Finite Group Databasebeta

← #30595813

Subgroups — #30595813

Order 15616.

440 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 90 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#21461734C211yesmaximal, characteristic, complemented
2#21461604C211yesmaximal, characteristic, complemented
3#21460640C212yesmaximal, complemented
4#21460635C212yesmaximal
5#21454190C211yesmaximal, characteristic
6C2×#14519844—21no—
7C2×C4⋊#92420—21no—
8C2×#14519832—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519768C411yescharacteristic
13C2×#14519764C411yescharacteristic
14#16839668C411yescharacteristic
15#16839667C2211yescharacteristic
16#16839665C2211yescharacteristic
17#16839655—21no—
18#16839655—21no—
19#16839650—21no—
20#16839650—21no—
21C42×Dic61C2211yescharacteristic
22C61×#390C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25#14519844—24no—
26C4⋊#92420—42no—
27#14519832—42no—
28C22×Dic122C2×C411yescharacteristic
29#14519768—42no—
30#14519764—42no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61C2×C411yescharacteristic
33C2×#11758196—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C4×Dic61C2×C411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43C2×C4×Dic61C2311yescharacteristic
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48C22×D4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61—21no—
51C2×C22⋊C4×C61—21no—
52C2×C42×C61C2×C411yescharacteristic
53C24×C61—21no—
54C2×D4×C61—22no—
55C2×D4×C61—41no—
56C2×D4×C61—41no—
57C22×C4×C61—21no—
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22×C411yescharacteristic
61C22×Dic61—21no—
62C22×Dic61—21no—
63C22×Dic61C22⋊C411yescharacteristic
64C22×Dic61C22⋊C411yescharacteristic
65C22×Dic61C2×D411yescharacteristic
66C22×Dic61C2×D411yescharacteristic
67C2×Dic122—22no—
68C2×Dic122—41no—
69C2×Dic122—41no—
70C2×#92420—24no—
71C2×#92420—42no—
72C2×#92420—42no—
73C2×#92420—42no—
74C2×#92420—42no—
75C2×#92420—42no—
76C4⋊C4×C61—41no—
77C22⋊C4×C61—42no—
78C22⋊C4×C61—42no—
79C42×C61—22no—
80C22⋊Dic61—42no—
81C22⋊Dic61—42no—
82C22⋊Dic61—41no—
83C22⋊Dic61—41no—
84C22⋊Dic61—41no—
85C22⋊Dic61—41no—
86C4⋊Dic61—42no—
87C4⋊Dic61—41no—
88#11758196—42no—
89C4×Dic61—22no—
90C4×Dic61—22no—
91C4×Dic61—21no—
92C4×Dic61—21no—
93C4×Dic61—21no—
94C4×Dic61—21no—
95C4×Dic61—41no—
96C4×Dic61—41no—
97C23×C61—41no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61C2×C22⋊C411yescharacteristic
105D4×C61—82no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61#10412yes—
117C2×C4×C61#13112yes—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61#10012yes—
134C2×Dic61AGL(1,ℤ/8)11yescharacteristic
135C2×Dic61AGL(1,ℤ/8)11yescharacteristic
136C2×Dic61#13711yescharacteristic
137C2×Dic61#13711yescharacteristic
138Dic122—82no—
139#92420—44no—
140#92420—82no—
141#7696—611nomaximal, Sylow
142C22×C61—81no—
143C22×C61—81no—
144C22×C61—81no—
145C22×C61—81no—
146C22×C61—81no—
147C22×C61—81no—
148C22×C61—81no—
149C22×C61#40811yescharacteristic
150C22×C61#41011yescharacteristic
151C22×C61C2×#10011yescharacteristic
152C22×C61#41711yescharacteristic
153C22×C61#41711yescharacteristic
154C22×C61C2×#10411yescharacteristic
155C22×C61#42111yescharacteristic
156C4×C61—22no—
157C4×C61—22no—
158C4×C61—41no—
159C4×C61—81no—
160Dic61—21no—
161Dic61—21no—
162Dic61—21no—
163Dic61—21no—
164Dic61—22no—
165Dic61—41no—
166Dic61—81no—
167Dic61—81no—
168#2861—611no—
169#1831—611no—
170#1767—611no—
171#1272—612no—
172#1270—612no—
173C2×C61—161no—
174C2×C61#146911yescharacteristic
175C2×C61#148311yescharacteristic
176C2×C61#149511yescharacteristic
177C2×C61#150911yescharacteristic
178C2×C61#151211yescharacteristic
179C2×C61#153011yescharacteristic
180C2×C61#156011yescharacteristic
181C2×#125—611no—
182C2×#124—611no—
183C2×C4⋊C8—1221no—
184C2×C4⋊C8—1221no—
185C2×#97—1221no—
186#391—611no—
187#391—611no—
188#390—611no—
189#390#692311yescharacteristic
190#388—1221no—
191#388—1221no—
192#388—1221no—
193#388—1221no—
194C43—611no—
195#336—612no—
196#336—612no—
197#336—1221no—
198C61#769611yescharacteristic, complemented, Sylow
199C22×Q8—611no—
200C22×D4C2×#692311yescharacteristic
201C22×C8—1222no—
202C22×C8—1222no—
203#125—2442no—
204#124—2442no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4—1221no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4C2×#692311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4—1221no—
213C2×C22⋊C4—1221no—
214C2×C22⋊C4—1221no—
215C2×C42—611no—
216C2×C42—611no—
217C2×C42—611no—
218C2×C42—611no—
219C2×C42—611no—
220C2×C42—611no—
221C2×C42C2×#692311yescharacteristic
222C4⋊C8—1224no—
223C4⋊C8—2442no—
224#97—2442no—
225#95—1221no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229C24—21no—
230C2×Q8—1222no—
231C2×Q8—2441no—
232C2×Q8—2441no—
233C2×D4—22no—
234C2×D4—41no—
235C2×D4—41no—
236C22×C4—21no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4C22⋊#692311yescharacteristic
244C22×C4C22⋊#692311yescharacteristic
245C22×C4C22×#692311yescharacteristic
246C2×C8—1224no—
247C2×C8—2442no—
248C2×C8—2442no—
249C2×C8—2442no—
250C2×C8—2442no—
251C2×C8—2442no—
252C4⋊C4—41no—
253C4⋊C4—2441no—
254C4⋊C4—2442no—
255C4⋊C4—2442no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—2441no—
259C22⋊C4—2441no—
260C22⋊C4—2441no—
261C22⋊C4—2441no—
262C22⋊C4—2442no—
263C22⋊C4—2442no—
264C42—22no—
265C42—1221no—
266C42—1221no—
267C42—1221no—
268C42—1221no—
269C42—1222no—
270C42—1222no—
271C42—2441no—
272C42—2441no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊#692311yescharacteristic
281Q8—4882no—
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—612no—
294C2×C4—611no—
295C2×C4—611no—
296C2×C4—611no—
297C2×C4—611no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1221no—
306C2×C4—1221no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4#1451984112yes—
314C2×C4#1451986412yes—
315C8—4882no—
316C8—2444no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#1684073511yescharacteristic
325C22#1684074111yescharacteristic
326C22D61⋊#10011yescharacteristic
327C22#1684075311yescharacteristic
328C22#1684075311yescharacteristic
329C22C2×#1451984111yescharacteristic
330C22#1684076011yescharacteristic
331C4—22no—
332C4—22no—
333C4—41no—
334C4—81no—
335C4—1222no—
336C4—1221no—
337C4—1221no—
338C4—1221no—
339C4—1221no—
340C4—4881no—
341C4—4881no—
342C4—2441no—
343C2—161no—
344C2#2146108811yescharacteristic
345C2#2146113411yescharacteristic
346C2#2146117411yescharacteristic
347C2#2146122111yescharacteristic
348C2#2146123011yescharacteristic
349C2#2146128911yescharacteristic
350C2#2146135711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#7696
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461734C2semidirect
2. #21461604C2semidirect
3. #21460640C2semidirect
198. C61#7696semidirect

Maximal subgroups

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

subgroupindex
#214617342
#214616042
#214606402
#214606352
#214541902
#769661