Finite Group Databasebeta

← #34060531

Subgroups — #34060531

Order 19072.

612 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011974C212yesmaximal, complemented
2C2×#18011973C212yesmaximal, complemented
3D149⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
4#23750567C211yesmaximal, characteristic, complemented
5C2×#41×D149C211yesmaximal, characteristic
6C2×D149⋊C8C2212yes—
7C2×C4⋊#10597010C2212yes—
8C2×C22⋊#10597010—21no—
9#18011974—24no—
10#18011973—24no—
11#18011968—21no—
12C22⋊#11924753—21no—
13(C2×#10597010)⋊C4—21no—
14C2×#14558737C411yescharacteristic, complemented
15C2×#14558736—21no—
16C2×#14558736C411yescharacteristic, complemented
17C2×#14558735—21no—
18C2×#14558735C411yescharacteristic, complemented
19#41×D149—24no—
20#41×D149—24no—
21C2×Q8×D149C2211yescharacteristic
22C2×D4×D149—21no—
23C2×D4×D149C2211yescharacteristic
24C22×C4×D149—21no—
25C22×C4×D149C2211yescharacteristic
26C2×#41×C149C411yescharacteristic, complemented
27C22×#10597010—22no—
28D149⋊C8—22no—
29D149⋊C8—22no—
30C2×#11924753—22no—
31C4⋊#10597010—22no—
32C4⋊#10597010—22no—
33C22⋊#10597010—42no—
34C23×D149—21no—
35C23×D149D411yescharacteristic
36#14558737—22no—
37#14558737—22no—
38#14558736—28no—
39#14558736—22no—
40#14558736—22no—
41#14558735—28no—
42#14558735—22no—
43#14558735—22no—
44C2×Dic298—21no—
45C2×Dic298C2×C411yescharacteristic
46Q8×D149—22no—
47Q8×D149D414yes—
48C2×D596—21no—
49C2×D596C2×C411yescharacteristic
50C2×C22⋊D149—21no—
51C2×C22⋊D149—21no—
52C2×C22⋊D149—21no—
53D4×D149—28no—
54D4×D149—22no—
55D4×D149D414yes—
56C22×Dic149—21no—
57C22×Dic149D411yescharacteristic
58C2×C4×D149—28no—
59C2×C4×D149—21no—
60C2×C4×D149—21no—
61C2×C4×D149—22no—
62C2×C4×D149—22no—
63C2×C4×D149—21no—
64C2×C4×D149C2×C411yescharacteristic
65C2×C4×D149D411yescharacteristic
66C2×C4×D149D411yescharacteristic
67C2×C4×D149C2311yescharacteristic
68#41×C149—22no—
69#41×C149—22no—
70C2×Q8×C149C2×C411yescharacteristic
71C2×D4×C149—21no—
72C2×D4×C149C2×C411yescharacteristic
73C22×C4×C149—21no—
74C22×C4×C149C2×C411yescharacteristic
75C2×#10597010—42no—
76C2×#10597010—42no—
77C2×#10597010—42no—
78#11924753—42no—
79C22×D149—21no—
80C22×D149—22no—
81C22×D149—24no—
82C22×D149—22no—
83C22×D149—21no—
84C22×D149—21no—
85C22×D149—42no—
86C22×D149—41no—
87C22×D149—41no—
88C22×D149C22⋊C411yescharacteristic
89C22×D149C2×D411yescharacteristic
90Dic298—21no—
91Dic298—24no—
92Dic298C22⋊C412yescomplemented
93D596—24no—
94D596—21no—
95D596C22⋊C412yescomplemented
96C22⋊D149—24no—
97C22⋊D149—24no—
98C22⋊D149—24no—
99C2×Dic149—24no—
100C2×Dic149—21no—
101C2×Dic149—22no—
102C2×Dic149C22⋊C411yescharacteristic
103C2×Dic149C2×D411yescharacteristic
104C4×D149—24no—
105C4×D149—24no—
106C4×D149—22no—
107C4×D149—24no—
108C4×D149—21no—
109C4×D149—21no—
110C4×D149—22no—
111C4×D149C2×D412yes—
112C4×D149C2×D412yes—
113C23×C149—21no—
114C23×C149C22⋊C411yescharacteristic
115Q8×C149—21no—
116Q8×C149C22⋊C412yescomplemented
117D4×C149—24no—
118D4×C149—21no—
119D4×C149C22⋊C412yescomplemented
120C2×C4×C149—24no—
121C2×C4×C149—21no—
122C2×C4×C149—21no—
123C2×C4×C149—21no—
124C2×C4×C149C22⋊C411yescharacteristic
125C2×C4×C149C22×C411yescharacteristic
126C22×C149—21no—
127C22×C149—22no—
128C22×C149—21no—
129C22×C149—21no—
130C22×C149—41no—
131C22×C149—41no—
132C22×C149C2×C22⋊C411yescharacteristic
133C2×D149—21no—
134C2×D149—22no—
135C2×D149—21no—
136C2×D149—21no—
137C2×D149—41no—
138C2×D149—41no—
139C2×D149—42no—
140C2×D149—44no—
141C2×D149—42no—
142C2×D149—44no—
143C2×D149#12012yes—
144C2×D149#13512yes—
145C2×D149AGL(1,ℤ/8)11yescharacteristic
146C2×D149#13711yescharacteristic
147#10597010—82no—
148C4×C149—21no—
149C4×C149—22no—
150C4×C149C2×C22⋊C411yescharacteristic
151C4×C149C2×C22⋊C411yescharacteristic
152Dic149—21no—
153Dic149—22no—
154Dic149#12012yes—
155C2×C149—41no—
156C2×C149—42no—
157C2×C149#37911yescharacteristic
158C2×C149#41611yescharacteristic
159C2×C149#41711yescharacteristic
160D149—41no—
161D149—42no—
162D149#44912yes—
163D149#45212yes—
164C149#178811yescharacteristic, complemented, Sylow
165#1788—1491nomaximal, Sylow
166C22×#41—1491no—
167C2×#103—1492no—
168C2×#102—1492no—
169C2×C22⋊C8—1491no—
170#380—1491no—
171C2×#41—2981no—
172C2×#41—2984no—
173C2×#41—2984no—
174C2×#41—2981no—
175C2×#41—1491no—
176C2×#41—1491no—
177C2×#41—1491no—
178C2×#41#1059701011yescharacteristic, complemented
179C22×Q8—1491no—
180C22×D4—2981no—
181C22×D4—1491no—
182C23×C4—2981no—
183C23×C4—1491no—
184C22×C8—1492no—
185C2×C4⋊C4—1492no—
186C2×C22⋊C4—2981no—
187#103—2984no—
188#102—2984no—
189C22⋊C8—2981no—
190C22⋊C8—2981no—
191#95—2981no—
192C24—2981no—
193C24—1491no—
194#41—22no—
195#41—22no—
196#41—2988no—
197#41—2982no—
198#41—2982no—
199#41—2988no—
200#41—2982no—
201#41—2982no—
202#41—2982no—
203#41—2982no—
204C2×Q8—2981no—
205C2×Q8—2982no—
206C2×Q8—1494no—
207C2×Q8—1491no—
208C2×Q8C2×#1059701011yescharacteristic
209C2×D4—21no—
210C2×D4—2981no—
211C2×D4—2988no—
212C2×D4—2981no—
213C2×D4—2981no—
214C2×D4—2982no—
215C2×D4—2981no—
216C2×D4—1494no—
217C2×D4—1491no—
218C2×D4C2×#1059701011yescharacteristic
219C22×C4—21no—
220C22×C4—2988no—
221C22×C4—2981no—
222C22×C4—2981no—
223C22×C4—2982no—
224C22×C4—2982no—
225C22×C4—2981no—
226C22×C4—2981no—
227C22×C4—2982no—
228C22×C4—1491no—
229C22×C4—1491no—
230C22×C4—1491no—
231C22×C4—1491no—
232C22×C4—1491no—
233C22×C4C2×#1059701011yescharacteristic
234C2×C8—2982no—
235C2×C8—2982no—
236C2×C8—2982no—
237C4⋊C4—2982no—
238C4⋊C4—2982no—
239C22⋊C4—5962no—
240C23—21no—
241C23—2981no—
242C23—2982no—
243C23—2984no—
244C23—2982no—
245C23—2981no—
246C23—2981no—
247C23—5962no—
248C23—5961no—
249C23—5961no—
250C23—1491no—
251C23—1491no—
252C23C22⋊#1059701011yescharacteristic
253Q8—21no—
254Q8—2984no—
255Q8—2981no—
256Q8—1492no—
257Q8C22⋊#1059701012yescomplemented
258D4—24no—
259D4—21no—
260D4—2984no—
261D4—2984no—
262D4—2984no—
263D4—2984no—
264D4—2981no—
265D4—1492no—
266D4C22⋊#1059701012yescomplemented
267C2×C4—24no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—2984no—
272C2×C4—2984no—
273C2×C4—2982no—
274C2×C4—2984no—
275C2×C4—2982no—
276C2×C4—2984no—
277C2×C4—2981no—
278C2×C4—2982no—
279C2×C4—2981no—
280C2×C4—2981no—
281C2×C4—5962no—
282C2×C4—5962no—
283C2×C4—5962no—
284C2×C4—1491no—
285C2×C4—1491no—
286C2×C4—1492no—
287C2×C4—1492no—
288C2×C4C22⋊#1059701011yescharacteristic
289C2×C4C22×#1059701011yescharacteristic
290C8—5962no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—21no—
295C22—41no—
296C22—41no—
297C22—2981no—
298C22—2982no—
299C22—2981no—
300C22—2981no—
301C22—5962no—
302C22—5962no—
303C22—5964no—
304C22—5964no—
305C22—5961no—
306C22—5961no—
307C22—1492no—
308C22—1492no—
309C22—1491no—
310C22—1491no—
311C22C2×C22⋊#1059701011yescharacteristic
312C4—21no—
313C4—22no—
314C4—2981no—
315C4—2982no—
316C4—1492no—
317C4—11922no—
318C4C2×C22⋊#1059701011yescharacteristic
319C4C2×C22⋊#1059701011yescharacteristic
320C2—41no—
321C2—42no—
322C2—5961no—
323C2—5962no—
324C2—1492no—
325C2—1492no—
326C2#2375056611yescharacteristic
327C2#2375061511yescharacteristic
328C2#2375061611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10597010
commutatorC2×C4×C149C22×C4
FrattiniC2×C4C22×#10597010
FittingC2×#41×C149C4
socleC22×C149C2×C22⋊C4
prime radicalC2×#41×D149C2
cosocleC2×C4×D149C23
hypercenterC2×#41#10597010
hypocenterC149#1788
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#18011974C2semidirect
2. C2×#18011973C2semidirect
3. D149⋊(C22⋊C8)C2semidirect
4. #23750567C2semidirect
14. C2×#14558737C4semidirect
16. C2×#14558736C4semidirect
18. C2×#14558735C4semidirect
26. C2×#41×C149C4semidirect
92. Dic298C22⋊C4semidirect
95. D596C22⋊C4semidirect
116. Q8×C149C22⋊C4semidirect
119. D4×C149C22⋊C4semidirect
164. C149#1788semidirect
178. C2×#41#10597010semidirect
257. Q8C22⋊#10597010semidirect
266. D4C22⋊#10597010semidirect

Maximal subgroups

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

subgroupindex
C2×#180119742
C2×#180119732
D149⋊(C22⋊C8)2
#237505672
C2×#41×D1492
#1788149