Finite Group Databasebeta

← #29592379

Subgroups — #29592379

Order 15616.

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

normal subgroups only

This group has 380 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#21447043C211yesmaximal, characteristic
2#21444821C211yesmaximal, characteristic
3#21444787C211yesmaximal, characteristic
4#21443849C211yesmaximal, characteristic, complemented
5#21443825C211yesmaximal, characteristic, complemented
6#21443765C211yesmaximal, characteristic, complemented
7C61×#2792C211yesmaximal, characteristic, complemented
8#16840343C2211yescharacteristic
9#16840075—21no—
10#16839901—21no—
11C2×#14519705—21no—
12#16839761—21no—
13#16839531C2211yescharacteristic
14#16839495—21no—
15#16839495—21no—
16#16839488C2211yescharacteristic
17#16839488C2211yescharacteristic
18C61×#398—21no—
19C61×#398C2211yescharacteristic
20C61×#395—21no—
21C61×#395C2211yescharacteristic
22C61×#389—21no—
23C61×#389—21no—
24C61×#387—21no—
25C61×#387—21no—
26C61×#384C2211yescharacteristic
27C2×#11758222—21no—
28C2×#11758201—41no—
29C2×#11758194—21no—
30#14519757—41no—
31#14519752—41no—
32#14519751—21no—
33#14519751—21no—
34#14519710—41no—
35#14519705—42no—
36#14519701—41no—
37#14519700—41no—
38#14519697—41no—
39#14519697—41no—
40C22⋊#92418—21no—
41C22⋊#92418—21no—
42C22×Q8×C61D411yescharacteristic
43C2×C4⋊C4×C61—21no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61—21no—
49C2×C4⋊C4×C61—21no—
50C2×C4⋊C4×C61D411yescharacteristic
51C2×C4⋊C4×C61D411yescharacteristic
52C2×C4⋊C4×C61C2311yescharacteristic
53C2×C42×C61—21no—
54C2×C42×C61D411yescharacteristic
55#95×C61—21no—
56#95×C61—21no—
57#95×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61D411yescharacteristic
62#95×C61D411yescharacteristic
63C2×Q8×C61—41no—
64C2×Q8×C61—81no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C2×C22⋊D61—41no—
75#11758222—41no—
76#11758222—41no—
77#11758222—41no—
78#11758222—41no—
79C2×D244—21no—
80C2×D244—21no—
81C2×D244—21no—
82C2×C4×D61—41no—
83C2×Dic122—21no—
84C4⋊C4×C61—24no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—81no—
91C4⋊C4×C61—81no—
92C4⋊C4×C61—81no—
93C4⋊C4×C61—81no—
94C4⋊C4×C61—81no—
95C42×C61—21no—
96C42×C61—21no—
97C42×C61—81no—
98#11758201—82no—
99#11758198—41no—
100#11758198—41no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103#11758194—41no—
104#11758194—41no—
105C2×#92418—41no—
106C2×#92418—41no—
107C2×#92418—41no—
108C23×C61#12011yescharacteristic
109C22×D61—41no—
110Q8×C61—42no—
111Q8×C61—81no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—22no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
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—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—41no—
136C2×C4×C61#13511yescharacteristic
137C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
138C22⋊D61—81no—
139C22⋊D61—81no—
140C2×Dic61—41no—
141D244—41no—
142D244—41no—
143D244—81no—
144D244—81no—
145C4×D61—81no—
146C4×D61—81no—
147Dic122—41no—
148Dic122—41no—
149#92418—81no—
150#92418—81no—
151#92418—81no—
152#12146—611nomaximal, Sylow
153C22×C61—21no—
154C22×C61—21no—
155C22×C61#44911yescharacteristic
156C22×C61#45411yescharacteristic
157C22×C61B(4,𝔽2)11yescharacteristic
158C2×D61—81no—
159C2×D61—81no—
160C2×D61—81no—
161C4×C61—21no—
162C4×C61—21no—
163C4×C61—42no—
164C4×C61—42no—
165C4×C61—42no—
166C4×C61—41no—
167C4×C61—41no—
168C4×C61—81no—
169C4×C61—81no—
170C4×C61—81no—
171Dic61—81no—
172#2792D6111yescharacteristic, complemented
173#2039—611no—
174#1610—611no—
175#1604—611no—
176#1354—611no—
177#1346—611no—
178#1325—611no—
179C2×C61—21no—
180C2×C61—21no—
181C2×C61#159711yescharacteristic
182C2×C61#220011yescharacteristic
183C2×C61#220511yescharacteristic
184D61—161no—
185#537—611no—
186#480—1221no—
187#448—1221no—
188C2×#104—1221no—
189#417—1221no—
190#398—21no—
191#398C2×D6111yescharacteristic
192#395—21no—
193#395C2×D6111yescharacteristic
194#389—21no—
195#389—21no—
196#387—21no—
197#387—21no—
198#384C2×D6111yescharacteristic
199#343—611no—
200#327—1221no—
201#327—1221no—
202#324—611no—
203#324—611no—
204C61#1214611yescharacteristic, complemented, Sylow
205C2×#41—1221no—
206C22×Q8C22⋊D6111yescharacteristic
207C2×SD16—2441no—
208C2×M16—1221no—
209C22⋊Q8—2441no—
210#121—1221no—
211#121—1221no—
212#121—2441no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C22⋊D6111yescharacteristic
221C2×C4⋊C4C22⋊D6111yescharacteristic
222C2×C4⋊C4C22×D6111yescharacteristic
223C2×C42—21no—
224C2×C42C22⋊D6111yescharacteristic
225#107—2441no—
226#104—2442no—
227#103—2441no—
228#103—2441no—
229#102—2441no—
230#102—2441no—
231C22⋊C8—1221no—
232C22⋊C8—1221no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95C22⋊D6111yescharacteristic
240#95C22⋊D6111yescharacteristic
241#41—2441no—
242#41—2441no—
243#41—2441no—
244#41—2441no—
245C2×Q8—41no—
246C2×Q8—81no—
247C2×Q8—1221no—
248C2×D4—1221no—
249C2×D4—1221no—
250C2×D4—1221no—
251C2×D4—2441no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—2441no—
259C22×C4C2×C22⋊D6111yescharacteristic
260C22×C4C2×C22⋊D6111yescharacteristic
261C22×C4C2×C22⋊D6111yescharacteristic
262SD16—4882no—
263M16—2441no—
264M16—2441no—
265C2×C8—2441no—
266C2×C8—2441no—
267C2×C8—2441no—
268C4⋊C4—24no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—81no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C4⋊C4—81no—
278C4⋊C4—81no—
279C4⋊C4—1221no—
280C4⋊C4—1221no—
281C22⋊C4—2441no—
282C22⋊C4—2441no—
283C42—21no—
284C42—21no—
285C42—81no—
286C23—2441no—
287C23#1451974611yescharacteristic
288Q8—42no—
289Q8—81no—
290Q8—2441no—
291Q8—2441no—
292D4—4881no—
293D4—4881no—
294D4—4881no—
295D4—4881no—
296D4—2441no—
297D4—2441no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
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—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—4881no—
323C2×C4—4881no—
324C2×C4—2441no—
325C2×C4#1451979711yescharacteristic
326C2×C4#1451980211yescharacteristic
327C8—4881no—
328C8—4881no—
329C8—4881no—
330C22—21no—
331C22—21no—
332C22—4881no—
333C22—4881no—
334C22—4881no—
335C22#1683990711yescharacteristic
336C22#1683993511yescharacteristic
337C22#1683994911yescharacteristic
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—42no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—4881no—
349C2—21no—
350C2—21no—
351C2—9761no—
352C2#2144474811yescharacteristic
353C2#2144782611yescharacteristic
354C2#2144784911yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2792C2
socleC22×C61B(4,𝔽2)
prime radical#16840343C22
hypercenter#2792D61
hypocenterC61#12146
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2792

Complements

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

NH ≅ G/Ntype
4. #21443849C2semidirect
5. #21443825C2semidirect
6. #21443765C2semidirect
7. C61×#2792C2semidirect
172. #2792D61semidirect
204. C61#12146semidirect

Maximal subgroups

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

subgroupindex
#214470432
#214448212
#214447872
#214438492
#214438252
#214437652
C61×#27922
#1214661