Finite Group Databasebeta

← #29593227

Subgroups — #29593227

Order 15616.

348 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 63 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#21456612C211yesmaximal, characteristic
2#21448017C211yesmaximal, characteristic, complemented
3#21448000C211yesmaximal, characteristic, complemented
4#21447242C211yesmaximal, characteristic, complemented
5#21443832C211yesmaximal, characteristic, complemented
6#21443829C211yesmaximal, characteristic, complemented
7C61×#1314C211yesmaximal, characteristic, complemented
8#16840273—21no—
9#16840104—21no—
10#16840087—21no—
11#16840003C2211yescharacteristic
12#16839998C2211yescharacteristic
13#16839991—21no—
14#16839926—21no—
15#16839909—21no—
16C2×#14519711C2211yescharacteristic
17#16839622—21no—
18#16839590—21no—
19#16839588—21no—
20#16839567—21no—
21#16839564—21no—
22#16839525—21no—
23#16839525C2211yescharacteristic
24C2×C22⋊C8×C61C2211yescharacteristic
25C61×C22⋊C16C2211yescharacteristic
26C61×C22⋊C16C2211yescharacteristic
27C2×#11758222—21no—
28C2×Dic244D411yescharacteristic
29C2×#11758190—21no—
30C2×D488D411yescharacteristic
31#14519760—41no—
32#14519756—21no—
33#14519756—21no—
34#14519751—21no—
35#14519751—21no—
36#14519746—41no—
37C4×C22⋊D61—41no—
38C4×C22⋊D61—41no—
39#14519736—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C22⋊Dic61—41no—
43#14519711—21no—
44#14519711—21no—
45#14519711D411yescharacteristic
46#14519711D411yescharacteristic
47#14519709—21no—
48#14519702—21no—
49#14519702—21no—
50#14519699—21no—
51#14519699—21no—
52#14519684—41no—
53C23×C4×C61D411yescharacteristic
54C22×C8×C61D411yescharacteristic
55C22×C8×C61C2311yescharacteristic
56C2×C16×C61—21no—
57C2×C16×C61—21no—
58C22⋊C8×C61—21no—
59C22⋊C8×C61—21no—
60C24×C61D811yescharacteristic
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61D811yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C2×C22⋊D61—41no—
68C2×C22⋊D61—41no—
69C2×C22⋊D61—41no—
70C22×Dic61—41no—
71C22×Dic61—41no—
72#11758222—81no—
73#11758222—81no—
74C2×D244—21no—
75C2×C4×D61—41no—
76C2×Dic122—21no—
77C2×C8×C61—21no—
78C2×C8×C61—41no—
79C2×C8×C61—41no—
80C2×C8×C61C2×D411yescharacteristic
81C2×C8×C61C2×D411yescharacteristic
82C22⋊Dic61—41no—
83C22⋊Dic61—41no—
84C22⋊Dic61—41no—
85C22⋊Dic61—82no—
86#11758198—41no—
87#11758198—41no—
88C4⋊Dic61—21no—
89C4⋊Dic61—21no—
90C4⋊Dic61—41no—
91C4⋊Dic61—41no—
92C4⋊Dic61—41no—
93C4⋊Dic61—41no—
94#11758196—41no—
95#11758196—41no—
96C4×Dic61—41no—
97Dic244—41no—
98D488—41no—
99#11758190—81no—
100C16×C61—41no—
101C16×C61—41no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61C2×D811yescharacteristic
108C22×D61—41no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—81no—
116C2×C4×C61#12011yescharacteristic
117C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
118C22⋊D61—82no—
119C22⋊D61—82no—
120C22⋊D61—81no—
121C22⋊D61—81no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—81no—
126C2×Dic61—81no—
127C2×Dic61—81no—
128C2×Dic61—81no—
129C2×Dic61—81no—
130C2×Dic61—81no—
131D244—41no—
132D244—41no—
133C4×D61—82no—
134Dic122—41no—
135Dic122—41no—
136C8×C61—21no—
137C8×C61—21no—
138C8×C61—41no—
139#12280—611nomaximal, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—42no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61—41no—
146C22×C61—81no—
147C22×C61#44711yescharacteristic
148C22×C61#50811yescharacteristic
149C22×C61#50911yescharacteristic
150C2×D61—81no—
151C2×D61—81no—
152C2×D61—81no—
153C4×C61—21no—
154C4×C61—41no—
155C4×C61#45511yescharacteristic
156C4×C61#45811yescharacteristic
157Dic61—81no—
158Dic61—81no—
159Dic61—81no—
160Dic61—161no—
161Dic61—161no—
162#3191—611no—
163#2235—611no—
164#2232—611no—
165#2075—611no—
166#1348—611no—
167#1347—611no—
168#1314D6111yescharacteristic, complemented
169C2×C61—21no—
170C2×C61—21no—
171C2×C61—81no—
172C2×C61#160711yescharacteristic
173C2×C61#218611yescharacteristic
174C2×C61#219111yescharacteristic
175D61—161no—
176#525—1221no—
177#485—1221no—
178#482—1221no—
179#467—611no—
180#466—611no—
181#465—1221no—
182#452—1221no—
183#449—1221no—
184C2×#107—611no—
185C2×C22⋊C8C2×D6111yescharacteristic
186#380—1221no—
187#367—1221no—
188#366—1221no—
189#358—1221no—
190#357—1221no—
191C22⋊C16C2×D6111yescharacteristic
192C22⋊C16C2×D6111yescharacteristic
193#340—611no—
194#340—1221no—
195C61#1228011yescharacteristic, complemented, Sylow
196C2×#41—1221no—
197C23×C4D24411yescharacteristic
198C2×Q16—611no—
199C2×SD16—1221no—
200C2×D8—611no—
201C22×C8D24411yescharacteristic
202C22×C8C22×D6111yescharacteristic
203#123—2441no—
204C22⋊Q8—1221no—
205C22⋊Q8—1221no—
206#121—1221no—
207#121—1221no—
208#120—2441no—
209C4×D4—2441no—
210C4×D4—2441no—
211#117—1221no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4—1221no—
214C2×C22⋊C4—2441no—
215C2×C16—21no—
216C2×C16—21no—
217#107—611no—
218#107—611no—
219#107—1221no—
220#107—1221no—
221#106—1221no—
222#103—1221no—
223#103—1221no—
224#102—1221no—
225#102—1221no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228#95—2441no—
229C24D48811yescharacteristic
230#41—4881no—
231#41—4881no—
232C2×Q8—1221no—
233C2×D4—1221no—
234C2×D4—2441no—
235C2×D4—2441no—
236C2×D4—2441no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—2441no—
242C22×C4—2441no—
243C22×C4—2441no—
244C22×C4D48811yescharacteristic
245C22×C4C2×D24411yescharacteristic
246Q16—2441no—
247SD16—4881no—
248D8—2441no—
249C2×C8—21no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8D4×D6111yescharacteristic
253C2×C8D4×D6111yescharacteristic
254C4⋊C4—1221no—
255C4⋊C4—1221no—
256C4⋊C4—2441no—
257C4⋊C4—2441no—
258C4⋊C4—2441no—
259C4⋊C4—2441no—
260C4⋊C4—2441no—
261C4⋊C4—2441no—
262C22⋊C4—4882no—
263C22⋊C4—2441no—
264C22⋊C4—2441no—
265C22⋊C4—2441no—
266C22⋊C4—2441no—
267C22⋊C4—2441no—
268C42—2441no—
269C16—41no—
270C16—41no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23—2441no—
277C23C2×D48811yescharacteristic
278Q8—2441no—
279Q8—2441no—
280D4—4882no—
281D4—4882no—
282D4—4881no—
283D4—4881no—
284D4—2441no—
285D4—2441no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—4882no—
294C2×C4—4881no—
295C2×C4—4881no—
296C2×C4—4881no—
297C2×C4—4881no—
298C2×C4—4881no—
299C2×C4—4881no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4#1451974711yescharacteristic
304C2×C4#1451980511yescharacteristic
305C8—21no—
306C8—21no—
307C8—41no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—81no—
315C22—4881no—
316C22—4881no—
317C22—4881no—
318C22#1683989811yescharacteristic
319C22#1684019411yescharacteristic
320C22#1684020011yescharacteristic
321C4—21no—
322C4—41no—
323C4—9761no—
324C4—9761no—
325C4—4881no—
326C4—4881no—
327C4—4881no—
328C4#1683994211yescharacteristic
329C4#1683995311yescharacteristic
330C2—21no—
331C2—21no—
332C2—81no—
333C2—9761no—
334C2#2144480711yescharacteristic
335C2#2144774411yescharacteristic
336C2#2144777211yescharacteristic
subgroupG/H
centerC22#16839898
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#1314C2
socleC22×C61#447
prime radical#21456612C2
hypercenter#1314D61
hypocenterC61#12280
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #21448017C2semidirect
3. #21448000C2semidirect
4. #21447242C2semidirect
5. #21443832C2semidirect
6. #21443829C2semidirect
7. C61×#1314C2semidirect
168. #1314D61semidirect
195. C61#12280semidirect

Maximal subgroups

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

subgroupindex
#214566122
#214480172
#214480002
#214472422
#214438322
#214438292
C61×#13142
#1228061