Finite Group Databasebeta

← #30601257

Subgroups — #30601257

Order 15616.

524 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 82 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#21462670C211yesmaximal, characteristic
2#21461718C212yesmaximal, complemented
3#21461075C211yesmaximal, characteristic
4#21460720C212yesmaximal, complemented
5#21450133C211yesmaximal, characteristic
6#16840793—24no—
7#16840782—21no—
8#16840732C2212yes—
9#16840732C2212yes—
10C2×C22⋊#92420—21no—
11#16840642—24no—
12#16840607—21no—
13#16840605—21no—
14C2×D4×Dic61C2211yescharacteristic
15C22×#11758196C2211yescharacteristic
16#16839704—21no—
17#16839698—21no—
18#16839659—21no—
19#16839642C411yescharacteristic
20#16839637C411yescharacteristic
21#16839635C411yescharacteristic
22#16839620—21no—
23#16839620C2211yescharacteristic
24C61×#386C411yescharacteristic
25C2×#11758217—22no—
26C2×#11758217D414yes—
27C22×#92420—22no—
28#14519844—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31C22⋊#92420—22no—
32C22⋊#92420—22no—
33C22⋊#92420—22no—
34#14519832—22no—
35C23×Dic61C2×C411yescharacteristic
36C23×Dic61D411yescharacteristic
37C23×Dic61C2311yescharacteristic
38D4×Dic61—22no—
39D4×Dic61—22no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758196—21no—
43C2×#11758196—22no—
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×#11758196—22no—
47C2×#11758196—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61C2×C411yescharacteristic
50C2×C22⋊Dic61C2×C411yescharacteristic
51C2×C22⋊Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C22×D4×C61C2×C411yescharacteristic
58C23×C4×C61C2×C411yescharacteristic
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61C2×C411yescharacteristic
61#95×C61—21no—
62C24×C61C22⋊C411yescharacteristic
63C24×C61C22×C411yescharacteristic
64C2×D4×C61—22no—
65C2×D4×C61—22no—
66C2×D4×C61—41no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—22no—
70C22×C4×C61—81no—
71C22×C4×C61C22⋊C411yescharacteristic
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—22no—
77C22×Dic61—42no—
78C22×Dic61—42no—
79C22×Dic61—81no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C2×D412yes—
83C22×Dic61C2×D411yescharacteristic
84C22×Dic61C2×D411yescharacteristic
85C22×Dic61#4112yes—
86#11758217—44no—
87#11758217—44no—
88C2×#92420—24no—
89C2×#92420—42no—
90C2×#92420—82no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—81no—
93C22⋊Dic61—22no—
94C22⋊Dic61—22no—
95C22⋊Dic61—41no—
96C22⋊Dic61—81no—
97C4⋊Dic61—22no—
98C4⋊Dic61—81no—
99#11758196—42no—
100#11758196—41no—
101#11758196—42no—
102#11758196—41no—
103#11758196—81no—
104C4×Dic61—21no—
105C4×Dic61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61#10112yes—
115C23×C61C2×C22⋊C411yescharacteristic
116C23×C61C4×D412yes—
117D4×C61—44no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—82no—
128C2×Dic61—22no—
129C2×Dic61—24no—
130C2×Dic61—21no—
131C2×Dic61—21no—
132C2×Dic61—41no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—81no—
144C2×Dic61—82no—
145C2×Dic61#12011yescharacteristic
146C2×Dic61#12112yes—
147C2×Dic61#12311yescharacteristic
148#92420—44no—
149#92420—82no—
150#9509—611nomaximal, Sylow
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—22no—
154C22×C61—22no—
155C22×C61—22no—
156C22×C61—22no—
157C22×C61—42no—
158C22×C61—42no—
159C22×C61—42no—
160C22×C61—41no—
161C22×C61—41no—
162C22×C61—81no—
163C22×C61#38611yescharacteristic
164C22×C61C2×#10111yescharacteristic
165C22×C61#41311yescharacteristic
166C4×C61—41no—
167C4×C61—81no—
168C4×C61—81no—
169Dic61—22no—
170Dic61—21no—
171Dic61—41no—
172Dic61—41no—
173Dic61—81no—
174Dic61—81no—
175Dic61#45812yes—
176#2973—611no—
177#2487—611no—
178#1821—612no—
179#1465—611no—
180#1305—612no—
181C2×C61—21no—
182C2×C61—21no—
183C2×C61—42no—
184C2×C61—42no—
185C2×C61#189011yescharacteristic
186C2×C61#190111yescharacteristic
187C2×C61#191411yescharacteristic
188C2×C4×D4—611no—
189C22×C4⋊C4—611no—
190#435—1224no—
191#432—1221no—
192#407—612no—
193#407—612no—
194C2×C22⋊C8—1221no—
195#400—1221no—
196#399—1221no—
197#389—1221no—
198#386—611no—
199#386#692311yescharacteristic
200#385—611no—
201#385—611no—
202#380—611no—
203#380—1221no—
204#343—1224no—
205#324—1221no—
206#323—1221no—
207C61#950911yescharacteristic, complemented, Sylow
208C22×D4C2×#692311yescharacteristic
209C23×C4—611no—
210C23×C4—611no—
211C23×C4—611no—
212C23×C4C2×#692311yescharacteristic
213C2×M16—614no—
214C2×M16—1222no—
215C22×C8—1222no—
216C4×D4—1222no—
217C4×D4—1222no—
218C2×C4⋊C4—611no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4—1221no—
221C2×C4⋊C4—1222no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4—1221no—
224C2×C4⋊C4—1222no—
225C2×C4⋊C4—1221no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—611no—
228C2×C22⋊C4—611no—
229C2×C22⋊C4—611no—
230C2×C22⋊C4—1221no—
231C2×C22⋊C4C2×#692311yescharacteristic
232C2×C42—611no—
233C4⋊C8—1222no—
234C22⋊C8—1222no—
235C22⋊C8—1222no—
236C22⋊C8—1222no—
237C22⋊C8—1222no—
238C22⋊C8—1222no—
239#97—1222no—
240#95—21no—
241#95—1221no—
242#95—1221no—
243#95—1221no—
244#95—1221no—
245C24C22⋊#692311yescharacteristic
246C24C22×#692311yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—81no—
254C22×C4—611no—
255C22×C4—612no—
256C22×C4—611no—
257C22×C4—612no—
258C22×C4—611no—
259C22×C4—611no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—1222no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—4881no—
266C22×C4—2442no—
267C22×C4—2442no—
268C22×C4C22⋊#692311yescharacteristic
269M16—2444no—
270M16—2444no—
271C2×C8—1224no—
272C2×C8—4882no—
273C2×C8—2442no—
274C4⋊C4—1222no—
275C4⋊C4—4881no—
276C4⋊C4—4881no—
277C4⋊C4—2441no—
278C4⋊C4—2442no—
279C4⋊C4—2441no—
280C4⋊C4—2442no—
281C22⋊C4—41no—
282C22⋊C4—81no—
283C22⋊C4—1222no—
284C22⋊C4—1222no—
285C22⋊C4—4881no—
286C22⋊C4—2441no—
287C42—1221no—
288C42—1221no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23#1451983812yes—
298C23C2×C22⋊#692311yescharacteristic
299C23D4×#692312yes—
300D4—44no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—82no—
311C2×C4—611no—
312C2×C4—612no—
313C2×C4—611no—
314C2×C4—1222no—
315C2×C4—1224no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—4881no—
319C2×C4—4882no—
320C2×C4—2442no—
321C2×C4—2441no—
322C2×C4—2442no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4—2441no—
331C8—4882no—
332C8—2444no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—81no—
345C22#1684071011yescharacteristic
346C22C2×#1451983811yescharacteristic
347C22#1684074911yescharacteristic
348C4—41no—
349C4—81no—
350C4—81no—
351C4—612no—
352C4—1222no—
353C4—1221no—
354C4—4881no—
355C4—4881no—
356C4—2441no—
357C4—2441no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—42no—
362C2#2146182911yescharacteristic
363C2#2146184611yescharacteristic
364C2#2146187011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC61×#386C4
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC23×Dic61C23
hypercenter#386#6923
hypocenterC61#9509
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #21461718C2semidirect
4. #21460720C2semidirect
207. C61#9509semidirect

Maximal subgroups

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

subgroupindex
#214626702
#214617182
#214610752
#214607202
#214501332
#950961