Finite Group Databasebeta

← #29592611

Subgroups — #29592611

Order 15616.

360 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 57 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#21454092C211yesmaximal, characteristic, complemented
2#21447045C211yesmaximal, characteristic
3#21444856C211yesmaximal, characteristic
4#21443847C211yesmaximal, characteristic, complemented
5#21443846C211yesmaximal, characteristic, complemented
6#21443762C211yesmaximal, characteristic, complemented
7C61×#1616C211yesmaximal, characteristic
8#16840078—21no—
9C2×#14519704—21no—
10#16839763—21no—
11#16839703—21no—
12#16839702—21no—
13#16839701C2211yescharacteristic
14#16839683—21no—
15#16839683—21no—
16#16839682—21no—
17#16839627C2211yescharacteristic
18#16839626—21no—
19#16839531C2211yescharacteristic
20#16839496—21no—
21#16839488C2211yescharacteristic
22C61×#537C2211yescharacteristic
23C61×#481—21no—
24C61×#400C2211yescharacteristic
25C61×#327—21no—
26C61×#324C2211yescharacteristic
27C22×Dic122—21no—
28C2×#11758194—21no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61D411yescharacteristic
31C2×C4⋊Dic61D411yescharacteristic
32C2×#11758196—21no—
33C2×#11758196—21no—
34C2×#11758196—21no—
35C2×C4×Dic61D411yescharacteristic
36#14519708—41no—
37#14519704—42no—
38#14519701—41no—
39#14519698—41no—
40#14519698—41no—
41C22⋊#92418—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684D411yescharacteristic
50C2×#41×C61—21no—
51C22⋊Q8×C61—41no—
52#121×C61—21no—
53#121×C61—21no—
54#121×C61—41no—
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C4⋊C4×C61C2311yescharacteristic
57#106×C61—41no—
58#102×C61—41no—
59C22⋊C8×C61—21no—
60#95×C61—21no—
61#95×C61—21no—
62#95×C61D411yescharacteristic
63#41×C61—41no—
64#41×C61—41no—
65#41×C61—41no—
66#41×C61—41no—
67C2×Q8×C61—21no—
68C2×D4×C61—21no—
69C2×D4×C61—21no—
70C2×D4×C61—21no—
71C2×D4×C61—41no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—41no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C2×Dic122—81no—
83C2×Dic122—81no—
84C2×Dic122—81no—
85C2×C8×C61—41no—
86C4⋊C4×C61—21no—
87C4⋊C4×C61—21no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C22⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C4⋊Dic61—41no—
93C4⋊Dic61—41no—
94C4⋊Dic61—81no—
95#11758196—81no—
96#11758196—81no—
97#11758196—81no—
98C4×Dic61—21no—
99C4×Dic61SD1612yes—
100#11758194—41no—
101#11758194—41no—
102C2×#92418—41no—
103C2×#92418—41no—
104C23×C61—41no—
105C23×C61#12011yescharacteristic
106Q8×C61—41no—
107Q8×C61—41no—
108D4×C61—41no—
109D4×C61—41no—
110D4×C61—81no—
111D4×C61—81no—
112D4×C61—81no—
113D4×C61—81no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—81no—
128C2×C4×C61—81no—
129C2×C4×C61C2×SD1611yescharacteristic
130C2×C4×C61#13711yescharacteristic
131C2×Dic61—22no—
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—
141C2×Dic61—41no—
142C2×Dic61—41no—
143Dic122—84no—
144C8×C61—81no—
145#92418—81no—
146#92418—81no—
147#12185—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—81no—
151C22×C61—81no—
152C22×C61—81no—
153C22×C61#44811yescharacteristic
154C22×C61#45411yescharacteristic
155C22×C61B(4,𝔽2)11yescharacteristic
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—41no—
159C4×C61—41no—
160C4×C61—81no—
161C4×C61—81no—
162C4×C61—81no—
163Dic61—42no—
164Dic61—81no—
165Dic61—81no—
166Dic61—81no—
167#2844—611no—
168#2040—611no—
169#1616—611no—
170#1616D6111yescharacteristic
171#1353—611no—
172#1353—611no—
173#1324—611no—
174C2×C61—21no—
175C2×C61—21no—
176C2×C61—161no—
177C2×C61#159911yescharacteristic
178C2×C61#220111yescharacteristic
179C2×C61#220211yescharacteristic
180#537C2×D6111yescharacteristic
181#481—21no—
182#481—1221no—
183C2×#104—1221no—
184#417—1221no—
185#400—611no—
186#400—1221no—
187#400—1221no—
188#400C2×D6111yescharacteristic
189#395—1221no—
190#395—1221no—
191#395—1221no—
192#382—611no—
193#382—1221no—
194#343—611no—
195#327—21no—
196#327—1221no—
197#324—611no—
198#324C2×D6111yescharacteristic
199C61#1218511yescharacteristic, complemented, Sylow
200C2×#41—21no—
201C22×Q8—1221no—
202C2×M16—1221no—
203C22⋊Q8—41no—
204#121—21no—
205#121—21no—
206#121—41no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4C22⋊D6111yescharacteristic
214C2×C4⋊C4C22×D6111yescharacteristic
215C2×C42—611no—
216#106—41no—
217#106—2441no—
218#104—2442no—
219#103—2441no—
220#102—41no—
221#102—2441no—
222#102—2441no—
223C22⋊C8—21no—
224C22⋊C8—1221no—
225#95—21no—
226#95—21no—
227#95—611no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234#95—1221no—
235#95C22⋊D6111yescharacteristic
236#41—41no—
237#41—41no—
238#41—41no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—4881no—
242C2×Q8—4881no—
243C2×Q8—4881no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—1221no—
252C22×C4—1221no—
253C22×C4—1221no—
254C22×C4—1221no—
255C22×C4D4×D6111yescharacteristic
256C22×C4D4×D6111yescharacteristic
257C22×C4C2×C22⋊D6111yescharacteristic
258M16—2441no—
259M16—2441no—
260C2×C8—41no—
261C2×C8—2441no—
262C2×C8—2441no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—4881no—
268C4⋊C4—4881no—
269C4⋊C4—4881no—
270C4⋊C4—4881no—
271C4⋊C4—2441no—
272C4⋊C4—2441no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C42—612no—
276C42—1221no—
277C23—41no—
278C23#1451974511yescharacteristic
279Q8—41no—
280Q8—41no—
281Q8—4884no—
282D4—41no—
283D4—41no—
284D4—81no—
285D4—81no—
286D4—81no—
287D4—81no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—1222no—
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×C4SD16×D6111yescharacteristic
316C2×C4#1451980611yescharacteristic
317C8—81no—
318C8—4881no—
319C8—4881no—
320C22—21no—
321C22—21no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22#1683989911yescharacteristic
326C22#1683993311yescharacteristic
327C22#1683994811yescharacteristic
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—4881no—
336C4—4881no—
337C4—4881no—
338C4—2442no—
339C2—21no—
340C2—21no—
341C2—161no—
342C2#2144475711yescharacteristic
343C2#2144783011yescharacteristic
344C2#2144783511yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1616C2
socleC22×C61B(4,𝔽2)
prime radicalC61×#537C22
hypercenter#1616D61
hypocenterC61#12185
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1616

Complements

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

NH ≅ G/Ntype
1. #21454092C2semidirect
4. #21443847C2semidirect
5. #21443846C2semidirect
6. #21443762C2semidirect
199. C61#12185semidirect

Maximal subgroups

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

subgroupindex
#214540922
#214470452
#214448562
#214438472
#214438462
#214437622
C61×#16162
#1218561