Finite Group Databasebeta

← #29593239

Subgroups — #29593239

Order 15616.

348 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 61 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#21456606C211yesmaximal, characteristic
2#21448021C211yesmaximal, characteristic, complemented
3#21447248C211yesmaximal, characteristic, complemented
4#21443832C211yesmaximal, characteristic, complemented
5#21443828C211yesmaximal, characteristic, complemented
6#21443742C211yesmaximal, characteristic, complemented
7C61×#2233C211yesmaximal, characteristic, complemented
8#16840272—21no—
9#16840105—21no—
10#16840084—21no—
11#16840003C2211yescharacteristic
12#16839989—21no—
13#16839925—21no—
14#16839904—21no—
15C2×#14519689C2211yescharacteristic
16#16839619—21no—
17#16839587—21no—
18#16839565—21no—
19C22⋊#11758185C2211yescharacteristic
20#16839525C2211yescharacteristic
21#16839524—21no—
22C61×#466C2211yescharacteristic
23C2×#107×C61C2211yescharacteristic
24C61×#367—21no—
25C61×#357—21no—
26C61×C22⋊C16C2211yescharacteristic
27C2×#11758224—21no—
28C22×C4×D61D411yescharacteristic
29C2×Dic244D411yescharacteristic
30C2×#11758200—21no—
31C22×#92418D411yescharacteristic
32#14519759—41no—
33#14519757—21no—
34#14519756—21no—
35#14519748—21no—
36#14519745—41no—
37D4×Dic61—41no—
38#14519737—41no—
39#14519735—21no—
40C2×C4⋊Dic61—21no—
41C2×#11758198—41no—
42C2×#11758185—21no—
43#14519711D411yescharacteristic
44#14519708—21no—
45#14519702—21no—
46#14519700—21no—
47#14519698—21no—
48#14519689—21no—
49#14519689—21no—
50#14519684—41no—
51C2×D8×C61D411yescharacteristic
52C22×C8×C61C2311yescharacteristic
53#121×C61—21no—
54C2×C4⋊C4×C61—21no—
55C2×C16×C61—21no—
56#107×C61—21no—
57#107×C61—21no—
58#107×C61D411yescharacteristic
59#102×C61—21no—
60C23×D61D811yescharacteristic
61C2×D4×C61—21no—
62C2×D4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61C2×D411yescharacteristic
65C2×C22⋊D61—41no—
66C2×C22⋊D61—41no—
67C22×Dic61—41no—
68C22×Dic61—41no—
69C22×Dic61D811yescharacteristic
70#11758224—81no—
71#11758224—81no—
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—41no—
75C2×C4×D61—41no—
76C2×Dic122—21no—
77D8×C61—41no—
78C2×C8×C61—41no—
79C2×C8×C61C2×D411yescharacteristic
80C2×C8×C61C2×D411yescharacteristic
81C4⋊C4×C61—21no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—41no—
85C22⋊Dic61—41no—
86C22⋊Dic61—41no—
87#11758200—81no—
88#11758198—41no—
89#11758198—41no—
90#11758198—82no—
91C4⋊Dic61—21no—
92C4⋊Dic61—41no—
93C4⋊Dic61—41no—
94C4⋊Dic61—41no—
95#11758196—41no—
96C4×Dic61—41no—
97C2×#92418—21no—
98C2×#92418—41no—
99Dic244—41no—
100C16×C61—41no—
101#11758185—41no—
102C23×C61—41no—
103C23×C61C2×D811yescharacteristic
104C22×D61—21no—
105C22×D61—21no—
106C22×D61—21no—
107C22×D61—41no—
108C22×D61—41no—
109D4×C61—41no—
110D4×C61—41no—
111D4×C61—81no—
112D4×C61—81no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—41no—
116C2×C4×C61—81no—
117C2×C4×C61—81no—
118C2×C4×C61—81no—
119C2×C4×C61#12011yescharacteristic
120C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
121C22⋊D61—82no—
122C22⋊D61—82no—
123C2×Dic61—21no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—82no—
128C2×Dic61—81no—
129C2×Dic61—81no—
130C2×Dic61—81no—
131C4×D61—41no—
132C4×D61—41no—
133C4×D61—81no—
134Dic122—41no—
135Dic122—41no—
136C8×C61—21no—
137C8×C61—21no—
138#92418—41no—
139#12283—611nomaximal, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—81no—
143C22×C61—81no—
144C22×C61—81no—
145C22×C61#44711yescharacteristic
146C22×C61#50811yescharacteristic
147C22×C61#51011yescharacteristic
148C2×D61—42no—
149C2×D61—41no—
150C2×D61—41no—
151C2×D61—41no—
152C2×D61—81no—
153C4×C61—21no—
154C4×C61—81no—
155C4×C61—161no—
156C4×C61#45511yescharacteristic
157C4×C61#45811yescharacteristic
158Dic61—41no—
159Dic61—81no—
160Dic61—81no—
161Dic61—161no—
162#3191—611no—
163#2236—611no—
164#2233D6111yescharacteristic, complemented
165#2076—611no—
166#1348—611no—
167#1347—611no—
168#1314—611no—
169C2×C61—21no—
170C2×C61—21no—
171C2×C61—161no—
172C2×C61#160711yescharacteristic
173C2×C61#218911yescharacteristic
174C2×C61#219011yescharacteristic
175D61—81no—
176#525—1221no—
177#485—1221no—
178#482—1221no—
179#467—611no—
180#466C2×D6111yescharacteristic
181#465—1221no—
182#452—1221no—
183#449—1221no—
184C2×#107C2×D6111yescharacteristic
185C2×C22⋊C8—611no—
186#380—1221no—
187#367—21no—
188#366—1221no—
189#358—1221no—
190#357—21no—
191C22⋊C16—611no—
192C22⋊C16C2×D6111yescharacteristic
193#340—611no—
194#340—1221no—
195C61#1228311yescharacteristic, complemented, Sylow
196C2×#41—1221no—
197C23×C4—611no—
198C2×Q16—611no—
199C2×SD16—1221no—
200C2×D8C22⋊D6111yescharacteristic
201C22×C8—611no—
202C22×C8C22×D6111yescharacteristic
203#123—2441no—
204C22⋊Q8—1221no—
205C22⋊Q8—1221no—
206#121—21no—
207#121—1221no—
208#120—2441no—
209C4×D4—2441no—
210C4×D4—2441no—
211#117—1221no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—1221no—
214C2×C22⋊C4—2441no—
215C2×C16—21no—
216C2×C16—1221no—
217#107—21no—
218#107—21no—
219#107—611no—
220#107C22⋊D6111yescharacteristic
221#106—1221no—
222#103—1221no—
223#103—1221no—
224#102—21no—
225#102—1221no—
226C22⋊C8—1221no—
227C22⋊C8—1221no—
228#95—2441no—
229C24—611no—
230#41—4881no—
231#41—4881no—
232C2×Q8—1221no—
233C2×D4—21no—
234C2×D4—41no—
235C2×D4—2441no—
236C2×D4—2441no—
237C22×C4—41no—
238C22×C4—611no—
239C22×C4—1221no—
240C22×C4—1221no—
241C22×C4—2441no—
242C22×C4—2441no—
243C22×C4—2441no—
244C22×C4—2441no—
245C22×C4D4×D6111yescharacteristic
246Q16—2441no—
247SD16—4881no—
248D8—41no—
249C2×C8—41no—
250C2×C8—1221no—
251C2×C8—2441no—
252C2×C8D4×D6111yescharacteristic
253C2×C8C2×C22⋊D6111yescharacteristic
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—1221no—
258C4⋊C4—2441no—
259C4⋊C4—2441no—
260C4⋊C4—2441no—
261C4⋊C4—2441no—
262C22⋊C4—41no—
263C22⋊C4—4882no—
264C22⋊C4—2441no—
265C22⋊C4—2441no—
266C22⋊C4—2441no—
267C22⋊C4—2441no—
268C42—2441no—
269C16—41no—
270C16—2441no—
271C23—41no—
272C23—1221no—
273C23—1221no—
274C23—1221no—
275C23—2441no—
276C23—2441no—
277C23D8×D6111yescharacteristic
278Q8—2441no—
279Q8—2441no—
280D4—41no—
281D4—41no—
282D4—81no—
283D4—81no—
284D4—4882no—
285D4—4882no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—1221no—
293C2×C4—4881no—
294C2×C4—4881no—
295C2×C4—4881no—
296C2×C4—4881no—
297C2×C4—4882no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4#1451974511yescharacteristic
304C2×C4#1451980111yescharacteristic
305C8—21no—
306C8—21no—
307C8—2441no—
308C22—21no—
309C22—21no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—4881no—
314C22—2442no—
315C22—2441no—
316C22—2441no—
317C22—2441no—
318C22#1683989411yescharacteristic
319C22#1684019311yescharacteristic
320C22#1684020111yescharacteristic
321C4—21no—
322C4—81no—
323C4—161no—
324C4—9761no—
325C4—4881no—
326C4—4881no—
327C4—2441no—
328C4#1683993811yescharacteristic
329C4#1683995111yescharacteristic
330C2—21no—
331C2—21no—
332C2—161no—
333C2—4881no—
334C2#2144480311yescharacteristic
335C2#2144775611yescharacteristic
336C2#2144776311yescharacteristic
subgroupG/H
centerC22#16839894
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#2233C2
socleC22×C61#447
prime radical#21456606C2
hypercenter#2233D61
hypocenterC61#12283
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2233

Complements

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

NH ≅ G/Ntype
2. #21448021C2semidirect
3. #21447248C2semidirect
4. #21443832C2semidirect
5. #21443828C2semidirect
6. #21443742C2semidirect
7. C61×#2233C2semidirect
164. #2233D61semidirect
195. C61#12283semidirect

Maximal subgroups

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

subgroupindex
#214566062
#214480212
#214472482
#214438322
#214438282
#214437422
C61×#22332
#1228361