Finite Group Databasebeta

← #30600351

Subgroups — #30600351

Order 15616.

626 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C22×#14519832C211yesmaximal, characteristic
2#21461545C211yesmaximal, characteristic
3C2×#16840619C212yesmaximal, complemented
4#21460695C212yesmaximal, complemented
5#21454127C211yesmaximal, characteristic
6C23×#92420—21no—
7C2×C22⋊#92420—21no—
8C2×#14519832—21no—
9C2×#14519832—24no—
10C2×#14519832C2212yes—
11C2×#14519832C2212yes—
12#16840628—21no—
13#16840619—24no—
14C2×#14519768C2211yescharacteristic
15C22×C4×Dic61C2211yescharacteristic
16#16839661C411yescharacteristic
17#16839658C411yescharacteristic
18#16839649—21no—
19#16839645—21no—
20#16839637—21no—
21#16839634—21no—
22#16839627C2211yescharacteristic
23#16839622C411yescharacteristic
24C61×#380C411yescharacteristic
25C22×#92420—24no—
26C22×#92420—24no—
27C22×#92420—24no—
28C22×#92420—22no—
29C22×#92420—22no—
30C22⋊#92420—24no—
31#14519832—24no—
32#14519832—22no—
33#14519832—22no—
34#14519832—22no—
35#14519832D414yes—
36C23×Dic61D411yescharacteristic
37C23×Dic61D411yescharacteristic
38#14519768—22no—
39#14519768—22no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×C22⋊Dic61—21no—
47C2×C4×Dic61—22no—
48C2×C4×Dic61—22no—
49C2×C4×Dic61C2×C411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61C2311yescharacteristic
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58C23×C4×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61—21no—
62#95×C61—21no—
63C24×C61C22⋊C411yescharacteristic
64C22×C4×C61—22no—
65C22×C4×C61—22no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—41no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×Dic61—22no—
72C22×Dic61—22no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—22no—
76C22×Dic61—22no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—41no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C22×Dic61C2×D411yescharacteristic
84C2×#92420—216no—
85C2×#92420—24no—
86C2×#92420—24no—
87C2×#92420—24no—
88C2×#92420—24no—
89C2×#92420—42no—
90C2×#92420—42no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—41no—
93C22⋊C4×C61—42no—
94C22⋊Dic61—42no—
95C4⋊Dic61—22no—
96C4⋊Dic61—42no—
97C4⋊Dic61—41no—
98C4⋊Dic61—41no—
99C4⋊Dic61—41no—
100#11758196—42no—
101#11758196—42no—
102C4×Dic61—22no—
103C4×Dic61—21no—
104C4×Dic61—22no—
105C4×Dic61—21no—
106C4×Dic61C2×D412yes—
107C4×Dic61#4112yes—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61#10112yes—
113C23×C61#10412yes—
114C23×C61#10412yes—
115C23×C61C2×C22⋊C411yescharacteristic
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—44no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61C4×D412yes—
129C2×Dic61—24no—
130C2×Dic61—24no—
131C2×Dic61—21no—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—21no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61—21no—
140C2×Dic61—44no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61#10412yes—
148C2×Dic61#12011yescharacteristic
149C2×Dic61#12112yes—
150C2×Dic61#12311yescharacteristic
151#92420—28no—
152#92420—44no—
153#9121—611nomaximal, Sylow
154C22×C61—22no—
155C22×C61—22no—
156C22×C61—22no—
157C22×C61—22no—
158C22×C61—22no—
159C22×C61—22no—
160C22×C61#33014yescomplemented
161C22×C61#38611yescharacteristic
162C22×C61C2×#10111yescharacteristic
163C22×C61#41311yescharacteristic
164C22×C61C2×#10411yescharacteristic
165C22×C61C2×#10411yescharacteristic
166C22×C61C2×#10411yescharacteristic
167C22×C61#42111yescharacteristic
168C4×C61—22no—
169C4×C61—41no—
170C4×C61—81no—
171C4×C61—81no—
172Dic61—22no—
173Dic61—21no—
174Dic61—22no—
175Dic61—21no—
176Dic61—41no—
177Dic61—81no—
178Dic61—81no—
179C22×#97—611no—
180#2851—611no—
181#1749—611no—
182C2×#330—612no—
183#1295—612no—
184C2×C61—24no—
185C2×C61C2×#33011yescharacteristic
186C2×C61#152411yescharacteristic
187C2×C61#153111yescharacteristic
188C2×C61#153211yescharacteristic
189C2×C61#187111yescharacteristic
190C2×C61#189611yescharacteristic
191C2×C61#190111yescharacteristic
192C23×C8—1221no—
193C2×#125—611no—
194C22×C42—611no—
195C2×C22⋊C8—1221no—
196C2×#97—612no—
197C2×#97—612no—
198C2×#97—1224no—
199C2×#97—1221no—
200#389—611no—
201#389—611no—
202#387—1221no—
203#387—1221no—
204#385—1221no—
205#385—1221no—
206#382—611no—
207#380—611no—
208#380#692311yescharacteristic
209#336—1221no—
210#330—1224no—
211C61#912111yescharacteristic, complemented, Sylow
212C23×C4—611no—
213C23×C4—611no—
214C23×C4C2×#692311yescharacteristic
215C22×C8—1222no—
216C22×C8—1224no—
217C22×C8—1224no—
218C22×C8—1224no—
219C22×C8—1222no—
220#125—1222no—
221#125—1222no—
222C2×C4⋊C4—611no—
223C2×C4⋊C4—611no—
224C2×C4⋊C4—1221no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1221no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4C2×#692311yescharacteristic
229C2×C4⋊C4C2×#692311yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—1221no—
232C2×C42—611no—
233C2×C42—611no—
234C2×C42—611no—
235C2×C42—611no—
236C2×C42—1222no—
237C2×C42—1222no—
238C22⋊C8—1224no—
239#97—614no—
240#97—1222no—
241#97—1222no—
242#97—1222no—
243#97—1224no—
244#95—21no—
245#95—1221no—
246#95—1221no—
247#95—1221no—
248#95—1221no—
249#95—1221no—
250C24C22⋊#692311yescharacteristic
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—611no—
257C22×C4—611no—
258C22×C4—611no—
259C22×C4—611no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—1222no—
263C22×C4—1222no—
264C22×C4—1222no—
265C22×C4—1222no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4—2441no—
269C22×C4C22⋊#692311yescharacteristic
270C22×C4C22×#692311yescharacteristic
271C2×C8—1224no—
272C2×C8—12216no—
273C2×C8—1224no—
274C2×C8—1224no—
275C2×C8—1224no—
276C2×C8—2442no—
277C2×C8—2442no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—1222no—
281C4⋊C4—2441no—
282C4⋊C4—2442no—
283C4⋊C4—2442no—
284C4⋊C4—2441no—
285C4⋊C4—2441no—
286C4⋊C4—2442no—
287C22⋊C4—42no—
288C22⋊C4—2442no—
289C42—612no—
290C42—612no—
291C42—1221no—
292C42—1222no—
293C42—1221no—
294C42—1222no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#1451983812yes—
300C23#1451984112yes—
301C23#1451984212yes—
302C23C2×C22⋊#692311yescharacteristic
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—44no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—611no—
316C2×C4—612no—
317C2×C4—611no—
318C2×C4—612no—
319C2×C4—1224no—
320C2×C4—1224no—
321C2×C4—1221no—
322C2×C4—1221no—
323C2×C4—1221no—
324C2×C4—1221no—
325C2×C4—1221no—
326C2×C4—1221no—
327C2×C4—1221no—
328C2×C4—1221no—
329C2×C4—1221no—
330C2×C4—2444no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4D4×#692312yes—
338C8—1228no—
339C8—2444no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22#1684061914yescomplemented
347C22#1684071011yescharacteristic
348C22C2×#1451983811yescharacteristic
349C22#1684074911yescharacteristic
350C22C2×#1451984111yescharacteristic
351C22D61⋊#10411yescharacteristic
352C22C2×#1451984211yescharacteristic
353C22#1684075911yescharacteristic
354C4—22no—
355C4—41no—
356C4—81no—
357C4—81no—
358C4—1222no—
359C4—1221no—
360C4—1222no—
361C4—1221no—
362C4—4881no—
363C4—4881no—
364C4—2441no—
365C2—24no—
366C2C2×#1684061911yescharacteristic
367C2#2146127011yescharacteristic
368C2#2146129111yescharacteristic
369C2#2146129711yescharacteristic
370C2#2146180211yescharacteristic
371C2#2146183611yescharacteristic
372C2#2146184611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#380#6923
hypocenterC61#9121
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C2×#16840619C2semidirect
4. #21460695C2semidirect
160. C22×C61#330semidirect
211. C61#9121semidirect
346. C22#16840619semidirect

Maximal subgroups

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

subgroupindex
C22×#145198322
#214615452
C2×#168406192
#214606952
#214541272
#912161