Finite Group Databasebeta

← #29580921

Subgroups — #29580921

Order 15616.

518 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 81 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#21456372C211yesmaximal, characteristic, complemented
2#21451687C211yesmaximal, characteristic
3#21444445C211yesmaximal, characteristic, complemented
4#21444370C211yesmaximal, characteristic, complemented
5#21444262C211yesmaximal, characteristic
6#21443685C211yesmaximal, characteristic, complemented
7C61×#3110C211yesmaximal, characteristic
8C2×#14519769C2211yescharacteristic
9#16840168—21no—
10#16840168—21no—
11#16840161—22no—
12#16839819C2211yescharacteristic
13#16839805—21no—
14#16839805—21no—
15C4×#11758194C2211yescharacteristic
16#16839698—21no—
17#16839693—21no—
18#16839687—21no—
19#16839676—21no—
20#16839635—22no—
21#16839630C2211yescharacteristic
22#16839503—22no—
23#16839499—22no—
24D4⋊#92418—22no—
25C61×#533C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic
27C61×#484—21no—
28C61×#474—21no—
29C61×#461—21no—
30C61×#450—21no—
31C61×#440—22no—
32C61×#432C2211yescharacteristic
33C2×#11758194—21no—
34#14519769—21no—
35#14519769—21no—
36#14519769—21no—
37#14519769—21no—
38C2×C4⋊Dic61D411yescharacteristic
39C2×C4⋊Dic61D411yescharacteristic
40C2×#11758196—21no—
41C2×#11758196—21no—
42C2×#11758196—22no—
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61—22no—
45#14519710—22no—
46#14519710—22no—
47#14519708—22no—
48#14519708—22no—
49C4⋊#92418—21no—
50C22⋊#92418—21no—
51#14519688—21no—
52#14519688—21no—
53C4×#92418D412yes—
54#14519684—22no—
55#14519684—21no—
56#14519684—21no—
57#14519684D412yes—
58C22×D4×C61D411yescharacteristic
59C23×C4×C61—21no—
60C2×SD16×C61—22no—
61C4⋊Q8×C61D411yescharacteristic
62C4⋊Q8×C61D411yescharacteristic
63#125×C61—21no—
64C22⋊Q8×C61—21no—
65C22⋊Q8×C61—21no—
66C4×Q8×C61—21no—
67C4×Q8×C61—21no—
68C4×D4×C61—21no—
69C4×D4×C61—21no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72#117×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C2×C42×C61C2311yescharacteristic
76#107×C61—22no—
77C4⋊C8×C61—21no—
78#103×C61—22no—
79#102×C61—22no—
80C22⋊C8×C61—21no—
81#97×C61D412yes—
82C24×C61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61—42no—
88C22×C4×C61—21no—
89C22×C4×C61—41no—
90C22×C4×C61—41no—
91C22×C4×C61—82no—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×Dic61—22no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98SD16×C61—44no—
99C2×C8×C61—22no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C42×C61—41no—
112C42×C61C2×D411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61C2×D411yescharacteristic
115C42×C61#4111yescharacteristic
116C22⋊Dic61—44no—
117C22⋊Dic61—82no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122#11758196—41no—
123#11758196—41no—
124#11758196—41no—
125#11758196—41no—
126#11758196—82no—
127#11758194—24no—
128C2×#92418—22no—
129C2×#92418—41no—
130C23×C61—22no—
131C23×C61—42no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—42no—
135C23×C61#12011yescharacteristic
136Q8×C61—42no—
137Q8×C61—81no—
138D4×C61—22no—
139D4×C61—22no—
140D4×C61—41no—
141D4×C61—41no—
142C2×C4×C61—22no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—82no—
150C2×C4×C61—82no—
151C2×C4×C61#12011yescharacteristic
152C2×C4×C61#12011yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12711yescharacteristic
157C2×Dic61—41no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—42no—
165C8×C61—42no—
166#92418—24no—
167#92418—81no—
168#10023—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—44no—
172C22×C61—42no—
173C22×C61—82no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61#39211yescharacteristic
177C22×C61#45612yes—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—81no—
185C4×C61—81no—
186C4×C61—81no—
187Dic61—82no—
188Dic61—81no—
189Dic61—81no—
190#3166—611no—
191#3110D6111yescharacteristic
192#2633—611no—
193#1534—611no—
194#1515—611no—
195#1490—611no—
196#1288—611no—
197C2×C61—22no—
198C2×C61—82no—
199C2×C61#200511yescharacteristic
200C2×C61#200811yescharacteristic
201C2×C61#201011yescharacteristic
202#533C2×D6111yescharacteristic
203C2×#125—611no—
204C2×C4×D4C2×D6111yescharacteristic
205#501—1221no—
206#501—1221no—
207#499—1222no—
208#484—21no—
209#474—21no—
210#461—21no—
211#450—21no—
212#440—22no—
213#432—611no—
214#432C2×D6111yescharacteristic
215#428—1221no—
216#428—1221no—
217C4×M16—611no—
218#399—1221no—
219#398—1221no—
220#397—1221no—
221#394—1221no—
222#385—1222no—
223#383—611no—
224#330—1222no—
225#328—1222no—
226D4⋊C8—1222no—
227C61#1002311yescharacteristic, complemented, Sylow
228C22×D4C22⋊D6111yescharacteristic
229C23×C4—21no—
230C2×SD16—22no—
231C2×M16—1221no—
232C4⋊Q8C22⋊D6111yescharacteristic
233C4⋊Q8C22⋊D6111yescharacteristic
234#125—21no—
235#125—1221no—
236#125—1221no—
237#125—1221no—
238#125—1221no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247#117—21no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—611no—
250C2×C4⋊C4—1222no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4C22⋊D6111yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—1222no—
256C2×C22⋊C4—1222no—
257C2×C42C22×D6111yescharacteristic
258#107—22no—
259#107—1222no—
260#107—1222no—
261#106—1222no—
262#106—1222no—
263C4⋊C8—21no—
264C4⋊C8—1221no—
265#103—22no—
266#102—22no—
267C22⋊C8—21no—
268C22⋊C8—1221no—
269#97—1221no—
270#97—1221no—
271#97C22⋊D6112yes—
272C4×C8—612no—
273#95—612no—
274#95—1222no—
275#95—1221no—
276#95—1221no—
277C24—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—42no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—1222no—
288C22×C4—1221no—
289C22×C4—1221no—
290C22×C4D4×D6111yescharacteristic
291C22×C4D4×D6111yescharacteristic
292C22×C4C2×C22⋊D6111yescharacteristic
293SD16—44no—
294M16—1224no—
295C2×C8—22no—
296C2×C8—1222no—
297C2×C8—2441no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—1221no—
307C4⋊C4—1221no—
308C4⋊C4—1221no—
309C4⋊C4—1221no—
310C4⋊C4—4882no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C4⋊C4—2441no—
314C4⋊C4—2441no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—4882no—
319C22⋊C4—2444no—
320C42—41no—
321C42D4×D6111yescharacteristic
322C42#1175822411yescharacteristic
323C42C2×C22⋊D6111yescharacteristic
324C42C2×C22⋊D6111yescharacteristic
325C23—22no—
326C23—42no—
327C23—41no—
328C23—42no—
329C23—41no—
330C23#1451974511yescharacteristic
331Q8—42no—
332Q8—81no—
333D4—22no—
334D4—22no—
335D4—41no—
336D4—41no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—2441no—
347C2×C4—2442no—
348C2×C4—2442no—
349C2×C4—2442no—
350C2×C4—2441no—
351C2×C4—2442no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4#1451974511yescharacteristic
355C2×C4#1451974611yescharacteristic
356C2×C4#1451974811yescharacteristic
357C2×C4#1451974911yescharacteristic
358C2×C4#1451974911yescharacteristic
359C2×C4#1451977311yescharacteristic
360C8—42no—
361C8—1224no—
362C8—4881no—
363C22—21no—
364C22—21no—
365C22—44no—
366C22—42no—
367C22—82no—
368C22—81no—
369C22—81no—
370C22#1683967011yescharacteristic
371C22#1683994312yes—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—4882no—
382C4—4881no—
383C4—4881no—
384C2—22no—
385C2—82no—
386C2#2144685211yescharacteristic
387C2#2144687011yescharacteristic
388C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3110C2
socleC22×C61#392
prime radicalC22×D4×C61D4
hypercenter#3110D61
hypocenterC61#10023
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #21456372C2semidirect
3. #21444445C2semidirect
4. #21444370C2semidirect
6. #21443685C2semidirect
227. C61#10023semidirect

Maximal subgroups

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

subgroupindex
#214563722
#214516872
#214444452
#214443702
#214442622
#214436852
C61×#31102
#1002361