Finite Group Databasebeta

← #29592557

Subgroups — #29592557

Order 15616.

486 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 63 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#21447070C211yesmaximal, characteristic
2#21444838C211yesmaximal, characteristic
3#21444770C211yesmaximal, characteristic
4#21443843C211yesmaximal, characteristic, complemented
5#21443840C211yesmaximal, characteristic, complemented
6#21443767C211yesmaximal, characteristic, complemented
7C61×#2677C211yesmaximal, characteristic, complemented
8#16840324C2211yescharacteristic
9#16840080—21no—
10#16839901—21no—
11C2×#14519691—21no—
12#16839641—21no—
13#16839529C2211yescharacteristic
14#16839495—21no—
15#16839495—21no—
16#16839486C2211yescharacteristic
17#16839486C2211yescharacteristic
18C2×C4⋊Q8×C61—21no—
19C2×C22⋊Q8×C61—21no—
20C61×#399C2211yescharacteristic
21C61×#398—21no—
22C61×#395—21no—
23C61×#393C2211yescharacteristic
24C61×#388—21no—
25C61×#387—21no—
26C61×#385C2211yescharacteristic
27C22×C22⋊D61—21no—
28C2×#11758201—41no—
29#14519761—41no—
30#14519751—21no—
31#14519751—21no—
32#14519747—41no—
33C2×C22⋊Dic61—21no—
34C2×#11758198—41no—
35#14519708—41no—
36#14519701—41no—
37#14519697—41no—
38#14519691—42no—
39C22⋊#92418—21no—
40C22⋊#92418—21no—
41#14519684—41no—
42C22×Q8×C61—21no—
43C22×Q8×C61D411yescharacteristic
44C23×C4×C61D411yescharacteristic
45C4⋊Q8×C61—44no—
46C22⋊Q8×C61—44no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61—21no—
49C2×C4⋊C4×C61—21no—
50C2×C4⋊C4×C61—21no—
51C2×C4⋊C4×C61—21no—
52C2×C4⋊C4×C61—21no—
53C2×C4⋊C4×C61D411yescharacteristic
54C2×C22⋊C4×C61—21no—
55C2×C22⋊C4×C61—21no—
56C2×C22⋊C4×C61D411yescharacteristic
57C2×C22⋊C4×C61C2311yescharacteristic
58C2×C42×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61—21no—
62#95×C61—21no—
63#95×C61D411yescharacteristic
64#95×C61D411yescharacteristic
65C24×C61C2×D411yescharacteristic
66C23×D61—21no—
67C2×Q8×C61—42no—
68C2×Q8×C61—42no—
69C2×Q8×C61—42no—
70C2×Q8×C61—41no—
71C2×Q8×C61—81no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—42no—
79C22×C4×C61—41no—
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C2×C22⋊D61—41no—
83C2×C22⋊D61—41no—
84C2×C22⋊D61—41no—
85C2×C22⋊D61—41no—
86C2×C22⋊D61—41no—
87C2×C22⋊D61—41no—
88C22×Dic61—21no—
89C22×Dic61—41no—
90C2×D244—21no—
91C2×D244—21no—
92C4⋊C4×C61—24no—
93C4⋊C4×C61—42no—
94C4⋊C4×C61—42no—
95C4⋊C4×C61—42no—
96C4⋊C4×C61—42no—
97C4⋊C4×C61—42no—
98C4⋊C4×C61—41no—
99C22⋊C4×C61—21no—
100C22⋊C4×C61—41no—
101C22⋊C4×C61—42no—
102C22⋊C4×C61—42no—
103C22⋊C4×C61SD1612yescomplemented
104C42×C61—42no—
105C22⋊Dic61—41no—
106C22⋊Dic61—41no—
107#11758201—82no—
108#11758198—41no—
109#11758198—41no—
110#11758198—82no—
111C4⋊Dic61—21no—
112C4⋊Dic61—21no—
113C2×#92418—41no—
114C2×#92418—41no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61#12011yescharacteristic
120C23×C61C2×SD1611yescharacteristic
121C23×C61#13711yescharacteristic
122C22×D61—41no—
123C22×D61—41no—
124C22×D61—41no—
125C22×D61—81no—
126C22×D61—81no—
127Q8×C61—42no—
128Q8×C61—81no—
129Q8×C61—84no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—22no—
133C2×C4×C61—22no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
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—42no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—81no—
154C22⋊D61—81no—
155C22⋊D61—81no—
156C22⋊D61—81no—
157C22⋊D61—81no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—81no—
162C2×Dic61—81no—
163C2×Dic61—81no—
164D244—81no—
165D244—81no—
166#92418—81no—
167#92418—81no—
168#12176—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—41no—
174C22×C61—41no—
175C22×C61—41no—
176C22×C61—41no—
177C22×C61—81no—
178C22×C61#44811yescharacteristic
179C22×C61#45411yescharacteristic
180C22×C61B(4,𝔽2)11yescharacteristic
181C2×D61—82no—
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C2×D61—161no—
186C4×C61—42no—
187C4×C61—42no—
188C4×C61—41no—
189C4×C61—42no—
190C4×C61—42no—
191C4×C61—41no—
192C4×C61—81no—
193C4×C61—81no—
194Dic61—81no—
195Dic61—161no—
196#2677D6111yescharacteristic, complemented
197#2044—611no—
198#1613—611no—
199#1601—611no—
200#1352—611no—
201#1351—611no—
202#1326—611no—
203C2×C61—21no—
204C2×C61—21no—
205C2×C61—41no—
206C2×C61—41no—
207C2×C61#159911yescharacteristic
208C2×C61#219611yescharacteristic
209C2×C61#220111yescharacteristic
210D61—161no—
211#534—611no—
212C2×C4⋊Q8—21no—
213C2×C22⋊Q8—21no—
214#481—1221no—
215#448—1221no—
216C2×#99—1221no—
217#399C2×D6111yescharacteristic
218#398—21no—
219#395—21no—
220#393C2×D6111yescharacteristic
221#388—21no—
222#387—21no—
223#386—1221no—
224#385C2×D6111yescharacteristic
225#342—611no—
226#327—1221no—
227#327—1221no—
228#323—611no—
229#323—611no—
230C61#1217611yescharacteristic, complemented, Sylow
231C22×Q8—21no—
232C22×Q8C22⋊D6111yescharacteristic
233C22×D4—1221no—
234C23×C4C22⋊D6111yescharacteristic
235C2×SD16—2441no—
236C4⋊Q8—44no—
237#123—2441no—
238C22⋊Q8—44no—
239#121—1221no—
240#121—1221no—
241#120—2441no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C22⋊D6111yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—1221no—
252C2×C22⋊C4—2441no—
253C2×C22⋊C4C22⋊D6111yescharacteristic
254C2×C22⋊C4C22×D6111yescharacteristic
255C2×C42—21no—
256#106—2441no—
257#103—2441no—
258#102—2441no—
259#99—2442no—
260C22⋊C8—1221no—
261C22⋊C8—1221no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—2441no—
267#95C22⋊D6111yescharacteristic
268#95C22⋊D6111yescharacteristic
269C24—1221no—
270C24C2×C22⋊D6111yescharacteristic
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—81no—
276C2×D4—1221no—
277C2×D4—1221no—
278C2×D4—2441no—
279C2×D4—2441no—
280C2×D4—2441no—
281C2×D4—2441no—
282C2×D4—2441no—
283C2×D4—2441no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—1221no—
293C22×C4—2441no—
294C22×C4C2×C22⋊D6111yescharacteristic
295C22×C4C2×C22⋊D6111yescharacteristic
296SD16—4882no—
297C2×C8—2441no—
298C2×C8—2441no—
299C4⋊C4—24no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—1221no—
307C4⋊C4—1221no—
308C22⋊C4—21no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—4882no—
313C22⋊C4—2441no—
314C22⋊C4—2441no—
315C22⋊C4—2441no—
316C22⋊C4—2441no—
317C22⋊C4#1175820112yescomplemented
318C42—42no—
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—4881no—
324C23—4881no—
325C23—2441no—
326C23—2441no—
327C23—2441no—
328C23#1451974611yescharacteristic
329C23C2×#1175820111yescharacteristic
330C23#1451980811yescharacteristic
331Q8—42no—
332Q8—81no—
333Q8—84no—
334D4—4881no—
335D4—4881no—
336D4—4881no—
337D4—4881no—
338D4—4881no—
339D4—4881no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—4881no—
365C2×C4—4881no—
366C2×C4—4881no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4—2441no—
370C8—4881no—
371C8—4881no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—9761no—
382C22—4882no—
383C22—4881no—
384C22—4881no—
385C22—4881no—
386C22#1683990111yescharacteristic
387C22#1683993511yescharacteristic
388C22#1683994911yescharacteristic
389C4—42no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—9761no—
398C4—4881no—
399C2—21no—
400C2—21no—
401C2—41no—
402C2—41no—
403C2—9761no—
404C2#2144475911yescharacteristic
405C2#2144780311yescharacteristic
406C2#2144783211yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C22⋊C4×C61C23
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2677C2
socleC22×C61B(4,𝔽2)
prime radical#16840324C22
hypercenter#2677D61
hypocenterC61#12176
LCSG ▷ C2×C22⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2677

Complements

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

NH ≅ G/Ntype
4. #21443843C2semidirect
5. #21443840C2semidirect
6. #21443767C2semidirect
7. C61×#2677C2semidirect
103. C22⋊C4×C61SD16semidirect
196. #2677D61semidirect
230. C61#12176semidirect
317. C22⋊C4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214470702
#214448382
#214447702
#214438432
#214438402
#214437672
C61×#26772
#1217661