Finite Group Databasebeta

← #30595404

Subgroups — #30595404

Order 15616.

500 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 106 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#21461747C211yesmaximal, characteristic, complemented
2#21461534C211yesmaximal, characteristic, complemented
3#21460637C212yesmaximal, complemented
4#21460636C212yesmaximal
5#21448277C211yesmaximal, characteristic
6C2×C4×#92420—21no—
7#16840628—21no—
8#16840628—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519739C411yescharacteristic
13C2×#14519735—21no—
14#16839664C411yescharacteristic
15#16839663C2211yescharacteristic
16#16839661C2211yescharacteristic
17#16839654—21no—
18#16839652—21no—
19C4⋊C4×Dic61C411yescharacteristic
20#16839609—21no—
21C42×Dic61C2211yescharacteristic
22C61×#389C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25C4×#92420—24no—
26C22×D244C2×C411yescharacteristic
27C22×C4×D61—21no—
28#14519739—22no—
29#14519739C814yescomplemented
30#14519735—44no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61C2×C411yescharacteristic
33C2×#11758198—21no—
34C2×#11758198—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×C4×Dic61—21no—
39C2×C4×Dic61C2×C411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43C2×C4×Dic61D411yescharacteristic
44C2×C4×Dic61C2311yescharacteristic
45#14519684—21no—
46#14519684—21no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C42×C61C2×C411yescharacteristic
51#95×C61—21no—
52C23×D61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22×C411yescharacteristic
58C22×Dic61—21no—
59C22×Dic61C22⋊C411yescharacteristic
60C22×Dic61C22⋊C411yescharacteristic
61C22×Dic61C2×D411yescharacteristic
62C22×Dic61C2×D411yescharacteristic
63C2×D244—21no—
64C2×D244—41no—
65C2×D244—41no—
66C2×D244C2×C812yes—
67C2×C4×D61—24no—
68C2×C4×D61—42no—
69C2×C4×D61—42no—
70C2×#92420—24no—
71C2×#92420—42no—
72C2×#92420—42no—
73C2×#92420—42no—
74C2×#92420—42no—
75C2×#92420—42no—
76C4⋊C4×C61—21no—
77C4⋊C4×C61—42no—
78C4⋊C4×C61—41no—
79C4⋊C4×C61C2×C812yes—
80C42×C61—41no—
81#11758198—24no—
82#11758198—42no—
83#11758198—42no—
84#11758198—42no—
85#11758198—42no—
86C4⋊Dic61—42no—
87C4⋊Dic61—41no—
88C4×Dic61—22no—
89C4×Dic61—22no—
90C4×Dic61—22no—
91C4×Dic61—41no—
92C4×Dic61—41no—
93C4×Dic61—42no—
94C4×Dic61C2×C812yes—
95C4×Dic61M1612yes—
96C23×C61C2×C22⋊C411yescharacteristic
97C22×D61—22no—
98C22×D61—41no—
99C22×D61—41no—
100C22×D61—41no—
101C22×D61—41no—
102C22×D61—41no—
103C22×D61—41no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—22no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61#10412yes—
118C2×C4×C61C22×C811yescharacteristic
119C2×C4×C61C2×M1611yescharacteristic
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61C22⋊C812yes—
132C2×Dic61C22⋊C812yes—
133C2×Dic61#13512yes—
134C2×Dic61AGL(1,ℤ/8)12yes—
135D244—42no—
136D244—42no—
137C4×D61—48no—
138#92420—44no—
139#92420—82no—
140#7580—611nomaximal, Sylow
141C22×C61C2×C22⋊C811yescharacteristic
142C22×C61#41011yescharacteristic
143C22×C61#41311yescharacteristic
144C22×C61#41611yescharacteristic
145C22×C61#41811yescharacteristic
146C22×C61C2×#10411yescharacteristic
147C22×C61#42111yescharacteristic
148C2×D61—42no—
149C2×D61—42no—
150C2×D61—42no—
151C2×D61—81no—
152C2×D61—81no—
153C2×D61—81no—
154C2×D61—81no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—22no—
158C4×C61—41no—
159C4×C61—42no—
160C4×C61—81no—
161Dic61—22no—
162Dic61—22no—
163Dic61—22no—
164Dic61—22no—
165Dic61—81no—
166#2293—611no—
167#1839—611no—
168#1744—611no—
169#1271—612no—
170#1270—612no—
171C2×C61#145811yescharacteristic
172C2×C61#147611yescharacteristic
173C2×C61#147811yescharacteristic
174C2×C61#150511yescharacteristic
175C2×C61#152111yescharacteristic
176C2×C61#153211yescharacteristic
177C2×C61#155311yescharacteristic
178D61—82no—
179C2×C4×D4—611no—
180C2×#117—1221no—
181C2×C4×C8—1221no—
182#390—611no—
183#390—611no—
184#389—611no—
185#389#692311yescharacteristic
186#388—1221no—
187#388—1221no—
188C4×C4⋊C4—611no—
189C4×C22⋊C4—1221no—
190C43—611no—
191#336—612no—
192#336—612no—
193#336—1221no—
194#336—1221no—
195#336—1221no—
196C61#758011yescharacteristic, complemented, Sylow
197C22×D4—611no—
198C23×C4—1221no—
199C22×C8—1222no—
200C22×C8—1222no—
201C4×D4—614no—
202C4×D4—1222no—
203#117—2444no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4—1221no—
207C2×C4⋊C4C2×#692311yescharacteristic
208C2×C4⋊C4C2×#692311yescharacteristic
209C2×C22⋊C4—1221no—
210C2×C22⋊C4—1221no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4—1221no—
213C2×C22⋊C4—1221no—
214C2×C42—611no—
215C2×C42—611no—
216C2×C42—611no—
217C2×C42—611no—
218C2×C42—611no—
219C2×C42—611no—
220C2×C42—1221no—
221C2×C42C2×#692311yescharacteristic
222C4×C8—1224no—
223#95—21no—
224#95—1221no—
225#95—1221no—
226C24—1221no—
227C2×D4—612no—
228C2×D4—1221no—
229C2×D4—2441no—
230C2×D4—2441no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—1224no—
238C22×C4—1221no—
239C22×C4—2442no—
240C22×C4—2442no—
241C22×C4C22⋊#692311yescharacteristic
242C22×C4C22⋊#692311yescharacteristic
243C22×C4C22×#692311yescharacteristic
244C2×C8—1224no—
245C2×C8—2442no—
246C2×C8—2442no—
247C2×C8—2442no—
248C2×C8—2442no—
249C2×C8—2442no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—2441no—
254C4⋊C4—2442no—
255C4⋊C4D61⋊C812yes—
256C22⋊C4—1224no—
257C22⋊C4—2442no—
258C22⋊C4—2442no—
259C22⋊C4—2442no—
260C22⋊C4—2442no—
261C42—41no—
262C42—612no—
263C42—612no—
264C42—1222no—
265C42—1222no—
266C42—1222no—
267C42—2441no—
268C42—2441no—
269C42—2442no—
270C23—1222no—
271C23—2441no—
272C23—2441no—
273C23—2441no—
274C23—2441no—
275C23—2441no—
276C23—2441no—
277C23C2×C22⋊#692311yescharacteristic
278D4—2442no—
279D4—2442no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—612no—
294C2×C4—612no—
295C2×C4—612no—
296C2×C4—612no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—2448no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4#1451984212yes—
310C2×C4C2×D61⋊C811yescharacteristic
311C2×C4D61⋊M1611yescharacteristic
312C8—4882no—
313C8—2444no—
314C22—4881no—
315C22—4881no—
316C22—4881no—
317C22—4881no—
318C22—2442no—
319C22—2442no—
320C22—2442no—
321C22D61⋊(C22⋊C8)11yescharacteristic
322C22#1684074311yescharacteristic
323C22#1684074911yescharacteristic
324C22#1684075211yescharacteristic
325C22#1684075411yescharacteristic
326C22C2×#1451984211yescharacteristic
327C22#1684075911yescharacteristic
328C4—22no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—42no—
333C4—81no—
334C4—1222no—
335C4—1222no—
336C4—1222no—
337C4—1222no—
338C4—4881no—
339C2—4882no—
340C2#2146105311yescharacteristic
341C2#2146111311yescharacteristic
342C2#2146111911yescharacteristic
343C2#2146121011yescharacteristic
344C2#2146126211yescharacteristic
345C2#2146129711yescharacteristic
346C2#2146134311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#389C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC2×C4×Dic61C23
hypercenter#389#6923
hypocenterC61#7580
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #21461747C2semidirect
2. #21461534C2semidirect
3. #21460637C2semidirect
29. #14519739C8semidirect
196. C61#7580semidirect

Maximal subgroups

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

subgroupindex
#214617472
#214615342
#214606372
#214606362
#214482772
#758061