Finite Group Databasebeta

← #34058778

Subgroups — #34058778

Order 19072.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750351C211yesmaximal, characteristic, complemented
2#23750350C213yesmaximal, complemented
3#23750346C213yesmaximal, complemented
4#23750317C213yesmaximal, complemented
5#23750315C213yesmaximal, complemented
6Q8×C22⋊D149C211yesmaximal, characteristic, complemented
7Q8×C22⋊D149C211yesmaximal, characteristic, complemented
8#23750236C213yesmaximal, complemented
9#23750235C213yesmaximal, complemented
10#23750177C213yesmaximal, complemented
11#23750155C213yesmaximal
12#23750103C211yesmaximal, characteristic, complemented
13#23750101C211yesmaximal, characteristic, complemented
14C2×Q8×Dic149C211yesmaximal, characteristic, complemented
15D4×Q8×C149C211yesmaximal, characteristic, complemented
16C2×Q8×D149C2211yescharacteristic
17#18011910C2213yes—
18#18011909C2213yes—
19#18011909C2213yes—
20#18011906C2213yes—
21#18011906C2216yes—
22#18011904C2213yes—
23#18011904C2216yes—
24#18011903C2213yes—
25#18011902C2216yes—
26#18011902C2213yes—
27#18011901C2213yes—
28#18011901C2216yes—
29#18011901C2216yes—
30#18011897C2213yes—
31#18011895C2213yes—
32#18011892C2213yes—
33#18011890C2213yes—
34#18011890C2213yes—
35#18011888C2213yes—
36#18011887C2213yes—
37#18011887C2213yes—
38#18011883C2211yescharacteristic
39Q8×Dic149C2213yes—
40Q8×Dic149C2213yes—
41Q8×Dic149C2211yescharacteristic
42Q8×Dic149C2211yescharacteristic
43Q8×Dic149C2213yes—
44Q8×Dic149C2213yes—
45Q8×Dic149C2211yescharacteristic
46C4×Dic298C2213yes—
47C4×Dic298C2213yes—
48#18011876C2213yes—
49#18011876C2216yes—
50C4×C22⋊D149C2213yes—
51C4×C22⋊D149C2213yes—
52#18011870C2213yes—
53#18011870C2213yes—
54#18011869C2213yes—
55#18011868C2213yes—
56#18011867C2216yes—
57C2×C4⋊Dic149C2213yes—
58C2×C4×Dic149C2213yes—
59C22×Q8×C149C2211yescharacteristic
60C22×Q8×C149C2211yescharacteristic
61C4⋊Q8×C149C2213yes—
62C22⋊Q8×C149C2213yes—
63C22⋊Q8×C149C2213yes—
64C4×Q8×C149C2211yescharacteristic
65C4×D4×C149C2213yes—
66C2×Dic298C2313yes—
67C2×Dic298C2313yes—
68Q8×D149—24no—
69C2×C22⋊D149C2311yescharacteristic
70C2×C22⋊D149C2311yescharacteristic
71C22×Dic149C2311yescharacteristic
72C22×Dic149C2313yes—
73C2×C4×D149C2313yes—
74C4⋊Dic149C2316yes—
75C4⋊Dic149C2313yes—
76C4⋊Dic149C2313yes—
77C4⋊Dic149C2313yes—
78C4⋊Dic149C2313yes—
79C4⋊Dic149C2313yes—
80C4⋊Dic149C2313yes—
81#14558715C2316yes—
82#14558715C2313yes—
83#14558715C2316yes—
84#14558715C2313yes—
85#14558715C2313yes—
86#14558715C2313yes—
87#14558715C2316yes—
88#14558715C2313yes—
89#14558715C2311yescharacteristic
90C22⋊Dic149C2311yescharacteristic
91C22⋊Dic149C2313yes—
92C22⋊Dic149C2313yes—
93C22⋊Dic149C2313yes—
94#14558713C2311yescharacteristic
95#14558713C2313yes—
96#14558713C2313yes—
97#14558713C2313yes—
98C4×Dic149C2313yes—
99C4×Dic149C2316yes—
100C4×Dic149C2316yes—
101C4×Dic149C2313yes—
102C4×Dic149C2313yes—
103C4×Dic149C2313yes—
104C4×Dic149C2313yes—
105C2×Q8×C149—24no—
106C2×Q8×C149—24no—
107C2×Q8×C149C2313yes—
108C2×Q8×C149C2313yes—
109C2×Q8×C149C2311yescharacteristic
110C2×D4×C149C2311yescharacteristic
111C22×C4×C149C2313yes—
112C22×C4×C149C2313yes—
113C4⋊C4×C149C2313yes—
114C4⋊C4×C149C2316yes—
115C4⋊C4×C149C2313yes—
116C22⋊C4×C149C2313yes—
117C22⋊C4×C149C2313yes—
118C42×C149C2313yes—
119C22×D149C2411yescharacteristic
120Dic298—26no—
121Dic298—26no—
122C22⋊D149—22no—
123C22⋊D149—22no—
124C2×Dic149—26no—
125C2×Dic149—22no—
126C2×Dic149C2413yes—
127C2×Dic149C2411yescharacteristic
128C2×Dic149C2413yes—
129C2×Dic149C2413yes—
130C2×Dic149C2411yescharacteristic
131C2×Dic149C2413yes—
132C2×Dic149C2411yescharacteristic
133C4×D149—26no—
134C23×C149C2411yescharacteristic
135C23×C149C2411yescharacteristic
136Q8×C149—26no—
137Q8×C149—26no—
138Q8×C149#4114yes—
139D4×C149—22no—
140C2×C4×C149—26no—
141C2×C4×C149—26no—
142C2×C4×C149C2413yes—
143C2×C4×C149C2411yescharacteristic
144C2×C4×C149C2413yes—
145C2×C4×C149C2413yes—
146C2×C4×C149C2413yes—
147C22×C149—21no—
148C22×C149—21no—
149C22×C149—21no—
150C22×C149—21no—
151C22×C149—21no—
152C22×C1492−1+412yes—
153C22×C149C2511yescharacteristic
154C2×D149—21no—
155C2×D149—21no—
156C2×D149—21no—
157C4×C149—23no—
158C4×C149—23no—
159C4×C149—21no—
160C4×C149—23no—
161C4×C149C2×#4116yes—
162Dic149—23no—
163Dic149—21no—
164Dic149—23no—
165Dic149—23no—
166Dic149—21no—
167Dic149—23no—
168Dic1492−1+412yes—
169C2×C149—22no—
170C2×C149—41no—
171C2×C149C22×#4111yescharacteristic
172C2×C149C2×2−1+411yescharacteristic
173C2×C149C2×2−1+411yescharacteristic
174D149—41no—
175C149#352311yescharacteristic, complemented, Sylow
176#3523—1491nomaximal, Sylow
177#557—1493no—
178#557—1491no—
179#556—1493no—
180#552—1493no—
181#552—1493no—
182D4×Q8—1491no—
183D4×Q8—1491no—
184D4×Q8D14911yescharacteristic, complemented
185#541—1493no—
186#541—1493no—
187#533—1493no—
188#529—1493no—
189#519—1491no—
190#519—1491no—
191C2×C4×Q8—1491no—
192C22×Q8—1491no—
193C22×Q8C2×D14911yescharacteristic
194C22×Q8C2×D14911yescharacteristic
195C4⋊Q8—1493no—
196C4⋊Q8—1493no—
197C4⋊Q8—1493no—
198C4⋊Q8C2×D14913yes—
199#126—1493no—
200#126—1496no—
201#126—1493no—
202#126—1496no—
203#125—1493no—
204#125—1496no—
205#125—1496no—
206#125—1496no—
207#125—1493no—
208#125—1493no—
209#124—1493no—
210#123—1493no—
211#123—1493no—
212C22⋊Q8—1493no—
213C22⋊Q8—1493no—
214C22⋊Q8—1493no—
215C22⋊Q8—1493no—
216C22⋊Q8—1493no—
217C22⋊Q8C2×D14913yes—
218C22⋊Q8C2×D14913yes—
219#121—1491no—
220C4×Q8—1491no—
221C4×Q8—1493no—
222C4×Q8—1491no—
223C4×Q8—1493no—
224C4×Q8—1493no—
225C4×Q8—1491no—
226C4×Q8—1493no—
227C4×Q8—1493no—
228C4×Q8—1496no—
229C4×Q8—1493no—
230C4×Q8—1493no—
231C4×Q8C2×D14911yescharacteristic
232C4×D4—1493no—
233C4×D4—1493no—
234C4×D4C2×D14913yes—
235#117—1493no—
236#117—1493no—
237#117—1493no—
238#117—1496no—
239#117—1493no—
240C2×C4⋊C4—1493no—
241C2×C42—1493no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—2984no—
245C2×Q8—1493no—
246C2×Q8—1493no—
247C2×Q8C22×D14913yes—
248C2×Q8C22×D14913yes—
249C2×Q8C22×D14911yescharacteristic
250C2×D4—1491no—
251C2×D4—1491no—
252C2×D4C22×D14911yescharacteristic
253C22×C4—1491no—
254C22×C4—1493no—
255C22×C4—1493no—
256C22×C4C22×D14913yes—
257C22×C4C22×D14913yes—
258C4⋊C4—1493no—
259C4⋊C4—1493no—
260C4⋊C4—1496no—
261C4⋊C4—1493no—
262C4⋊C4—1493no—
263C4⋊C4—1493no—
264C4⋊C4—1493no—
265C4⋊C4—1493no—
266C4⋊C4—1496no—
267C4⋊C4—1496no—
268C4⋊C4—1493no—
269C4⋊C4—1493no—
270C4⋊C4—1491no—
271C4⋊C4—1496no—
272C4⋊C4—1493no—
273C4⋊C4—1493no—
274C4⋊C4C22×D14913yes—
275C4⋊C4C22×D14913yes—
276C4⋊C4C22×D14916yes—
277C22⋊C4—1491no—
278C22⋊C4—1493no—
279C22⋊C4—1493no—
280C22⋊C4—1493no—
281C22⋊C4—1491no—
282C22⋊C4—1493no—
283C22⋊C4—1493no—
284C22⋊C4—1493no—
285C22⋊C4C22×D14913yes—
286C22⋊C4C22×D14913yes—
287C42—1493no—
288C42—1493no—
289C42—1493no—
290C42—1493no—
291C42—1496no—
292C42—1496no—
293C42—1493no—
294C42C22×D14913yes—
295C23—1491no—
296C23C23×D14911yescharacteristic
297C23C23×D14911yescharacteristic
298Q8—26no—
299Q8—26no—
300Q8—2986no—
301Q8—2986no—
302Q8#1455873514yes—
303D4—22no—
304D4—2982no—
305D4—2982no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—2986no—
309C2×C4—2982no—
310C2×C4—2986no—
311C2×C4—1491no—
312C2×C4—1493no—
313C2×C4—1493no—
314C2×C4—1491no—
315C2×C4—1493no—
316C2×C4—1493no—
317C2×C4—1491no—
318C2×C4C23×D14913yes—
319C2×C4C23×D14913yes—
320C2×C4C23×D14913yes—
321C2×C4C23×D14911yescharacteristic
322C2×C4C23×D14913yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—2981no—
329C22—2981no—
330C22—2981no—
331C22#1801195912yes—
332C22C24×D14911yescharacteristic
333C4—23no—
334C4—23no—
335C4—23no—
336C4—21no—
337C4—2983no—
338C4—2981no—
339C4—2983no—
340C4—2983no—
341C4—2983no—
342C4—2981no—
343C4—1492no—
344C4C2×#1455873516yes—
345C2—22no—
346C2—41no—
347C2—5961no—
348C2C22×#1455873511yescharacteristic
349C22−1+4×D14911yescharacteristic
350C2C2×#1801195911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
FittingD4×Q8×C149C2
prime radical#18011883C22
hypercenterD4×Q8D149
hypocenterC149#3523
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #23750351C2semidirect
2. #23750350C2semidirect
3. #23750346C2semidirect
4. #23750317C2semidirect
5. #23750315C2semidirect
6. Q8×C22⋊D149C2semidirect
7. Q8×C22⋊D149C2semidirect
8. #23750236C2semidirect
9. #23750235C2semidirect
10. #23750177C2semidirect
12. #23750103C2semidirect
13. #23750101C2semidirect
14. C2×Q8×Dic149C2semidirect
15. D4×Q8×C149C2semidirect
175. C149#3523semidirect
184. D4×Q8D149semidirect

Maximal subgroups

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

subgroupindex
#237503512
#237503502
#237503462
#237503172
#237503152
Q8×C22⋊D1492
#237502362
#237502352
#237501772
#237501552
#237501032
#237501012
C2×Q8×Dic1492
D4×Q8×C1492
#3523149