Finite Group Databasebeta

← #30599208

Subgroups — #30599208

Order 15616.

1158 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 168 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#21461641C212yesmaximal
2#21460657C214yesmaximal, complemented
3C22×#14519684C211yesmaximal, characteristic
4C2×C22⋊#92420—22no—
5C2×C22⋊#92420—22no—
6C2×C22⋊#92420C2214yes—
7C24×Dic61C411yescharacteristic
8C24×Dic61C2211yescharacteristic
9C2×#14519684—22no—
10C2×#14519684—22no—
11C2×#14519684C412yes—
12C2×#14519684C2212yes—
13C24×C4×C61C411yescharacteristic
14C22×#92420—24no—
15C22⋊#92420—28no—
16C22⋊#92420—44no—
17C22⋊#92420—44no—
18C23×Dic61—21no—
19C23×Dic61—21no—
20C23×Dic61—22no—
21C23×Dic61—24no—
22C23×Dic61—42no—
23C23×Dic61—42no—
24C23×Dic61—42no—
25C23×Dic61—42no—
26C23×Dic61C814yescomplemented
27C23×Dic61C2×C411yescharacteristic
28C23×Dic61C2×C412yes—
29C23×Dic61D412yes—
30C23×Dic61D412yes—
31C23×Dic61C2311yescharacteristic
32#14519684—24no—
33#14519684—24no—
34C25×C61C2×C411yescharacteristic
35C23×C4×C61—21no—
36C23×C4×C61—21no—
37C23×C4×C61—24no—
38C23×C4×C61—42no—
39C23×C4×C61—42no—
40C23×C4×C61C2×C412yes—
41C24×C61—22no—
42C24×C61—22no—
43C24×C61—22no—
44C24×C61—22no—
45C24×C61—22no—
46C24×C61—22no—
47C24×C61C22⋊C411yescharacteristic
48C24×C61C22⋊C411yescharacteristic
49C24×C61C2×C812yes—
50C24×C61M1612yes—
51C24×C61C22×C411yescharacteristic
52C22×C4×C61—24no—
53C22×C4×C61—24no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—416no—
57C22×C4×C61—42no—
58C22×C4×C61—42no—
59C22×C4×C61—42no—
60C22×C4×C61—42no—
61C22×C4×C61—42no—
62C22×C4×C61—42no—
63C22×C4×C61M1614yes—
64C22×Dic61—24no—
65C22×Dic61—24no—
66C22×Dic61—24no—
67C22×Dic61—24no—
68C22×Dic61—22no—
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—24no—
72C22×Dic61—24no—
73C22×Dic61—41no—
74C22×Dic61—41no—
75C22×Dic61—42no—
76C22×Dic61—42no—
77C22×Dic61—41no—
78C22×Dic61—41no—
79C22×Dic61—416no—
80C22×Dic61—48no—
81C22×Dic61—42no—
82C22×Dic61—42no—
83C22×Dic61—42no—
84C22×Dic61—42no—
85C22×Dic61—48no—
86C22×Dic61—42no—
87C22×Dic61—42no—
88C22×Dic61C22⋊C412yes—
89C22×Dic61C2×C814yes—
90C22×Dic61C2×D412yes—
91C22×Dic61#4114yes—
92C2×#92420—44no—
93C2×#92420—44no—
94C2×#92420—44no—
95C23×C61—28no—
96C23×C61—28no—
97C23×C61—28no—
98C23×C61—28no—
99C23×C61—24no—
100C23×C61—24no—
101C23×C61—24no—
102C23×C61—24no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—24no—
116C23×C61—24no—
117C23×C61C22⋊C812yes—
118C23×C61C22⋊C812yes—
119C23×C61#9912yes—
120C23×C61#9912yes—
121C23×C61#10012yes—
122C23×C61#10112yes—
123C23×C61C2×C22⋊C411yescharacteristic
124C23×C61#11712yes—
125C23×C61C22×C811yescharacteristic
126C23×C61C2×M1611yescharacteristic
127C23×C61C2×M1612yes—
128C2×C4×C61—24no—
129C2×C4×C61—48no—
130C2×C4×C61—48no—
131C2×C4×C61—44no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—44no—
135C2×Dic61—22no—
136C2×Dic61—22no—
137C2×Dic61—22no—
138C2×Dic61—24no—
139C2×Dic61—22no—
140C2×Dic61—22no—
141C2×Dic61—44no—
142C2×Dic61—48no—
143C2×Dic61—48no—
144C2×Dic61—44no—
145C2×Dic61—48no—
146C2×Dic61—44no—
147C2×Dic61—48no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—44no—
151C2×Dic61#12314yes—
152#92420—84no—
153#8698—611nomaximal, Sylow
154C22×C61—216no—
155C22×C61—216no—
156C22×C61—22no—
157C22×C61—22no—
158C22×C61—22no—
159C22×C61—22no—
160C22×C61—22no—
161C22×C61—22no—
162C22×C61—22no—
163C22×C61—22no—
164C22×C61—22no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—22no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—22no—
171C22×C61—22no—
172C22×C61—22no—
173C22×C61—22no—
174C22×C61—22no—
175C22×C61—22no—
176C22×C61#32314yes—
177C22×C61#32414yes—
178C22×C61#38111yescharacteristic
179C22×C61C2×C22⋊C811yescharacteristic
180C22×C61#40711yescharacteristic
181C22×C61C2×#9911yescharacteristic
182C22×C61C2×#9911yescharacteristic
183C22×C61C2×#10011yescharacteristic
184C22×C61C2×#10111yescharacteristic
185C22×C61#43112yes—
186C22×C61#43212yes—
187C4×C61—44no—
188Dic61—24no—
189Dic61—42no—
190Dic61—44no—
191C22×#95—611no—
192#1779—612no—
193#1280—614no—
194C2×C61—24no—
195C2×C61—24no—
196C2×C61—22no—
197C2×C61—22no—
198C2×C61C2×#32311yescharacteristic
199C2×C61C2×#32411yescharacteristic
200C2×C61#146211yescharacteristic
201C2×C61#146311yescharacteristic
202C2×C61#181711yescharacteristic
203C2×C61#181911yescharacteristic
204C2×C61#182111yescharacteristic
205C24×C4—611no—
206C24×C4—611no—
207C24×C4#692311yescharacteristic
208C2×C22⋊C8—614no—
209C2×C22⋊C8—1222no—
210C2×C22⋊C8—1222no—
211C2×#95—612no—
212C2×#95—612no—
213C2×#95—1222no—
214C2×#95—1222no—
215C61#869811yescharacteristic, complemented, Sylow
216C25C2×#692311yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—24no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—611no—
223C23×C4—612no—
224C23×C4—612no—
225C23×C4—614no—
226C23×C4—611no—
227C23×C4—612no—
228C23×C4—1224no—
229C23×C4—1222no—
230C23×C4—1221no—
231C23×C4—1221no—
232C23×C4—2442no—
233C23×C4—2442no—
234C23×C4—2442no—
235C23×C4—2442no—
236C23×C4C2×#692312yes—
237C22×C8—1224no—
238C22⋊C8—1228no—
239C22⋊C8—2444no—
240C22⋊C8—2444no—
241#95—1224no—
242#95—1224no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24D61⋊C812yes—
250C24#1175821312yes—
251C24C22⋊#692311yescharacteristic
252C24C22⋊#692311yescharacteristic
253C24C22×#692311yescharacteristic
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—416no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—612no—
266C22×C4—614no—
267C22×C4—612no—
268C22×C4—614no—
269C22×C4—1224no—
270C22×C4—1224no—
271C22×C4—1224no—
272C22×C4—1224no—
273C22×C4—1224no—
274C22×C4—1224no—
275C22×C4—1222no—
276C22×C4—1222no—
277C22×C4—1222no—
278C22×C4—2441no—
279C22×C4—2442no—
280C22×C4—2441no—
281C22×C4—2442no—
282C22×C4—2441no—
283C22×C4—2441no—
284C22×C4—24416no—
285C22×C4—2442no—
286C22×C4—2442no—
287C22×C4—2448no—
288C22×C4—2442no—
289C22×C4—2442no—
290C22×C4—2442no—
291C22×C4—2442no—
292C22×C4—2448no—
293C22×C4#1175821714yes—
294C2×C8—2444no—
295C2×C8—2444no—
296C2×C8—2444no—
297C23—28no—
298C23—28no—
299C23—28no—
300C23—28no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—24no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—24no—
319C23#1451983412yes—
320C23#1451983412yes—
321C23#1451983512yes—
322C23#1451983612yes—
323C23#1451983712yes—
324C23#1451983812yes—
325C23C2×C22⋊#692311yescharacteristic
326C23#1451985612yes—
327C23C2×D61⋊C811yescharacteristic
328C23C2×#1175821712yes—
329C23C2×#1175821311yescharacteristic
330C2×C4—24no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—614no—
338C2×C4—1222no—
339C2×C4—1222no—
340C2×C4—1222no—
341C2×C4—1224no—
342C2×C4—1222no—
343C2×C4—1222no—
344C2×C4—2444no—
345C2×C4—2448no—
346C2×C4—2448no—
347C2×C4—2448no—
348C2×C4—2444no—
349C2×C4—2444no—
350C2×C4—2442no—
351C2×C4—2442no—
352C2×C4—2448no—
353C2×C4—2444no—
354C8—4884no—
355C22—216no—
356C22—216no—
357C22—22no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22#1684060614yes—
378C22#1684060814yes—
379C22#1684070511yescharacteristic
380C22C2×#1451983411yescharacteristic
381C22#1684073211yescharacteristic
382C22C2×#1451983511yescharacteristic
383C22C2×#1451983611yescharacteristic
384C22C2×#1451983711yescharacteristic
385C22C2×#1451983811yescharacteristic
386C22#1684077912yes—
387C22#1684078312yes—
388C4—44no—
389C4—1224no—
390C4—2444no—
391C4—2442no—
392C2—22no—
393C2—22no—
394C2—24no—
395C2—24no—
396C2C2×#1684060611yescharacteristic
397C2C2×#1684060811yescharacteristic
398C2#2146106411yescharacteristic
399C2#2146106711yescharacteristic
400C2#2146171011yescharacteristic
401C2#2146171411yescharacteristic
402C2#2146171811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC24×C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC25×C61C2×C4
cosocleC23×Dic61C23
hypercenterC24×C4#6923
hypocenterC61#8698
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C25 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
2. #21460657C2semidirect
26. C23×Dic61C8semidirect
215. C61#8698semidirect

Maximal subgroups

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

subgroupindex
#214616412
#214606572
C22×#145196842
#869861