Finite Group Databasebeta

← #29592344

Subgroups — #29592344

Order 15616.

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

normal subgroups only

This group has 412 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#21447048C211yesmaximal, characteristic, complemented
2#21444855C211yesmaximal, characteristic, complemented
3#21444753C211yesmaximal, characteristic
4#21443846C211yesmaximal, characteristic, complemented
5#21443822C211yesmaximal, characteristic, complemented
6#21443763C211yesmaximal, characteristic, complemented
7C61×#2782C211yesmaximal, characteristic, complemented
8#16840343C2211yescharacteristic
9#16840080—21no—
10#16839896—21no—
11C2×#14519705—21no—
12#16839761—21no—
13#16839531C2211yescharacteristic
14#16839495—21no—
15#16839495—21no—
16#16839488C2211yescharacteristic
17#16839488C2211yescharacteristic
18C61×#400—21no—
19C61×#400C2211yescharacteristic
20C61×#397—21no—
21C61×#394C2211yescharacteristic
22C61×#388—21no—
23C61×#386—21no—
24C61×#385—21no—
25C61×#385—21no—
26C61×#382C2211yescharacteristic
27C2×#11758222—21no—
28C2×#11758199—41no—
29C2×#11758194—21no—
30#14519757—41no—
31#14519752—41no—
32#14519751—21no—
33#14519751—21no—
34#14519708—41no—
35#14519705—42no—
36#14519700—41no—
37#14519698—41no—
38#14519697—41no—
39#14519697—41no—
40C22⋊#92418—21no—
41C22⋊#92418—21no—
42C22×D4×C61D411yescharacteristic
43C23×C4×C61—21no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61D411yescharacteristic
47C2×C4⋊C4×C61D411yescharacteristic
48C2×C4⋊C4×C61C2311yescharacteristic
49C2×C22⋊C4×C61—21no—
50C2×C22⋊C4×C61—21no—
51C2×C22⋊C4×C61—21no—
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×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
63C24×C61—21no—
64C2×D4×C61—41no—
65C2×D4×C61—42no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—81no—
72C22×C4×C61—81no—
73C22×C4×C61—81no—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C2×C22⋊D61—41no—
78#11758222—41no—
79#11758222—41no—
80#11758222—41no—
81#11758222—41no—
82C2×D244—21no—
83C2×D244—21no—
84C2×D244—21no—
85C2×C4×D61—41no—
86C2×Dic122—21no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—81no—
91C4⋊C4×C61—81no—
92C22⋊C4×C61—42no—
93C22⋊C4×C61—42no—
94C22⋊C4×C61—41no—
95C22⋊C4×C61—41no—
96C22⋊C4×C61—81no—
97C22⋊C4×C61—81no—
98C42×C61—21no—
99C42×C61—21no—
100#11758199—82no—
101#11758198—41no—
102#11758198—41no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105#11758194—41no—
106#11758194—41no—
107C2×#92418—41no—
108C2×#92418—41no—
109C2×#92418—41no—
110C23×C61—22no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61#12011yescharacteristic
118C22×D61—41no—
119D4×C61—42no—
120D4×C61—81no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
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—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—84no—
143C2×C4×C61#13511yescharacteristic
144C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
145C22⋊D61—81no—
146C22⋊D61—81no—
147C2×Dic61—41no—
148D244—41no—
149D244—41no—
150D244—81no—
151D244—81no—
152C4×D61—81no—
153C4×D61—81no—
154Dic122—41no—
155Dic122—41no—
156#92418—81no—
157#92418—81no—
158#92418—81no—
159#12141—611nomaximal, Sylow
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—42no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61—81no—
166C22×C61—81no—
167C22×C61—81no—
168C22×C61—81no—
169C22×C61#44911yescharacteristic
170C22×C61#45411yescharacteristic
171C22×C61B(4,𝔽2)11yescharacteristic
172C2×D61—81no—
173C2×D61—81no—
174C2×D61—81no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—41no—
178C4×C61—41no—
179C4×C61—81no—
180C4×C61—81no—
181C4×C61—81no—
182C4×C61—81no—
183C4×C61—81no—
184Dic61—81no—
185#2782D6111yescharacteristic, complemented
186#2040—611no—
187#1616—611no—
188#1598—611no—
189#1353—611no—
190#1345—611no—
191#1324—611no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—82no—
195C2×C61#159711yescharacteristic
196C2×C61#219711yescharacteristic
197C2×C61#220211yescharacteristic
198D61—161no—
199#537—611no—
200#481—1221no—
201#447—1221no—
202C2×#104—1221no—
203#417—1221no—
204#400—21no—
205#400C2×D6111yescharacteristic
206#397—21no—
207#394C2×D6111yescharacteristic
208#388—21no—
209#386—21no—
210#385—21no—
211#385—21no—
212#382C2×D6111yescharacteristic
213#343—611no—
214#327—1221no—
215#327—1221no—
216#324—611no—
217#324—611no—
218C61#1214111yescharacteristic, complemented, Sylow
219C2×#41—1221no—
220C22×D4C22⋊D6111yescharacteristic
221C23×C4—21no—
222C2×D8—2441no—
223C2×M16—1221no—
224C22⋊Q8—2441no—
225#121—1221no—
226#121—1221no—
227#121—2441no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C22⋊D6111yescharacteristic
231C2×C4⋊C4C22⋊D6111yescharacteristic
232C2×C4⋊C4C22×D6111yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42C22⋊D6111yescharacteristic
239#106—2441no—
240#104—2442no—
241#103—2441no—
242#102—2441no—
243#102—2441no—
244#102—2441no—
245C22⋊C8—1221no—
246C22⋊C8—1221no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95C22⋊D6111yescharacteristic
254#95C22⋊D6111yescharacteristic
255C24—21no—
256#41—2441no—
257#41—2441no—
258#41—2441no—
259#41—2441no—
260C2×Q8—1221no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—1221no—
264C2×D4—1221no—
265C2×D4—1221no—
266C2×D4—2441no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—2441no—
276C22×C4C2×C22⋊D6111yescharacteristic
277C22×C4C2×C22⋊D6111yescharacteristic
278C22×C4C2×C22⋊D6111yescharacteristic
279D8—4882no—
280M16—2441no—
281M16—2441no—
282C2×C8—2441no—
283C2×C8—2441no—
284C2×C8—2441no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—1221no—
291C4⋊C4—1221no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C22⋊C4—2441no—
299C22⋊C4—2441no—
300C42—21no—
301C42—21no—
302C23—22no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—2441no—
310C23#1451974611yescharacteristic
311Q8—2441no—
312Q8—2441no—
313D4—42no—
314D4—81no—
315D4—4881no—
316D4—4881no—
317D4—4881no—
318D4—4881no—
319D4—2441no—
320D4—2441no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—84no—
343C2×C4—4881no—
344C2×C4—4881no—
345C2×C4—2441no—
346C2×C4#1451979711yescharacteristic
347C2×C4#1451980211yescharacteristic
348C8—4881no—
349C8—4881no—
350C8—4881no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—4881no—
361C22—4881no—
362C22—4881no—
363C22#1683990711yescharacteristic
364C22#1683993511yescharacteristic
365C22#1683994911yescharacteristic
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—4881no—
376C2—21no—
377C2—21no—
378C2—82no—
379C2—9761no—
380C2#2144474811yescharacteristic
381C2#2144780911yescharacteristic
382C2#2144783611yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2782C2
socleC22×C61B(4,𝔽2)
prime radical#21444753C2
hypercenter#2782D61
hypocenterC61#12141
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2782

Complements

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

NH ≅ G/Ntype
1. #21447048C2semidirect
2. #21444855C2semidirect
4. #21443846C2semidirect
5. #21443822C2semidirect
6. #21443763C2semidirect
7. C61×#2782C2semidirect
185. #2782D61semidirect
218. C61#12141semidirect

Maximal subgroups

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

subgroupindex
#214470482
#214448552
#214447532
#214438462
#214438222
#214437632
C61×#27822
#1214161