Finite Group Databasebeta

← #29577304

Subgroups — #29577304

Order 15616.

424 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 89 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#21457021C211yesmaximal, characteristic
2#21448620C211yesmaximal, characteristic
3#21446764C211yesmaximal, characteristic
4#21444422C211yesmaximal, characteristic
5#21443683C211yesmaximal, characteristic
6#21443676C211yesmaximal, characteristic
7C61×#1515C211yesmaximal, characteristic
8C2×#14519769C2211yescharacteristic
9#16840168C412yescomplemented
10#16840130C412yescomplemented
11#16840095—21no—
12#16840090—21no—
13#16840069—22no—
14#16839819C2211yescharacteristic
15C4×#11758194C2211yescharacteristic
16#16839659—21no—
17#16839649—21no—
18#16839647—21no—
19#16839645—21no—
20#16839644—21no—
21C4×C4⋊Dic61C2211yescharacteristic
22#16839613—21no—
23#16839521—21no—
24#16839521—21no—
25#16839503—22no—
26C61×#533C2211yescharacteristic
27C61×#432C2211yescharacteristic
28C61×#383C2211yescharacteristic
29C61×#328—21no—
30C61×#328—21no—
31C2×#11758194—21no—
32#14519769—21no—
33#14519769—21no—
34#14519769C2×C412yes—
35#14519769D412yes—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61D411yescharacteristic
38C2×C4⋊Dic61D411yescharacteristic
39C2×#11758196—21no—
40C2×#11758196—21no—
41C2×#11758196—21no—
42C2×#11758196—21no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61—21no—
45C2×C4×Dic61D411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47#14519710—22no—
48#14519708—22no—
49C4⋊#92418—21no—
50#14519701—22no—
51#14519701—22no—
52C22⋊#92418—21no—
53#14519688—21no—
54#14519688C2×C412yes—
55C4×#92418—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61C4⋊Q8×C61C2×C411yescharacteristic
62C4⋊Q8×C61C2×C411yescharacteristic
63#125×C61—21no—
64C22⋊Q8×C61—21no—
65C22⋊Q8×C61—21no—
66C4×Q8×C61—21no—
67C4×Q8×C61—21no—
68#117×C61—21no—
69C2×C42×C61C2311yescharacteristic
70C4⋊C8×C61—21no—
71C22⋊C8×C61—21no—
72#97×C61—21no—
73#95×C61—21no—
74#95×C61—21no—
75#95×C61D411yescharacteristic
76#95×C61D411yescharacteristic
77C2×Q8×C61—21no—
78C2×Q8×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
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×C8×C61—41no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61—21no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—41no—
95C22⋊C4×C61—41no—
96C42×C61—41no—
97C42×C61C22×C411yescharacteristic
98C42×C61C2×D411yescharacteristic
99C42×C61#4111yescharacteristic
100C42×C61#4111yescharacteristic
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61—42no—
104C4⋊Dic61C22⋊C412yes—
105C4⋊Dic61C22⋊C412yes—
106#11758196—22no—
107#11758196—22no—
108#11758196—22no—
109#11758196—22no—
110#11758196—42no—
111#11758196—81no—
112C4×Dic61—42no—
113C4×Dic61—41no—
114C4×Dic61—41no—
115C4×Dic61—81no—
116#11758194—22no—
117#11758194—41no—
118C2×#92418—22no—
119C2×#92418—41no—
120C23×C61#12011yescharacteristic
121Q8×C61—42no—
122Q8×C61—81no—
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×C61C2×C22⋊C411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61C4×D411yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12311yescharacteristic
141C2×Dic61—22no—
142C2×Dic61—22no—
143C2×Dic61—22no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C8×C61—81no—
154#92418—42no—
155#92418—42no—
156#9380—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61#38611yescharacteristic
160C22×C61#45511yescharacteristic
161C22×C61#45611yescharacteristic
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—81no—
169C4×C61—81no—
170C4×C61—81no—
171C4×C61—81no—
172Dic61—42no—
173Dic61—42no—
174Dic61—42no—
175Dic61—81no—
176#3236—611no—
177#2336—611no—
178#1986—611no—
179#1528—611no—
180#1515D6111yescharacteristic
181#1288—611no—
182#1285—611no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61#188211yescharacteristic
186C2×C61#190611yescharacteristic
187C2×C61#191211yescharacteristic
188#533C2×D6111yescharacteristic
189C2×#125—611no—
190#501—612no—
191#491—612no—
192#484—1221no—
193#483—1221no—
194#479—1222no—
195#432—611no—
196#432C2×D6111yescharacteristic
197C4×M16—611no—
198#389—1221no—
199#387—1221no—
200#387—1221no—
201#387—1221no—
202#387—1221no—
203#383C2×D6111yescharacteristic
204C4×C4⋊C4—611no—
205C4×C4⋊C4—1221no—
206#339—1221no—
207#339—1221no—
208#330—1222no—
209#328—21no—
210#328—21no—
211C61#938011yescharacteristic, complemented, Sylow
212C2×M16—1221no—
213C4⋊Q8C4×D6111yescharacteristic
214C4⋊Q8C4×D6111yescharacteristic
215#125—21no—
216#125—612no—
217#125—612no—
218#125—1221no—
219#125—1221no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224#117—21no—
225C2×C4⋊C4—611no—
226C2×C4⋊C4—611no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4—1221no—
229C2×C4⋊C4—1221no—
230C2×C4⋊C4—1221no—
231C2×C4⋊C4—1221no—
232C2×C42—611no—
233C2×C42—611no—
234C2×C42—1221no—
235C2×C42—1221no—
236C2×C42C22×D6111yescharacteristic
237#107—1222no—
238#106—1222no—
239C4⋊C8—21no—
240C4⋊C8—1221no—
241#103—1222no—
242#103—1222no—
243C22⋊C8—21no—
244C22⋊C8—1221no—
245#97—21no—
246#97—612no—
247#97—1221no—
248C4×C8—1221no—
249#95—21no—
250#95—21no—
251#95—1221no—
252#95—1221no—
253#95—1221no—
254#95—1221no—
255#95—1221no—
256#95C22⋊D6111yescharacteristic
257#95C22⋊D6111yescharacteristic
258C2×Q8—21no—
259C2×Q8—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—1221no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4D4×D6111yescharacteristic
267C22×C4D4×D6111yescharacteristic
268C22×C4C2×C22⋊D6111yescharacteristic
269M16—1222no—
270M16—2441no—
271C2×C8—41no—
272C2×C8—1222no—
273C2×C8—2441no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—612no—
281C4⋊C4—612no—
282C4⋊C4—1222no—
283C4⋊C4—1222no—
284C4⋊C4—1222no—
285C4⋊C4—1222no—
286C4⋊C4—1221no—
287C4⋊C4—1221no—
288C4⋊C4—4881no—
289C4⋊C4—2442no—
290C4⋊C4—2442no—
291C22⋊C4—41no—
292C42—41no—
293C42—4881no—
294C42—2441no—
295C42—2441no—
296C42—2442no—
297C42C2×C4×D6111yescharacteristic
298C42#1175822211yescharacteristic
299C42D4×D6111yescharacteristic
300C42#1175822411yescharacteristic
301C23#1451974511yescharacteristic
302Q8—42no—
303Q8—81no—
304C2×C4—21no—
305C2×C4—21no—
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—1222no—
317C2×C4—1222no—
318C2×C4—1222no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4C22⋊C4×D6111yescharacteristic
329C2×C4#1451973711yescharacteristic
330C2×C4C4×C22⋊D6111yescharacteristic
331C2×C4#1451974811yescharacteristic
332C2×C4#1451975011yescharacteristic
333C2×C4#1451975911yescharacteristic
334C8—81no—
335C8—2442no—
336C8—2442no—
337C22—21no—
338C22—21no—
339C22#1683963811yescharacteristic
340C22#1683993811yescharacteristic
341C22#1683994311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—4881no—
353C4—2442no—
354C4—2442no—
355C4—2442no—
356C2—21no—
357C2—21no—
358C2#2144618511yescharacteristic
359C2#2144631711yescharacteristic
360C2#2144635311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1515C2
socleC22×C61#386
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1515D61
hypocenterC61#9380
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
9. #16840168C4semidirect
10. #16840130C4semidirect
211. C61#9380semidirect

Maximal subgroups

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

subgroupindex
#214570212
#214486202
#214467642
#214444222
#214436832
#214436762
C61×#15152
#938061