Finite Group Databasebeta

← #30600533

Subgroups — #30600533

Order 15616.

504 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 94 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#21462461C211yesmaximal, characteristic
2#21461332C212yesmaximal, complemented
3#21461051C211yesmaximal, characteristic
4#21460672C212yesmaximal, complemented
5#21451639C211yesmaximal, characteristic
6C22×#11758217—21no—
7#16840779C2212yes—
8#16840768—24no—
9#16840765—21no—
10#16840765C2212yes—
11#16840622—24no—
12#16840608—21no—
13#16840605—21no—
14C2×#14519775C2211yescharacteristic
15C2×#14519733C2211yescharacteristic
16#16839703—21no—
17#16839698—21no—
18#16839683C411yescharacteristic
19#16839681C411yescharacteristic
20#16839674C411yescharacteristic
21#16839659—21no—
22#16839626C2211yescharacteristic
23#16839620—21no—
24C61×#393C411yescharacteristic
25C2×#11758217—22no—
26C2×#11758217—22no—
27C2×#11758217—42no—
28C2×#11758217—42no—
29C22×#92420—22no—
30#14519844—22no—
31#14519844—22no—
32#14519844D414yes—
33C4⋊#92420—22no—
34C4⋊#92420—22no—
35C22⋊#92420—22no—
36C4×#92420—22no—
37C22×Dic122C2×C411yescharacteristic
38C23×Dic61D411yescharacteristic
39#14519775—22no—
40#14519775—22no—
41#14519733—22no—
42#14519733—22no—
43C2×C4⋊Dic61—21no—
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196C2×C411yescharacteristic
49C2×#11758196D411yescharacteristic
50C2×C22⋊Dic61—21no—
51C2×C22⋊Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61C2311yescharacteristic
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684C2×C411yescharacteristic
59C22×Q8×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61C2×C411yescharacteristic
62#95×C61—21no—
63#95×C61C2×C411yescharacteristic
64C24×C61C22⋊C411yescharacteristic
65C2×Q8×C61—22no—
66C2×Q8×C61—81no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61SD1612yes—
80C22×Dic61Q1612yes—
81C22×Dic61C2×D411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C2×Dic122—22no—
84C2×Dic122—81no—
85#11758217—44no—
86#11758217—42no—
87#11758217—42no—
88C2×#92420—24no—
89C2×#92420—44no—
90C2×#92420—42no—
91C22⋊C4×C61—22no—
92C22⋊C4×C61—81no—
93C22⋊Dic61—22no—
94C22⋊Dic61—81no—
95C4⋊Dic61—81no—
96#11758196—22no—
97#11758196—22no—
98#11758196—41no—
99#11758196—42no—
100#11758196—41no—
101#11758196—41no—
102#11758196—41no—
103#11758196—81no—
104C4×Dic61—21no—
105C4×Dic61—21no—
106C4×Dic61C2×D412yes—
107C4×Dic61#4112yes—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61#10312yes—
113C23×C61C2×C22⋊C411yescharacteristic
114Q8×C61—82no—
115C2×C4×C61—21no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61C4×D412yes—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—22no—
128C2×Dic61—22no—
129C2×Dic61—22no—
130C2×Dic61—22no—
131C2×Dic61—21no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61#10312yes—
141C2×Dic61#12011yescharacteristic
142C2×Dic61#12112yes—
143C2×Dic61#12311yescharacteristic
144C2×Dic61C2×SD1611yescharacteristic
145C2×Dic61C2×Q1611yescharacteristic
146C2×Dic61AGL(1,ℤ/8)11yescharacteristic
147C2×Dic61#13711yescharacteristic
148Dic122—82no—
149#92420—44no—
150#92420—82no—
151#9191—611nomaximal, Sylow
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—22no—
155C22×C61—42no—
156C22×C61—41no—
157C22×C61—41no—
158C22×C61—81no—
159C22×C61#38611yescharacteristic
160C22×C61C2×#10311yescharacteristic
161C22×C61#41711yescharacteristic
162C4×C61—22no—
163C4×C61—41no—
164C4×C61—81no—
165C4×C61—81no—
166Dic61—22no—
167Dic61—22no—
168Dic61—41no—
169Dic61—81no—
170Dic61—81no—
171Dic61#45011yescharacteristic
172Dic61#45111yescharacteristic
173Dic61#48311yescharacteristic
174Dic61#48411yescharacteristic
175#2905—611no—
176#2628—611no—
177#1548—612no—
178#1458—611no—
179#1286—612no—
180C2×C61—21no—
181C2×C61—21no—
182C2×C61—42no—
183C2×C61#187211yescharacteristic
184C2×C61#190911yescharacteristic
185C2×C61#191411yescharacteristic
186C22×M16—1221no—
187C2×C4⋊Q8—611no—
188C2×#117—611no—
189#431—612no—
190#424—1224no—
191#423—612no—
192#423—1221no—
193#400—1221no—
194#399—1221no—
195#395—611no—
196#395—611no—
197#393—611no—
198#393#692311yescharacteristic
199#389—1221no—
200#382—611no—
201#380—1221no—
202#333—1224no—
203#324—1221no—
204#323—1221no—
205C61#919111yescharacteristic, complemented, Sylow
206C22×Q8—611no—
207C22×Q8C2×#692311yescharacteristic
208C23×C4—611no—
209C2×M16—1222no—
210C2×M16—1222no—
211C2×M16—2442no—
212C2×M16—2442no—
213C22×C8—1222no—
214C4⋊Q8—1222no—
215C4⋊Q8—1222no—
216#117—1222no—
217#117—1222no—
218C2×C4⋊C4—611no—
219C2×C4⋊C4—611no—
220C2×C4⋊C4—1221no—
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4—1221no—
224C2×C4⋊C4—1221no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—611no—
227C2×C22⋊C4—1221no—
228C2×C22⋊C4C2×#692311yescharacteristic
229C2×C42—611no—
230C2×C42—611no—
231C4⋊C8—614no—
232C4⋊C8—1222no—
233C4⋊C8—1222no—
234C4⋊C8—1222no—
235C4⋊C8—1222no—
236C22⋊C8—1222no—
237C4×C8—1222no—
238#95—21no—
239#95—611no—
240#95—1221no—
241#95—1221no—
242#95—1221no—
243#95—1221no—
244#95C2×#692311yescharacteristic
245C24C22⋊#692311yescharacteristic
246C2×Q8—22no—
247C2×Q8—81no—
248C2×Q8—1222no—
249C2×Q8—4881no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—611no—
253C22×C4—612no—
254C22×C4—612no—
255C22×C4—611no—
256C22×C4—611no—
257C22×C4—611no—
258C22×C4—1221no—
259C22×C4—1221no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—1221no—
263C22×C4—1221no—
264C22×C4C22⋊#692311yescharacteristic
265C22×C4C22×#692311yescharacteristic
266M16—2442no—
267M16—2442no—
268M16—2444no—
269C2×C8—1224no—
270C2×C8—2442no—
271C2×C8—2444no—
272C4⋊C4—1222no—
273C4⋊C4—1222no—
274C4⋊C4—4881no—
275C4⋊C4—4881no—
276C4⋊C4—2441no—
277C4⋊C4—2441no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C4⋊C4—2442no—
281C22⋊C4—22no—
282C22⋊C4—81no—
283C22⋊C4—1222no—
284C22⋊C4—4881no—
285C42—612no—
286C42—612no—
287C42—1221no—
288C42—1221no—
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23#1451984012yes—
294C23C2×C22⋊#692311yescharacteristic
295Q8—82no—
296Q8—4882no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—611no—
307C2×C4—611no—
308C2×C4—611no—
309C2×C4—611no—
310C2×C4—611no—
311C2×C4—611no—
312C2×C4—612no—
313C2×C4—612no—
314C2×C4—1222no—
315C2×C4—1222no—
316C2×C4—1222no—
317C2×C4—1222no—
318C2×C4—1221no—
319C2×C4—1221no—
320C2×C4—1221no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4D4×#692312yes—
330C8—4882no—
331C8—2444no—
332C22—21no—
333C22—21no—
334C22—22no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22#1684071011yescharacteristic
340C22C2×#1451984011yescharacteristic
341C22#1684075311yescharacteristic
342C4—22no—
343C4—41no—
344C4—81no—
345C4—81no—
346C4—611no—
347C4—611no—
348C4—611no—
349C4—611no—
350C4—1222no—
351C4—1222no—
352C4—4881no—
353C4—4881no—
354C4—2441no—
355C2—21no—
356C2—21no—
357C2—42no—
358C2#2146180411yescharacteristic
359C2#2146186011yescharacteristic
360C2#2146187011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#393C4
socleC22×C61#386
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#393#6923
hypocenterC61#9191
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #21461332C2semidirect
4. #21460672C2semidirect
205. C61#9191semidirect

Maximal subgroups

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

subgroupindex
#214624612
#214613322
#214610512
#214606722
#214516392
#919161