Finite Group Databasebeta

← #30595734

Subgroups — #30595734

Order 15616.

550 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 92 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#21461546C211yesmaximal, characteristic, complemented
2C4×#14519834C211yesmaximal, characteristic, complemented
3#21460637C212yesmaximal, complemented
4#21460635C212yesmaximal, complemented
5#21454149C211yesmaximal, characteristic
6C2×#14519834—21no—
7C2×#14519834—21no—
8C2×#14519832—21no—
9C2×C4×#92420—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519759—21no—
13C2×#14519748C411yescharacteristic
14C2×C42×D61C2211yescharacteristic
15#16839665C2211yescharacteristic
16#16839661C2211yescharacteristic
17#16839660C411yescharacteristic
18#16839656—21no—
19#16839655—21no—
20#16839638—21no—
21#16839619C411yescharacteristic
22C61×#390C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25#14519834—24no—
26#14519834—24no—
27#14519832—42no—
28C4×#92420—42no—
29C22×C22⋊D61—21no—
30C22×C4×D61C2×C411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C22×C4×D61D411yescharacteristic
33#14519759—44no—
34#14519748—42no—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—21no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×#11758198—21no—
43C2×C4×Dic61D411yescharacteristic
44C2×C4×Dic61D411yescharacteristic
45C2×C4×Dic61C2311yescharacteristic
46C42×D61—42no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50C22×D4×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×C61—21no—
54C2×C42×C61C2×C411yescharacteristic
55C24×C61—21no—
56C23×D61C22⋊C411yescharacteristic
57C2×D4×C61—22no—
58C2×D4×C61—41no—
59C2×D4×C61—41no—
60C22×C4×C61—21no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C2×C22⋊D61—42no—
65C2×C22⋊D61—42no—
66C2×C22⋊D61—42no—
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71C22×Dic61C2×D411yescharacteristic
72C2×C4×D61—22no—
73C2×C4×D61—22no—
74C2×C4×D61—22no—
75C2×C4×D61—22no—
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—41no—
79C2×C4×D61—41no—
80C2×C4×D61—41no—
81C2×#92420—24no—
82C2×#92420—24no—
83C2×#92420—42no—
84C2×#92420—42no—
85C2×#92420—42no—
86C2×#92420—42no—
87C4⋊C4×C61—22no—
88C22⋊C4×C61—42no—
89C22⋊C4×C61—42no—
90C42×C61—41no—
91C22⋊Dic61—42no—
92C22⋊Dic61—42no—
93C22⋊Dic61—42no—
94#11758198—42no—
95C4⋊Dic61—42no—
96C4⋊Dic61—41no—
97C4⋊Dic61—41no—
98#11758196—42no—
99C4×Dic61—22no—
100C4×Dic61—22no—
101C4×Dic61—41no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61—41no—
109C23×C61C2×C22⋊C411yescharacteristic
110C22×D61—21no—
111C22×D61—21no—
112C22×D61—21no—
113C22×D61—21no—
114C22×D61—21no—
115C22×D61—21no—
116C22×D61#10412yes—
117D4×C61—82no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61#10412yes—
129C2×C4×C61#13112yes—
130C22⋊D61—84no—
131C2×Dic61—21no—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—21no—
136C2×Dic61—21no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61#9912yes—
144C2×Dic61#13512yes—
145C2×Dic61AGL(1,ℤ/8)12yes—
146C4×D61—44no—
147C4×D61—44no—
148C4×D61—44no—
149#92420—44no—
150#92420—44no—
151#7674—611nomaximal, Sylow
152C22×C61—81no—
153C22×C61—81no—
154C22×C61—81no—
155C22×C61—81no—
156C22×C61—81no—
157C22×C61—81no—
158C22×C61—81no—
159C22×C61#40811yescharacteristic
160C22×C61C2×#9911yescharacteristic
161C22×C61#41311yescharacteristic
162C22×C61#41611yescharacteristic
163C22×C61#41811yescharacteristic
164C22×C61C2×#10411yescharacteristic
165C22×C61C2×#10411yescharacteristic
166C2×D61—22no—
167C2×D61—22no—
168C2×D61—22no—
169C2×D61—41no—
170C2×D61—41no—
171C2×D61—41no—
172C2×D61—41no—
173C4×C61—22no—
174C4×C61—22no—
175C4×C61—41no—
176C4×C61—81no—
177Dic61—22no—
178Dic61—22no—
179Dic61—22no—
180Dic61—81no—
181Dic61—81no—
182#2854—611no—
183#1749—611no—
184C4×C22⋊C8—611no—
185#1271—612no—
186#1270—612no—
187C2×C61—161no—
188C2×C61#146811yescharacteristic
189C2×C61#148411yescharacteristic
190C2×C61#149411yescharacteristic
191C2×C61#150111yescharacteristic
192C2×C61#150711yescharacteristic
193C2×C61#152411yescharacteristic
194C2×C61#155311yescharacteristic
195D61—42no—
196C2×#123—1221no—
197C2×#121—611no—
198C22×C42—611no—
199C2×C22⋊C8—1221no—
200C2×C22⋊C8—1221no—
201C2×#97—1221no—
202C2×C4×C8—1221no—
203#390—611no—
204#390#692311yescharacteristic
205#389—611no—
206#389—611no—
207#388—1221no—
208#388—1221no—
209#386—1221no—
210#380—611no—
211#336—612no—
212#336—612no—
213C61#767411yescharacteristic, complemented, Sylow
214C22×D4—1221no—
215C22×D4C2×#692311yescharacteristic
216C23×C4—611no—
217C23×C4—611no—
218C23×C4—611no—
219C22×C8—1222no—
220C22×C8—1222no—
221#123—2444no—
222#121—2442no—
223C2×C4⋊C4—611no—
224C2×C4⋊C4—611no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1221no—
227C2×C4⋊C4C2×#692311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—1221no—
231C2×C22⋊C4—1221no—
232C2×C22⋊C4—1221no—
233C2×C22⋊C4—1221no—
234C2×C42—611no—
235C2×C42—611no—
236C2×C42—611no—
237C2×C42—2442no—
238C2×C42C2×#692311yescharacteristic
239C22⋊C8—1224no—
240C22⋊C8—1224no—
241#97—2442no—
242C4×C8—2442no—
243#95—1221no—
244#95—1221no—
245#95—1221no—
246C24—21no—
247C24—611no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—2442no—
252C2×D4—2442no—
253C2×D4—2442no—
254C22×C4—21no—
255C22×C4—611no—
256C22×C4—611no—
257C22×C4—611no—
258C22×C4—1222no—
259C22×C4—1222no—
260C22×C4—1222no—
261C22×C4—1222no—
262C22×C4—1222no—
263C22×C4—1222no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—2441no—
267C22×C4—2441no—
268C22×C4—2441no—
269C22×C4C22⋊#692311yescharacteristic
270C22×C4C22⋊#692311yescharacteristic
271C22×C4C22×#692311yescharacteristic
272C2×C8—1224no—
273C2×C8—1224no—
274C2×C8—2442no—
275C2×C8—2442no—
276C2×C8—2442no—
277C2×C8—2442no—
278C4⋊C4—22no—
279C4⋊C4—2441no—
280C4⋊C4—2441no—
281C4⋊C4—2442no—
282C4⋊C4—2442no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—2442no—
286C22⋊C4—2442no—
287C22⋊C4—2442no—
288C22⋊C4—2442no—
289C42—41no—
290C42—1222no—
291C42—1222no—
292C42—2441no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—612no—
301C23—1221no—
302C23—1221no—
303C23—1221no—
304C23—1221no—
305C23—1221no—
306C23—1221no—
307C23C2×C22⋊#692311yescharacteristic
308D4—82no—
309D4—4884no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—612no—
321C2×C4—612no—
322C2×C4—612no—
323C2×C4—1221no—
324C2×C4—1221no—
325C2×C4—1221no—
326C2×C4—1221no—
327C2×C4—1221no—
328C2×C4—1221no—
329C2×C4—2444no—
330C2×C4—2444no—
331C2×C4—2444no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4#1451984112yes—
339C2×C4#1451986412yes—
340C8—2444no—
341C8—2444no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—1222no—
350C22—1222no—
351C22—1222no—
352C22—2441no—
353C22—2441no—
354C22—2441no—
355C22—2441no—
356C22#1684073611yescharacteristic
357C22D61⋊#9911yescharacteristic
358C22#1684074811yescharacteristic
359C22#1684075211yescharacteristic
360C22#1684075411yescharacteristic
361C22C2×#1451984111yescharacteristic
362C22D61⋊#10411yescharacteristic
363C4—22no—
364C4—22no—
365C4—41no—
366C4—81no—
367C4—1222no—
368C4—1222no—
369C4—1222no—
370C4—4881no—
371C4—4881no—
372C2—161no—
373C2—2442no—
374C2#2146108511yescharacteristic
375C2#2146113911yescharacteristic
376C2#2146117111yescharacteristic
377C2#2146119511yescharacteristic
378C2#2146121411yescharacteristic
379C2#2146126911yescharacteristic
380C2#2146134211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519748C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#7674
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. #21461546C2semidirect
2. C4×#14519834C2semidirect
3. #21460637C2semidirect
4. #21460635C2semidirect
213. C61#7674semidirect

Maximal subgroups

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

subgroupindex
#214615462
C4×#145198342
#214606372
#214606352
#214541492
#767461