Finite Group Databasebeta

← #29568967

Subgroups — #29568967

Order 15616.

446 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 87 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#21447152C211yesmaximal, characteristic, complemented
2#21447065C211yesmaximal, characteristic, complemented
3#21446955C211yesmaximal, characteristic
4#21443642C211yesmaximal, characteristic, complemented
5#21443642C211yesmaximal, characteristic, complemented
6#21443640C211yesmaximal, characteristic
7C61×#2727C211yesmaximal, characteristic
8C2×#14519777C2211yescharacteristic
9C2×#14519708—21no—
10C2×#14519700—21no—
11C2×#14519700—21no—
12C2×#14519698—21no—
13#16839524—21no—
14#16839524—21no—
15#16839517C2211yescharacteristic
16#16839517C2211yescharacteristic
17#16839517C2211yescharacteristic
18C2×#126×C61—21no—
19C61×#399—21no—
20C61×#396—21no—
21C61×#394—21no—
22C61×#390C2211yescharacteristic
23C61×#389—21no—
24C61×#389C2211yescharacteristic
25C61×#389C2211yescharacteristic
26C61×#388—21no—
27C22×Dic122—21no—
28C22×#92418—21no—
29C22×#92418—21no—
30C22×#92418—21no—
31#14519777—42no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61—21no—
34#14519708—42no—
35#14519700—42no—
36#14519700—42no—
37#14519698—42no—
38C22×D4×C61D411yescharacteristic
39#126×C61—42no—
40#126×C61—42no—
41C2×C4⋊C4×C61—21no—
42C2×C4⋊C4×C61—21no—
43C2×C4⋊C4×C61—21no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61D411yescharacteristic
46C2×C4⋊C4×C61D411yescharacteristic
47C2×C4⋊C4×C61D411yescharacteristic
48C2×C4⋊C4×C61D411yescharacteristic
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C22⋊C4×C61—21no—
51C2×C22⋊C4×C61—21no—
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×C61—21no—
54C2×C22⋊C4×C61—21no—
55C2×C22⋊C4×C61—21no—
56C2×C42×C61—21no—
57C2×C42×C61C2311yescharacteristic
58#95×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61C24×C61—21no—
62C2×D4×C61—21no—
63C2×D4×C61—21no—
64C2×D4×C61—41no—
65C2×D4×C61—41no—
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
74C22×Dic61—41no—
75C2×Dic122—22no—
76C2×Dic122—41no—
77C2×Dic122—41no—
78C2×Dic122—41no—
79C2×Dic122—41no—
80C2×Dic122—41no—
81C4⋊C4×C61—21no—
82C4⋊C4×C61—21no—
83C4⋊C4×C61—21no—
84C4⋊C4×C61—42no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—41no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—81no—
93C4⋊C4×C61SD1612yes—
94C22⋊C4×C61—42no—
95C22⋊C4×C61—42no—
96C22⋊C4×C61—41no—
97C22⋊C4×C61—41no—
98C22⋊C4×C61—41no—
99C22⋊C4×C61—41no—
100C22⋊C4×C61—81no—
101C22⋊C4×C61—81no—
102C42×C61—41no—
103C42×C61—42no—
104C4⋊Dic61—22no—
105C4⋊Dic61—22no—
106C4⋊Dic61—41no—
107C4⋊Dic61—41no—
108C2×#92418—22no—
109C2×#92418—41no—
110C2×#92418—41no—
111C2×#92418—41no—
112C2×#92418—41no—
113C2×#92418—41no—
114C2×#92418—41no—
115C2×#92418—41no—
116C2×#92418—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61#12011yescharacteristic
125D4×C61—42no—
126D4×C61—81no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—22no—
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—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61C2×SD1611yescharacteristic
149C2×C4×C61#13511yescharacteristic
150C2×C4×C61#13511yescharacteristic
151C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C61#13711yescharacteristic
154C2×Dic61—42no—
155C2×Dic61—81no—
156C2×Dic61—81no—
157Dic122—42no—
158Dic122—42no—
159Dic122—81no—
160Dic122—81no—
161#92418—42no—
162#92418—81no—
163#92418—81no—
164#7776—611nomaximal, Sylow
165C22×C61—81no—
166C22×C61—81no—
167C22×C61—81no—
168C22×C61—81no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61—81no—
172C22×C61#44911yescharacteristic
173C22×C61#45011yescharacteristic
174C22×C61#45211yescharacteristic
175C22×C61#45411yescharacteristic
176C22×C61#45511yescharacteristic
177C22×C61#45611yescharacteristic
178C22×C61B(4,𝔽2)11yescharacteristic
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—42no—
186C4×C61—81no—
187C4×C61—81no—
188C4×C61—81no—
189C4×C61—81no—
190Dic61—82no—
191#2727D6111yescharacteristic
192#2058—611no—
193#2043—611no—
194#2023—611no—
195#1271—611no—
196#1271—611no—
197#1270—611no—
198C2×C61—161no—
199C2×C61#159711yescharacteristic
200C2×C61#160411yescharacteristic
201C2×C61#160611yescharacteristic
202C2×C61#163511yescharacteristic
203C2×C61#164211yescharacteristic
204C2×C61#165211yescharacteristic
205C2×C61#166411yescharacteristic
206C2×C4⋊Q8—611no—
207C2×#126—21no—
208C2×#106—1221no—
209C2×#103—1221no—
210C2×#103—1221no—
211C2×#102—1221no—
212#399—21no—
213#396—21no—
214#394—21no—
215#390C2×D6111yescharacteristic
216#389—21no—
217#389C2×D6111yescharacteristic
218#389C2×D6111yescharacteristic
219#388—21no—
220#340—1221no—
221#340—1221no—
222#336—611no—
223#336—611no—
224#336—611no—
225C61#777611yescharacteristic, complemented, Sylow
226C22×Q8—1221no—
227C22×D4C22⋊D6111yescharacteristic
228C22×C8—1221no—
229C22×C8—1221no—
230C22×C8—1221no—
231C4⋊Q8—2442no—
232#126—42no—
233#126—42no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4C22⋊D6111yescharacteristic
241C2×C4⋊C4C22⋊D6111yescharacteristic
242C2×C4⋊C4C22⋊D6111yescharacteristic
243C2×C4⋊C4C22⋊D6111yescharacteristic
244C2×C4⋊C4C22⋊D6111yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—21no—
252C2×C42C22×D6111yescharacteristic
253#106—2442no—
254#103—2442no—
255#103—2442no—
256#102—2442no—
257#95—21no—
258#95—21no—
259#95—21no—
260C24—21no—
261C2×Q8—1222no—
262C2×Q8—2441no—
263C2×Q8—2441no—
264C2×Q8—2441no—
265C2×Q8—2441no—
266C2×Q8—2441no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—2441no—
277C22×C4C2×C22⋊D6111yescharacteristic
278C22×C4C2×C22⋊D6111yescharacteristic
279C22×C4C2×C22⋊D6111yescharacteristic
280C2×C8—1222no—
281C2×C8—2441no—
282C2×C8—2441no—
283C2×C8—2441no—
284C2×C8—2441no—
285C2×C8—2441no—
286C2×C8—2441no—
287C2×C8—2441no—
288C2×C8—2441no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—81no—
301C4⋊C4—1222no—
302C4⋊C4—1222no—
303C4⋊C4—2441no—
304C4⋊C4—2441no—
305C4⋊C4#1175820012yes—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C42—41no—
315C42—42no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#1451974611yescharacteristic
324Q8—4881no—
325Q8—4881no—
326Q8—2442no—
327Q8—2442no—
328D4—42no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—4881no—
352C2×C4—4881no—
353C2×C4—2442no—
354C2×C4C2×#1175820011yescharacteristic
355C2×C4#1451979711yescharacteristic
356C2×C4#1451979711yescharacteristic
357C2×C4#1451980211yescharacteristic
358C2×C4#1451980311yescharacteristic
359C2×C4#1451980711yescharacteristic
360C8—4881no—
361C8—4881no—
362C8—2442no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#1683990711yescharacteristic
371C22#1683991311yescharacteristic
372C22#1683992411yescharacteristic
373C22#1683993511yescharacteristic
374C22#1683994011yescharacteristic
375C22#1683994511yescharacteristic
376C22#1683994911yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—4882no—
389C2—161no—
390C2#2144474811yescharacteristic
391C2#2144478811yescharacteristic
392C2#2144479911yescharacteristic
393C2#2144497011yescharacteristic
394C2#2144501911yescharacteristic
395C2#2144508911yescharacteristic
396C2#2144515311yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2727C2
socleC23×C61#120
prime radicalC22×D4×C61D4
hypercenter#2727D61
hypocenterC61#7776
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2727

Complements

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

NH ≅ G/Ntype
1. #21447152C2semidirect
2. #21447065C2semidirect
4. #21443642C2semidirect
5. #21443642C2semidirect
225. C61#7776semidirect

Maximal subgroups

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

subgroupindex
#214471522
#214470652
#214469552
#214436422
#214436402
C61×#27272
#777661