Finite Group Databasebeta

← #34057630

Subgroups — #34057630

Order 19072.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750335C211yesmaximal, characteristic
2#23750035C211yesmaximal, characteristic, complemented
3#23750031C211yesmaximal, characteristic, complemented
4#23750024C211yesmaximal, characteristic, complemented
5#23749932C211yesmaximal, characteristic, complemented
6#23749932C211yesmaximal, characteristic, complemented
7#23749922C211yesmaximal, characteristic
8#23749922C211yesmaximal, characteristic
9#23749912C211yesmaximal, characteristic
10#23749910C211yesmaximal, characteristic, complemented
11#23749756C211yesmaximal, characteristic, complemented
12#23749723C211yesmaximal, characteristic
13#23749723C211yesmaximal, characteristic
14C4×#14558723C211yesmaximal, characteristic
15#557×C149C211yesmaximal, characteristic, complemented
16C2×#14558723C2211yescharacteristic
17#18011911C2211yescharacteristic
18#18011903C2211yescharacteristic
19#18011903C2211yescharacteristic
20#18011889—21no—
21#18011889—21no—
22#18011887—21no—
23C4×D596C2211yescharacteristic
24C4×D596C2211yescharacteristic
25C4×D596C2211yescharacteristic
26C4⋊C4×D149—21no—
27#18011844C2211yescharacteristic
28#18011844C2211yescharacteristic
29#18011844C2211yescharacteristic
30#18011844C2211yescharacteristic
31#18011842C2211yescharacteristic
32#18011842C2211yescharacteristic
33#18011842C2211yescharacteristic
34#18011842C2211yescharacteristic
35#18011842C2211yescharacteristic
36C4⋊#11924746C2211yescharacteristic
37C4⋊#11924746C2211yescharacteristic
38C4⋊#11924746C2211yescharacteristic
39#18011835C2211yescharacteristic
40#18011835C2211yescharacteristic
41#18011835C2211yescharacteristic
42#18011831C2211yescharacteristic
43#18011831C2211yescharacteristic
44#18011831C2211yescharacteristic
45#18011822C2211yescharacteristic
46#18011822C2211yescharacteristic
47C4×#11924746C2211yescharacteristic
48C4⋊Q8×C149—21no—
49C4⋊Q8×C149C2211yescharacteristic
50#125×C149—21no—
51#125×C149—21no—
52#125×C149C2211yescharacteristic
53#125×C149C2211yescharacteristic
54C4×Q8×C149—21no—
55C4×Q8×C149C2211yescharacteristic
56C4×Q8×C149C2211yescharacteristic
57C4×Q8×C149C2211yescharacteristic
58C4×Q8×C149C2211yescharacteristic
59C2×Dic298—21no—
60C2×D596C2311yescharacteristic
61C2×C4×D149—21no—
62C2×C4×D149—21no—
63C2×C4×D149—21no—
64#14558723Q814yescomplemented
65C2×#11924746C2311yescharacteristic
66C2×#11924746C2311yescharacteristic
67C2×#11924746C2311yescharacteristic
68C2×#11924746C2311yescharacteristic
69C4⋊Dic149—21no—
70C4⋊Dic149C2311yescharacteristic
71C4⋊Dic149C2311yescharacteristic
72C4⋊Dic149C2311yescharacteristic
73#14558715—21no—
74#14558715—21no—
75#14558715—21no—
76#14558715—21no—
77#14558713—21no—
78#14558713—21no—
79#14558713—21no—
80C2×Q8×C149—21no—
81C2×Q8×C149D411yescharacteristic
82C2×Q8×C149C2311yescharacteristic
83C4⋊C4×C149—21no—
84C4⋊C4×C149—21no—
85C4⋊C4×C149—21no—
86C4⋊C4×C149—21no—
87C4⋊C4×C149—21no—
88C4⋊C4×C149—21no—
89C4⋊C4×C149—21no—
90C4⋊C4×C149—21no—
91C4⋊C4×C149D411yescharacteristic
92C4⋊C4×C149D411yescharacteristic
93C4⋊C4×C149D411yescharacteristic
94C4⋊C4×C149C2311yescharacteristic
95C4⋊C4×C149C2311yescharacteristic
96C4⋊C4×C149C2311yescharacteristic
97C42×C149—21no—
98C42×C149—21no—
99C42×C149—21no—
100C42×C149C2311yescharacteristic
101C42×C149C2311yescharacteristic
102C42×C149C2311yescharacteristic
103C22×D149—21no—
104Dic298—81no—
105D596—21no—
106D596C2×Q812yes—
107C2×Dic149—21no—
108C2×Dic149—21no—
109C2×Dic149—21no—
110C4×D149—42no—
111C4×D149—81no—
112C4×D149—81no—
113#11924746—21no—
114#11924746—21no—
115#11924746—21no—
116#11924746C2×Q812yes—
117Q8×C149—21no—
118Q8×C149—41no—
119Q8×C149—42no—
120Q8×C149C2×Q812yes—
121C2×C4×C149—21no—
122C2×C4×C149—21no—
123C2×C4×C149—21no—
124C2×C4×C149—21no—
125C2×C4×C149C2×D411yescharacteristic
126C2×C4×C149C2×D411yescharacteristic
127C2×C4×C149C2×D411yescharacteristic
128C2×C4×C149C2×D411yescharacteristic
129C2×C4×C149C2×D411yescharacteristic
130C2×C4×C149C2×D411yescharacteristic
131C2×C4×C149C2411yescharacteristic
132C22×C149C22×D411yescharacteristic
133C2×D149—22no—
134C2×D149—41no—
135C2×D149—41no—
136C4×C149—22no—
137C4×C149—21no—
138C4×C149—21no—
139C4×C149—21no—
140C4×C149—21no—
141C4×C149—21no—
142C4×C149—21no—
143C4×C149—41no—
144C4×C149—41no—
145C4×C149—41no—
146C4×C149C22×Q811yescharacteristic
147C4×C1492−1+411yescharacteristic
148Dic149—41no—
149Dic149—41no—
150Dic149—41no—
151C2×C149D4×Q811yescharacteristic
152C2×C149#57511yescharacteristic
153C2×C149#57711yescharacteristic
154D149—42no—
155C149#338811yescharacteristic, complemented, Sylow
156#3388—1491nomaximal, Sylow
157#557D14911yescharacteristic, complemented
158#554—1491no—
159#501—1491no—
160#501—1491no—
161#499—1491no—
162#479—1491no—
163#479—1491no—
164#478—1491no—
165#478—1491no—
166#476—1491no—
167#475—1491no—
168#446—1491no—
169#440—1491no—
170#440—1491no—
171C4×SD16—1491no—
172C2×SD16—1491no—
173C4⋊Q8—21no—
174C4⋊Q8—1491no—
175C4⋊Q8C2×D14911yescharacteristic
176#125—21no—
177#125—21no—
178#125—1491no—
179#125—1491no—
180#125C2×D14911yescharacteristic
181#125C2×D14911yescharacteristic
182C22⋊Q8—2981no—
183C22⋊Q8—2981no—
184C22⋊Q8—2981no—
185C4×Q8—21no—
186C4×Q8C2×D14911yescharacteristic
187C4×Q8C2×D14911yescharacteristic
188C4×Q8C2×D14911yescharacteristic
189C4×Q8C2×D14911yescharacteristic
190C4×D4—1491no—
191C4×D4—1491no—
192C4×D4—1491no—
193C2×C4⋊C4—2981no—
194#107—1491no—
195#107—1491no—
196#107—1491no—
197#107—1491no—
198#106—1491no—
199#106—1491no—
200#106—1491no—
201#106—1491no—
202#106—1491no—
203C4⋊C8—1491no—
204C4⋊C8—1491no—
205C4⋊C8—1491no—
206#103—1491no—
207#103—1491no—
208#103—1491no—
209#102—1491no—
210#102—1491no—
211#102—1491no—
212#97—1491no—
213#97—1491no—
214C4×C8—1491no—
215C2×Q8—21no—
216C2×Q8—2981no—
217C2×Q8C22⋊D14911yescharacteristic
218C2×Q8C22×D14911yescharacteristic
219C2×D4—1491no—
220C22×C4—2981no—
221C22×C4—2981no—
222C22×C4—2981no—
223SD16—1494no—
224C2×C8—1491no—
225C2×C8—1491no—
226C2×C8—1491no—
227C2×C8—1491no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—2981no—
237C4⋊C4—2981no—
238C4⋊C4—2981no—
239C4⋊C4—2981no—
240C4⋊C4—2981no—
241C4⋊C4—1491no—
242C4⋊C4—1491no—
243C4⋊C4—1491no—
244C4⋊C4C22⋊D14911yescharacteristic
245C4⋊C4C22⋊D14911yescharacteristic
246C4⋊C4C22⋊D14911yescharacteristic
247C4⋊C4C22×D14911yescharacteristic
248C4⋊C4C22×D14911yescharacteristic
249C4⋊C4C22×D14911yescharacteristic
250C22⋊C4—2981no—
251C22⋊C4—2981no—
252C22⋊C4—2981no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42C22×D14911yescharacteristic
257C42C22×D14911yescharacteristic
258C42C22×D14911yescharacteristic
259C23—2981no—
260Q8—21no—
261Q8—41no—
262Q8—42no—
263Q8—11921no—
264Q8Q8×D14912yes—
265D4—2981no—
266D4—1492no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—2981no—
272C2×C4—2981no—
273C2×C4—2981no—
274C2×C4—5962no—
275C2×C4—11921no—
276C2×C4—11921no—
277C2×C4C2×C22⋊D14911yescharacteristic
278C2×C4C2×C22⋊D14911yescharacteristic
279C2×C4C2×C22⋊D14911yescharacteristic
280C2×C4C2×C22⋊D14911yescharacteristic
281C2×C4C2×C22⋊D14911yescharacteristic
282C2×C4C2×C22⋊D14911yescharacteristic
283C2×C4C23×D14911yescharacteristic
284C8—2981no—
285C8—2981no—
286C8—2981no—
287C8—1492no—
288C22—2982no—
289C22—5961no—
290C22—5961no—
291C22C22×C22⋊D14911yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—22no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—5961no—
303C4—5961no—
304C4—5961no—
305C4C2×Q8×D14911yescharacteristic
306C4#1801195911yescharacteristic
307C2—5962no—
308C2Q8×C22⋊D14911yescharacteristic
309C2#2375042311yescharacteristic
310C2#2375043311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#557×C149C2
socleC22×C149C22×D4
prime radicalC2×D596C23
hypercenter#557D149
hypocenterC149#3388
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
2. #23750035C2semidirect
3. #23750031C2semidirect
4. #23750024C2semidirect
5. #23749932C2semidirect
6. #23749932C2semidirect
10. #23749910C2semidirect
11. #23749756C2semidirect
15. #557×C149C2semidirect
64. #14558723Q8semidirect
155. C149#3388semidirect
157. #557D149semidirect

Maximal subgroups

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

subgroupindex
#237503352
#237500352
#237500312
#237500242
#237499322
#237499222
#237499122
#237499102
#237497562
#237497232
C4×#145587232
#557×C1492
#3388149