Finite Group Databasebeta

← #29592255

Subgroups — #29592255

Order 15616.

408 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 59 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#21453703C211yesmaximal, characteristic, complemented
2#21444856C211yesmaximal, characteristic
3#21444788C211yesmaximal, characteristic
4#21443848C211yesmaximal, characteristic, complemented
5#21443845C211yesmaximal, characteristic, complemented
6#21443753C211yesmaximal, characteristic, complemented
7C61×#2007C211yesmaximal, characteristic
8#16840078—21no—
9#16839900—21no—
10#16839704—21no—
11#16839701C2211yescharacteristic
12#16839696—21no—
13#16839682C2211yescharacteristic
14#16839669—21no—
15#16839666—21no—
16#16839649—21no—
17#16839646—21no—
18#16839496—21no—
19#16839496—21no—
20#16839488C2211yescharacteristic
21#16839488C2211yescharacteristic
22C2×#137×C61—21no—
23C61×#537C2211yescharacteristic
24C2×#104×C61—21no—
25C61×#391C2211yescharacteristic
26C61×#343C2211yescharacteristic
27C22×Dic122D411yescharacteristic
28C2×#11758200—41no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61D411yescharacteristic
31C2×#11758196—21no—
32C2×#11758196—21no—
33C2×#11758196—21no—
34C2×C4×Dic61—21no—
35#14519708—41no—
36#14519700—41no—
37#14519698—41no—
38C22⋊#92418—21no—
39C22⋊#92418—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684D411yescharacteristic
48#14519684D411yescharacteristic
49C2×#41×C61—21no—
50C22×Q8×C61D411yescharacteristic
51#137×C61—42no—
52#137×C61—42no—
53C2×Q16×C61—41no—
54C2×SD16×C61—41no—
55C2×M16×C61—21no—
56C22⋊Q8×C61—41no—
57#121×C61—21no—
58#121×C61—21no—
59#121×C61—41no—
60C2×C4⋊C4×C61C2311yescharacteristic
61C2×C42×C61D411yescharacteristic
62#104×C61—42no—
63#95×C61—21no—
64#95×C61—21no—
65#41×C61—41no—
66#41×C61—41no—
67#41×C61—41no—
68#41×C61—41no—
69C2×Q8×C61—21no—
70C2×Q8×C61—21no—
71C2×Q8×C61—41no—
72C2×Q8×C61—41no—
73C2×Q8×C61SD1612yescomplemented
74C2×D4×C61—21no—
75C2×D4×C61—21no—
76C2×D4×C61—21no—
77C2×D4×C61—41no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—41no—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C2×Dic122—41no—
89C2×Dic122—42no—
90Q16×C61—44no—
91SD16×C61—44no—
92M16×C61—22no—
93M16×C61—41no—
94C2×C8×C61—41no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—41no—
98C22⋊C4×C61—41no—
99C22⋊C4×C61—41no—
100C42×C61—41no—
101#11758200—82no—
102C4⋊Dic61—41no—
103C4⋊Dic61—81no—
104#11758196—42no—
105#11758196—42no—
106#11758196—81no—
107C4×Dic61—81no—
108C2×#92418—41no—
109C2×#92418—41no—
110C23×C61—41no—
111C23×C61#12011yescharacteristic
112Q8×C61—42no—
113Q8×C61—41no—
114Q8×C61—41no—
115Q8×C61—81no—
116D4×C61—41no—
117D4×C61—41no—
118D4×C61—81no—
119D4×C61—81no—
120D4×C61—81no—
121D4×C61—81no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—22no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—81no—
136C2×C4×C61—81no—
137C2×C4×C61C2×SD1611yescharacteristic
138C2×C4×C61#13711yescharacteristic
139C2×Dic61—22no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—42no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150Dic122—42no—
151Dic122—81no—
152C8×C61—42no—
153#92418—81no—
154#92418—81no—
155#12128—611nomaximal, Sylow
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—81no—
159C22×C61—81no—
160C22×C61—81no—
161C22×C61#44811yescharacteristic
162C22×C61#45311yescharacteristic
163C22×C61B(4,𝔽2)11yescharacteristic
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—42no—
167C4×C61—41no—
168C4×C61—41no—
169C4×C61—81no—
170C4×C61—81no—
171Dic61—42no—
172Dic61—81no—
173Dic61—81no—
174Dic61—81no—
175#2799—611no—
176#2007D6111yescharacteristic
177#1616—611no—
178#1604—611no—
179#1354—611no—
180#1353—611no—
181#1319—611no—
182C2×C61—21no—
183C2×C61—21no—
184C2×C61—161no—
185C2×C61#159611yescharacteristic
186C2×C61#219711yescharacteristic
187C2×C61#220311yescharacteristic
188C2×#137—21no—
189#537C2×D6111yescharacteristic
190#481—1221no—
191#448—1221no—
192C2×#104—21no—
193#400—611no—
194#400—1221no—
195#398—1221no—
196#395—611no—
197#391—1221no—
198#391—1221no—
199#391C2×D6111yescharacteristic
200#387—1221no—
201#387—1221no—
202#343C2×D6111yescharacteristic
203#327—1221no—
204#327—1221no—
205#324—611no—
206#324—611no—
207C61#1212811yescharacteristic, complemented, Sylow
208C2×#41—21no—
209C22×Q8—611no—
210C22×Q8C22⋊D6111yescharacteristic
211#137—42no—
212#137—42no—
213C2×Q16—41no—
214C2×SD16—41no—
215C2×SD16—2441no—
216C2×M16—21no—
217C22⋊Q8—41no—
218#121—21no—
219#121—21no—
220#121—41no—
221C2×C4⋊C4—611no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4—1221no—
224C2×C4⋊C4—1221no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4C22×D6111yescharacteristic
227C2×C42—1221no—
228C2×C42C22⋊D6111yescharacteristic
229#106—2441no—
230#104—42no—
231#103—2441no—
232#102—2441no—
233C22⋊C8—1221no—
234C22⋊C8—1221no—
235#95—21no—
236#95—21no—
237#95—611no—
238#95—611no—
239#95—1221no—
240#95—1221no—
241#95—1221no—
242#95—1221no—
243#95—1221no—
244#95—1221no—
245#95—1221no—
246#41—41no—
247#41—41no—
248#41—41no—
249#41—41no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—2441no—
255C2×Q8—2442no—
256C2×Q8#1175820012yescomplemented
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4D4×D6111yescharacteristic
269C22×C4D4×D6111yescharacteristic
270C22×C4C2×C22⋊D6111yescharacteristic
271Q16—44no—
272SD16—44no—
273SD16—4882no—
274M16—22no—
275M16—41no—
276C2×C8—41no—
277C2×C8—2441no—
278C2×C8—2441no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—4881no—
283C4⋊C4—4881no—
284C4⋊C4—2441no—
285C4⋊C4—2442no—
286C4⋊C4—2442no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C42—41no—
290C42—4881no—
291C23—41no—
292C23#1451974511yescharacteristic
293Q8—42no—
294Q8—41no—
295Q8—41no—
296Q8—81no—
297Q8—4881no—
298Q8—2442no—
299D4—41no—
300D4—41no—
301D4—81no—
302D4—81no—
303D4—81no—
304D4—81no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—1222no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2442no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4C2×#1175820011yescharacteristic
332C2×C4#1451980711yescharacteristic
333C8—42no—
334C8—4881no—
335C8—4881no—
336C22—21no—
337C22—21no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#1683990011yescharacteristic
342C22#1683992911yescharacteristic
343C22#1683994811yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—42no—
349C4—81no—
350C4—81no—
351C4—4881no—
352C4—4881no—
353C4—4881no—
354C4—2442no—
355C2—21no—
356C2—21no—
357C2—161no—
358C2#2144474111yescharacteristic
359C2#2144780811yescharacteristic
360C2#2144783811yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2007C2
socleC22×C61B(4,𝔽2)
prime radicalC61×#537C22
hypercenter#2007D61
hypocenterC61#12128
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2007

Complements

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

NH ≅ G/Ntype
1. #21453703C2semidirect
4. #21443848C2semidirect
5. #21443845C2semidirect
6. #21443753C2semidirect
73. C2×Q8×C61SD16semidirect
207. C61#12128semidirect
256. C2×Q8#11758200semidirect

Maximal subgroups

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

subgroupindex
#214537032
#214448562
#214447882
#214438482
#214438452
#214437532
C61×#20072
#1212861