Finite Group Databasebeta

← #29573569

Subgroups — #29573569

Order 15616.

472 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 115 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#21445807C211yesmaximal, characteristic
2#21444137C211yesmaximal, characteristic
3#21443698C212yesmaximal, complemented
4#21443686C212yesmaximal, complemented
5C61×#2353C211yesmaximal, characteristic
6C2×#14519692—21no—
7C2×C22⋊#92418C2212yes—
8#16839622C2212yes—
9#16839529—21no—
10#16839488—21no—
11#16839486—21no—
12C2×C4×D4×C61C2211yescharacteristic
13C61×#388—21no—
14C61×#387—21no—
15C61×#386—21no—
16C61×#383C411yescharacteristic, complemented
17C61×#381C411yescharacteristic, complemented
18C61×#380C2211yescharacteristic
19C4×C22⋊C4×C61—21no—
20C4×C22⋊C4×C61C411yescharacteristic, complemented
21C61×#376C411yescharacteristic, complemented
22C2×#95×C61C2211yescharacteristic
23C22×#92418—22no—
24C2×C4⋊Dic61—22no—
25C2×C22⋊Dic61—22no—
26#14519692—42no—
27C22⋊#92418—22no—
28C22⋊#92418—22no—
29#14519684—22no—
30C22×D4×C61D411yescharacteristic
31C23×C4×C61—21no—
32C23×C4×C61C2×C411yescharacteristic
33C23×C4×C61C2311yescharacteristic
34C4×D4×C61—22no—
35C4×D4×C61—22no—
36C2×C4⋊C4×C61—21no—
37C2×C4⋊C4×C61D411yescharacteristic
38C2×C22⋊C4×C61—21no—
39C2×C22⋊C4×C61—21no—
40C2×C22⋊C4×C61—21no—
41C2×C22⋊C4×C61D411yescharacteristic
42C2×C22⋊C4×C61D411yescharacteristic
43C2×C42×C61—21no—
44C2×C42×C61—21no—
45C2×C42×C61C2×C411yescharacteristic
46#95×C61—21no—
47#95×C61—21no—
48#95×C61—21no—
49#95×C61—21no—
50#95×C61—21no—
51#95×C61C2×C411yescharacteristic
52#95×C61C2×C411yescharacteristic
53#95×C61C2×C411yescharacteristic
54#95×C61C2×C411yescharacteristic
55C24×C61C22⋊C411yescharacteristic
56C24×C61C2×D411yescharacteristic
57C2×D4×C61—22no—
58C2×D4×C61—22no—
59C2×D4×C61—21no—
60C2×D4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—42no—
68C22×C4×C61—41no—
69C22×C4×C61—41no—
70C22×C4×C61—81no—
71C22×C4×C61—81no—
72C22×C4×C61—81no—
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61#4112yes—
79C22×C4×C61#4112yes—
80C22×Dic61—42no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61—81no—
83C22⋊C4×C61—22no—
84C22⋊C4×C61—22no—
85C22⋊C4×C61—42no—
86C22⋊C4×C61—42no—
87C22⋊C4×C61—81no—
88C42×C61—21no—
89C42×C61—21no—
90C42×C61—42no—
91C42×C61—81no—
92C22⋊Dic61—42no—
93C22⋊Dic61—42no—
94C4⋊Dic61—82no—
95C2×#92418—42no—
96C2×#92418—82no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61C2×C22⋊C411yescharacteristic
105C23×C61#12312yes—
106C23×C61#13512yes—
107C23×C61AGL(1,ℤ/8)11yescharacteristic
108C23×C61#13711yescharacteristic
109D4×C61—42no—
110D4×C61—42no—
111C2×C4×C61—22no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
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—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—81no—
135C2×C4×C61—81no—
136C2×C4×C61—82no—
137C2×C4×C61—82no—
138C2×C4×C61#11711yescharacteristic
139C2×C4×C61#11711yescharacteristic
140C2×C4×C61#12312yes—
141C2×Dic61—82no—
142C2×Dic61—82no—
143C2×Dic61—82no—
144#92418—82no—
145#8762—611nomaximal, Sylow
146C22×C61—22no—
147C22×C61—21no—
148C22×C61—22no—
149C22×C61—21no—
150C22×C61—41no—
151C22×C61—41no—
152C22×C61—42no—
153C22×C61—41no—
154C22×C61—41no—
155C22×C61—41no—
156C22×C61—81no—
157C22×C61—81no—
158C22×C61#38111yescharacteristic
159C22×C61#41611yescharacteristic
160C22×C61#41711yescharacteristic
161C22×C61#48212yes—
162C22×C61#48512yes—
163C4×C61—22no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—42no—
167C4×C61—41no—
168C4×C61—41no—
169C4×C61—41no—
170C4×C61—81no—
171C4×C61—81no—
172C4×C61—81no—
173Dic61—162no—
174#2353D6111yescharacteristic
175#1799—611no—
176#1459—611no—
177#1294—612no—
178#1289—612no—
179C2×C61—22no—
180C2×C61—21no—
181C2×C61—21no—
182C2×C61—22no—
183C2×C61—81no—
184C2×C61#182511yescharacteristic
185C2×C61#182711yescharacteristic
186C2×C61#182811yescharacteristic
187C2×C4×D4C2×D6111yescharacteristic
188C2×#99—1221no—
189C2×C22⋊C8—612no—
190#388—21no—
191#387—21no—
192#386—21no—
193#383Dic6111yescharacteristic, complemented
194#381Dic6111yescharacteristic, complemented
195#380—612no—
196#380C2×D6111yescharacteristic
197C4×C22⋊C4—21no—
198C4×C22⋊C4Dic6111yescharacteristic, complemented
199#376Dic6111yescharacteristic, complemented
200C2×#95C2×D6111yescharacteristic
201#342—1221no—
202#324—1221no—
203#323—1221no—
204C61#876211yescharacteristic, complemented, Sylow
205C22×D4C22⋊D6111yescharacteristic
206C23×C4—21no—
207C23×C4C2×Dic6111yescharacteristic
208C23×C4C22×D6111yescharacteristic
209C22×C8—1222no—
210C4×D4—22no—
211C4×D4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—1222no—
214C2×C4⋊C4C22⋊D6111yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—1222no—
219C2×C22⋊C4C22⋊D6111yescharacteristic
220C2×C22⋊C4C22⋊D6111yescharacteristic
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C2×Dic6111yescharacteristic
224#99—2442no—
225C22⋊C8—1222no—
226C22⋊C8—1222no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—1222no—
233#95C2×Dic6111yescharacteristic
234#95C2×Dic6111yescharacteristic
235#95C2×Dic6111yescharacteristic
236#95C2×Dic6111yescharacteristic
237C24C22⋊Dic6111yescharacteristic
238C24C2×C22⋊D6111yescharacteristic
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—2442no—
256C22×C4C22⋊Dic6111yescharacteristic
257C22×C4C22⋊Dic6111yescharacteristic
258C22×C4C22⋊Dic6111yescharacteristic
259C22×C4#1175822212yes—
260C22×C4#1175822212yes—
261C22×C4C22×Dic6111yescharacteristic
262C22×C4C2×C22⋊D6111yescharacteristic
263C2×C8—4882no—
264C2×C8—2442no—
265C4⋊C4—22no—
266C4⋊C4—81no—
267C4⋊C4—4882no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—81no—
273C22⋊C4—2442no—
274C22⋊C4—2442no—
275C42—21no—
276C42—21no—
277C42—42no—
278C42—81no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23C2×C22⋊Dic6111yescharacteristic
287C23#1451976012yes—
288C23#1451979712yes—
289C23#1451980211yescharacteristic
290C23#1451980711yescharacteristic
291D4—42no—
292D4—42no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—4882no—
321C2×C4—4882no—
322C2×C4—4882no—
323C2×C4#1451973611yescharacteristic
324C2×C4#1451973611yescharacteristic
325C2×C4#1451976012yes—
326C8—4882no—
327C22—21no—
328C22—22no—
329C22—21no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22#1683962511yescharacteristic
340C22#1683975811yescharacteristic
341C22#1683976311yescharacteristic
342C22#1684008612yes—
343C22#1684010312yes—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—9762no—
355C2—21no—
356C2—21no—
357C2—22no—
358C2—22no—
359C2—81no—
360C2#2144591411yescharacteristic
361C2#2144592311yescharacteristic
362C2#2144592711yescharacteristic
subgroupG/H
centerC22#16839625
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2353C2
socleC22×C61#381
prime radicalC22×D4×C61D4
cosocleC23×C4×C61C23
hypercenter#2353D61
hypocenterC61#8762
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2353

Complements

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

NH ≅ G/Ntype
3. #21443698C2semidirect
4. #21443686C2semidirect
16. C61×#383C4semidirect
17. C61×#381C4semidirect
20. C4×C22⋊C4×C61C4semidirect
21. C61×#376C4semidirect
193. #383Dic61semidirect
194. #381Dic61semidirect
198. C4×C22⋊C4Dic61semidirect
199. #376Dic61semidirect
204. C61#8762semidirect

Maximal subgroups

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

subgroupindex
#214458072
#214441372
#214436982
#214436862
C61×#23532
#876261