Finite Group Databasebeta

← #29592587

Subgroups — #29592587

Order 15616.

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

normal subgroups only

This group has 372 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#21451057C211yesmaximal, characteristic, complemented
2#21447040C211yesmaximal, characteristic
3#21444856C211yesmaximal, characteristic
4#21443847C211yesmaximal, characteristic, complemented
5#21443846C211yesmaximal, characteristic, complemented
6#21443764C211yesmaximal, characteristic, complemented
7C61×#1616C211yesmaximal, characteristic
8#16840078—21no—
9C2×#14519704—21no—
10#16839763—21no—
11#16839704—21no—
12#16839701C2211yescharacteristic
13#16839659—21no—
14#16839657—21no—
15#16839631C2211yescharacteristic
16#16839626—21no—
17#16839626—21no—
18#16839613—21no—
19#16839531C2211yescharacteristic
20#16839496—21no—
21#16839488C2211yescharacteristic
22C61×#537C2211yescharacteristic
23C61×#481—21no—
24C61×#400C2211yescharacteristic
25C61×#327—21no—
26C61×#324C2211yescharacteristic
27C2×#11758194—21no—
28C2×C4⋊Dic61D411yescharacteristic
29C2×C4⋊Dic61D411yescharacteristic
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×#11758196—21no—
33C2×#11758196—21no—
34C2×#11758196—21no—
35C2×C4×Dic61—21no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61D411yescharacteristic
38#14519708—41no—
39#14519704—42no—
40#14519701—41no—
41#14519698—41no—
42#14519698—41no—
43C22⋊#92418—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×C8×C61—41no—
83C4⋊C4×C61—21no—
84C4⋊C4×C61—21no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C22⋊C4×C61—41no—
88C22⋊C4×C61—41no—
89C4⋊Dic61—41no—
90C4⋊Dic61—41no—
91#11758196—42no—
92#11758196—42no—
93#11758196—42no—
94#11758196—42no—
95#11758196—81no—
96C4×Dic61—21no—
97C4×Dic61—42no—
98C4×Dic61—42no—
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—22no—
133C2×Dic61—22no—
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—
143C8×C61—81no—
144#92418—81no—
145#92418—81no—
146#12181—611nomaximal, Sylow
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—81no—
150C22×C61—81no—
151C22×C61—81no—
152C22×C61#44811yescharacteristic
153C22×C61#45411yescharacteristic
154C22×C61B(4,𝔽2)11yescharacteristic
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—41no—
158C4×C61—41no—
159C4×C61—81no—
160C4×C61—81no—
161C4×C61—81no—
162Dic61—24no—
163Dic61—81no—
164Dic61—81no—
165Dic61—81no—
166#2571—611no—
167#2039—611no—
168#1616—611no—
169#1616D6111yescharacteristic
170#1353—611no—
171#1353—611no—
172#1325—611no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61—161no—
176C2×C61#159911yescharacteristic
177C2×C61#219811yescharacteristic
178C2×C61#220411yescharacteristic
179#537C2×D6111yescharacteristic
180#481—21no—
181#481—1221no—
182C2×#104—1221no—
183#417—1221no—
184#400—611no—
185#400—1221no—
186#400C2×D6111yescharacteristic
187#389—1221no—
188#389—1221no—
189#384—611no—
190#382—1221no—
191#382—1221no—
192C4×C4⋊C4—1221no—
193#343—611no—
194#327—21no—
195#327—1221no—
196#324—611no—
197#324C2×D6111yescharacteristic
198C61#1218111yescharacteristic, complemented, Sylow
199C2×#41—21no—
200C2×M16—1221no—
201C22⋊Q8—41no—
202#121—21no—
203#121—21no—
204#121—41no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4C22⋊D6111yescharacteristic
213C2×C4⋊C4C22×D6111yescharacteristic
214C2×C42—611no—
215C2×C42—1221no—
216C2×C42—1221no—
217#106—41no—
218#106—2441no—
219#104—2442no—
220#103—2441no—
221#102—41no—
222#102—2441no—
223#102—2441no—
224C22⋊C8—21no—
225C22⋊C8—1221no—
226#95—21no—
227#95—21no—
228#95—611no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234#95C22⋊D6111yescharacteristic
235#41—41no—
236#41—41no—
237#41—41no—
238#41—41no—
239C2×Q8—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4D4×D6111yescharacteristic
252C22×C4D4×D6111yescharacteristic
253C22×C4C2×C22⋊D6111yescharacteristic
254M16—2441no—
255M16—2441no—
256C2×C8—41no—
257C2×C8—2441no—
258C2×C8—2441no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—4881no—
264C4⋊C4—2441no—
265C4⋊C4—2441no—
266C4⋊C4—2442no—
267C4⋊C4—2442no—
268C4⋊C4—2442no—
269C4⋊C4—2442no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C42—612no—
273C42—1221no—
274C42—2442no—
275C42—2442no—
276C23—41no—
277C23#1451974511yescharacteristic
278Q8—41no—
279Q8—41no—
280D4—41no—
281D4—41no—
282D4—81no—
283D4—81no—
284D4—81no—
285D4—81no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—1222no—
302C2×C4—1222no—
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×C4SD16×D6111yescharacteristic
314C2×C4#1451980611yescharacteristic
315C8—81no—
316C8—4881no—
317C8—4881no—
318C22—21no—
319C22—21no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22#1683989911yescharacteristic
324C22#1683993311yescharacteristic
325C22#1683994811yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—1224no—
334C4—4881no—
335C4—4881no—
336C4—4881no—
337C2—21no—
338C2—21no—
339C2—161no—
340C2#2144475711yescharacteristic
341C2#2144781311yescharacteristic
342C2#2144784311yescharacteristic
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#12181
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. #21451057C2semidirect
4. #21443847C2semidirect
5. #21443846C2semidirect
6. #21443764C2semidirect
198. C61#12181semidirect

Maximal subgroups

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

subgroupindex
#214510572
#214470402
#214448562
#214438472
#214438462
#214437642
C61×#16162
#1218161