Finite Group Databasebeta

← #29590365

Subgroups — #29590365

Order 15616.

436 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#21451736C211yesmaximal, characteristic
2#21444140C211yesmaximal, characteristic
3#21443839C212yesmaximal, complemented
4#21443818C212yesmaximal, complemented
5C4×#99×C61C211yesmaximal, characteristic
6C2×#14519756C411yescharacteristic, complemented
7#16839815—21no—
8#16839692C411yescharacteristic, complemented
9#16839690—21no—
10#16839687C2211yescharacteristic
11#16839682C411yescharacteristic, complemented
12#16839676—21no—
13#16839674C2211yescharacteristic
14#16839646—21no—
15#16839635—21no—
16#16839625C411yescharacteristic, complemented
17#16839486—21no—
18#16839486C2212yes—
19C2×C4×D4×C61C2211yescharacteristic
20C2×#99×C61—21no—
21C4×C22⋊C4×C61—21no—
22C61×#342C2212yes—
23C22×Dic122C2×C411yescharacteristic
24#14519756—41no—
25#14519756—41no—
26C2×C4⋊Dic61C2×C411yescharacteristic
27C2×#11758196—21no—
28C2×#11758196—21no—
29C2×#11758196—21no—
30C2×C22⋊Dic61—21no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C4⋊#92418—41no—
35C22⋊#92418—22no—
36C4×#92418—41no—
37#14519684—21no—
38#14519684—21no—
39#14519684—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684C2×C411yescharacteristic
43#14519684C2×C411yescharacteristic
44C22×D4×C61D411yescharacteristic
45C23×C4×C61C2×C411yescharacteristic
46C23×C4×C61D411yescharacteristic
47C4×D4×C61—41no—
48C4×D4×C61—41no—
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61—22no—
51C2×C22⋊C4×C61D411yescharacteristic
52C2×C22⋊C4×C61C2311yescharacteristic
53C2×C42×C61—41no—
54C2×C42×C61D411yescharacteristic
55#99×C61—24no—
56#95×C61—41no—
57C24×C61C22⋊C411yescharacteristic
58C24×C61C2×D411yescharacteristic
59C2×D4×C61—22no—
60C2×D4×C61—22no—
61C2×D4×C61—41no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—42no—
67C22×C4×C61—81no—
68C22×C4×C61—81no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C2×Dic122—41no—
79C2×Dic122—81no—
80C4⋊C4×C61—41no—
81C22⋊C4×C61—24no—
82C22⋊C4×C61—41no—
83C22⋊C4×C61—41no—
84C22⋊C4×C61—42no—
85C42×C61—21no—
86C42×C61—21no—
87C42×C61—82no—
88C22⋊Dic61—42no—
89C22⋊Dic61—42no—
90C22⋊Dic61—81no—
91C22⋊Dic61—81no—
92C4⋊Dic61—41no—
93#11758196—41no—
94#11758196—41no—
95#11758196—42no—
96#11758196—81no—
97C2×#92418—42no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61C2×C22⋊C411yescharacteristic
108C23×C61#13512yes—
109D4×C61—44no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—41no—
116C2×C4×C61—42no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—44no—
121C2×C4×C61—81no—
122C2×C4×C61—81no—
123C2×C4×C61—81no—
124C2×C4×C61—82no—
125C2×C4×C61—82no—
126C2×C4×C61#9911yescharacteristic
127C2×C4×C61#9911yescharacteristic
128C2×C4×C61#10412yes—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141Dic122—82no—
142#92418—82no—
143#11731—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—22no—
147C22×C61—42no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61—41no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—81no—
154C22×C61—81no—
155C22×C61—81no—
156C22×C61C2×#9911yescharacteristic
157C22×C61#41611yescharacteristic
158C22×C61C2×#10411yescharacteristic
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—22no—
162C4×C61—41no—
163C4×C61—41no—
164C4×C61—41no—
165C4×C61—84no—
166Dic61—81no—
167Dic61—81no—
168Dic61—81no—
169Dic61—81no—
170#2638—611no—
171C4×#99D6111yescharacteristic
172#1460—611no—
173#1351—612no—
174#1344—612no—
175C2×C61—21no—
176C2×C61—21no—
177C2×C61—42no—
178C2×C61—81no—
179C2×C61#150111yescharacteristic
180C2×C61#212111yescharacteristic
181C2×C61#212911yescharacteristic
182C2×C22⋊Q8—611no—
183C2×C4×D4C2×D6111yescharacteristic
184#431—1221no—
185C2×#99—21no—
186#398—611no—
187#397—611no—
188#397—1221no—
189#395—611no—
190#394—1221no—
191#393—611no—
192#387—1221no—
193#385—1221no—
194#381—611no—
195C4×C22⋊C4—21no—
196#342C2×D6112yes—
197#323—612no—
198#323—1221no—
199C61#1173111yescharacteristic, complemented, Sylow
200C22×Q8—611no—
201C22×D4C22⋊D6111yescharacteristic
202C23×C4C4×D6111yescharacteristic
203C23×C4C22⋊D6111yescharacteristic
204C22⋊Q8—2441no—
205C22⋊Q8—2441no—
206C4×D4—41no—
207C4×D4—41no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4C4×D6111yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—1221no—
215C2×C22⋊C4—1221no—
216C2×C22⋊C4—1221no—
217C2×C22⋊C4—1221no—
218C2×C22⋊C4D24411yescharacteristic
219C2×C22⋊C4C22×D6111yescharacteristic
220C2×C42—41no—
221C2×C42D24411yescharacteristic
222C4⋊C8—2441no—
223#99—24no—
224C22⋊C8—1222no—
225C4×C8—2441no—
226#95—41no—
227#95—611no—
228#95—611no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234C24#1175819811yescharacteristic
235C24C2×C22⋊D6111yescharacteristic
236C2×Q8—4881no—
237C2×Q8—2441no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4—1221no—
252C22×C4#1175819811yescharacteristic
253C22×C4#1175819811yescharacteristic
254C22×C4#1175819811yescharacteristic
255C22×C4C2×C4×D6111yescharacteristic
256C22×C4C2×D24411yescharacteristic
257C2×C8—2442no—
258C4⋊C4—41no—
259C4⋊C4—4881no—
260C4⋊C4—2441no—
261C4⋊C4—2441no—
262C4⋊C4—2441no—
263C4⋊C4—2442no—
264C22⋊C4—24no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—4881no—
269C22⋊C4—4881no—
270C22⋊C4—2442no—
271C22⋊C4—2442no—
272C42—21no—
273C42—21no—
274C42—82no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×#1175819811yescharacteristic
285C23#1451979712yes—
286Q8—4882no—
287D4—44no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—44no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4#1451969111yescharacteristic
317C2×C4#1451969111yescharacteristic
318C2×C4#1451970512yes—
319C8—4882no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22C2×#1451969111yescharacteristic
333C22#1683975611yescharacteristic
334C22C2×#1451970511yescharacteristic
335C4—21no—
336C4—21no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—84no—
342C4—4881no—
343C4—4881no—
344C4—4881no—
345C4—4881no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—81no—
350C2#2144430811yescharacteristic
351C2#2144746311yescharacteristic
352C2#2144749311yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC4×#99×C61C2
socleC22×C61C2×#99
prime radicalC22×D4×C61D4
cosocleC2×C22⋊C4×C61C23
hypercenterC4×#99D61
hypocenterC61#11731
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #21443839C2semidirect
4. #21443818C2semidirect
6. C2×#14519756C4semidirect
8. #16839692C4semidirect
11. #16839682C4semidirect
16. #16839625C4semidirect
199. C61#11731semidirect

Maximal subgroups

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

subgroupindex
#214517362
#214441402
#214438392
#214438182
C4×#99×C612
#1173161