteristic
306C2×C4C23×D14913yes—
307C22—2981no—
308C22—2981no—
309C22—1492no—
310C22C24×D14911yescharacteristic
311C4—21no—
312C4—23no—
313C4—21no—
314C4—23no—
315C4—21no—
316C4—23no—
317C4—2981no—
318C4—2983no—
319C4—2983no—
320C4—2981no—
321C4—2983no—
322C4—1496no—
323C4—1492no—
324C4C2×Q8×D14916yes—
325C2—2982no—
326C2C22×Q8×D14911yescharacteristic
327C22−1+4×D14911yescharacteristic
328C2#2375046111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#557×C149C2
prime radicalC22×D149C24
hypercenter#557D149
hypocenterC149#3502
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. Q8×Dic298C2semidirect
2. #23750355C2semidirect
3. #23750353C2semidirect
4. #23750352C2semidirect
5. #23750352C2semidirect
6. #23750351C2semidirect
7. #23750351C2semidirect
15. #557×C149C2semidirect
71. Q8×D149Q8semidirect
164. C149#3502semidirect
173. #557D149semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2982
#237503552
#237503532
#237503522
#237503512
#237502552
#237502522
#237501812
#237501792
#237501012
C4×Q8×D1492
#557×C1492
#3502149