Finite Group Databasebeta

← #29576936

Subgroups — #29576936

Order 15616.

492 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 103 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#21455918C211yesmaximal, characteristic
2#21446762C211yesmaximal, characteristic, complemented
3#21444330C211yesmaximal, characteristic, complemented
4#21444262C211yesmaximal, characteristic
5#21443712C211yesmaximal, characteristic, complemented
6#21443675C211yesmaximal, characteristic, complemented
7C61×#2319C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840097—21no—
10#16840042C412yescomplemented
11#16839967C412yescomplemented
12#16839913—21no—
13#16839862—22no—
14#16839819C2211yescharacteristic
15#16839819C2211yescharacteristic
16C4×#11758194C2211yescharacteristic
17#16839522—22no—
18#16839498—21no—
19#16839498—21no—
20D4⋊#92418—22no—
21C2×C4×D4×C61C2211yescharacteristic
22C61×#390—21no—
23C61×#389—21no—
24C61×#388—22no—
25C61×#381—21no—
26C61×#380—21no—
27C4×C4⋊C4×C61C2211yescharacteristic
28C61×#376C2211yescharacteristic
29C2×#11758200—22no—
30C2×#11758194—21no—
31#14519777C2×C411yescharacteristic
32#14519777C2×C411yescharacteristic
33#14519769—21no—
34#14519756—21no—
35#14519756—21no—
36C4×Dic122—21no—
37C4×Dic122—21no—
38#14519736—21no—
39#14519710—22no—
40C4⋊#92418—21no—
41C4⋊#92418C2×C412yes—
42#14519700—22no—
43#14519698—22no—
44C22⋊#92418—21no—
45C22⋊#92418—21no—
46#14519688—21no—
47#14519688—21no—
48#14519688—21no—
49C4×#92418—21no—
50C22×D4×C61D411yescharacteristic
51C23×C4×C61—21no—
52C4×D4×C61—21no—
53C4×D4×C61—21no—
54C4×D4×C61C2×C412yes—
55C4×D4×C61D412yes—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61—22no—
58C2×C4⋊C4×C61D411yescharacteristic
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61—22no—
61C2×C22⋊C4×C61—22no—
62C2×C42×C61—21no—
63C2×C42×C61D412yes—
64C2×C42×C61C2311yescharacteristic
65#95×C61—22no—
66#95×C61—21no—
67#95×C61—21no—
68#95×C61D411yescharacteristic
69#95×C61D411yescharacteristic
70C24×C61—21no—
71C2×D4×C61—21no—
72C2×D4×C61—42no—
73C2×D4×C61C22⋊C412yes—
74C22×C4×C61—21no—
75C22×C4×C61—22no—
76C22×C4×C61—22no—
77C22×C4×C61—21no—
78C22×C4×C61—22no—
79C22×C4×C61—21no—
80C22×C4×C61—82no—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C2×Dic122—21no—
85C2×Dic122—21no—
86C4⋊C4×C61—21no—
87C4⋊C4×C61—42no—
88C4⋊C4×C61—82no—
89C4⋊C4×C61C22⋊C412yes—
90C22⋊C4×C61—22no—
91C22⋊C4×C61—22no—
92C22⋊C4×C61—82no—
93C22⋊C4×C61—82no—
94C42×C61—42no—
95C42×C61—42no—
96C42×C61C22×C411yescharacteristic
97C42×C61C2×D411yescharacteristic
98C42×C61#4111yescharacteristic
99C42×C61#4111yescharacteristic
100C22⋊Dic61—41no—
101#11758200—44no—
102C4⋊Dic61—21no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105C4⋊Dic61—21no—
106#11758196—41no—
107#11758196—41no—
108C4×Dic61—41no—
109#11758194—22no—
110#11758194—41no—
111C2×#92418—22no—
112C2×#92418—41no—
113C2×#92418—41no—
114C23×C61—22no—
115C23×C61—42no—
116C23×C61—42no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61#12011yescharacteristic
120D4×C61—22no—
121D4×C61—22no—
122D4×C61—42no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—41no—
128C2×C4×C61—42no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—82no—
139C2×C4×C61—82no—
140C2×C4×C61C2×C22⋊C411yescharacteristic
141C2×C4×C61C4×D411yescharacteristic
142C2×C4×C61C4×D411yescharacteristic
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12311yescharacteristic
146C2×Dic61—41no—
147C2×Dic61—41no—
148Dic122—42no—
149Dic122—81no—
150#92418—42no—
151#92418—42no—
152#92418—81no—
153#9327—611nomaximal, Sylow
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—42no—
157C22×C61—42no—
158C22×C61—42no—
159C22×C61—82no—
160C22×C61—82no—
161C22×C61#38611yescharacteristic
162C22×C61#45511yescharacteristic
163C22×C61#45611yescharacteristic
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—21no—
169C4×C61—42no—
170C4×C61—81no—
171C4×C61—82no—
172C4×C61—81no—
173C4×C61#42112yes—
174Dic61—81no—
175Dic61—81no—
176#3110—611no—
177#2319D6111yescharacteristic
178#1986—611no—
179#1506—611no—
180#1490—611no—
181#1300—611no—
182#1285—611no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—82no—
186C2×C61#188211yescharacteristic
187C2×C61#189711yescharacteristic
188C2×C61#190411yescharacteristic
189#533—611no—
190C2×C4×D4C2×D6111yescharacteristic
191#484—1221no—
192#474—612no—
193#461—612no—
194#450—1221no—
195#440—1222no—
196#432—611no—
197#432—611no—
198C4×M16—611no—
199#390—21no—
200#389—21no—
201#388—22no—
202#381—21no—
203#380—21no—
204C4×C4⋊C4C2×D6111yescharacteristic
205#376C2×D6111yescharacteristic
206#339—1222no—
207#328—1221no—
208#328—1221no—
209D4⋊C8—1222no—
210C61#932711yescharacteristic, complemented, Sylow
211C22×D4C22⋊D6111yescharacteristic
212C23×C4—21no—
213C2×SD16—1222no—
214C2×M16—1221no—
215C4⋊Q8—611no—
216C4⋊Q8—611no—
217#125—1221no—
218C22⋊Q8—1221no—
219C22⋊Q8—1221no—
220C4×Q8—1221no—
221C4×Q8—1221no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4C4×D6112yes—
225C4×D4D24412yes—
226#117—1221no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4C22⋊D6111yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C42—21no—
234C2×C42C22⋊D6112yes—
235C2×C42C22×D6111yescharacteristic
236#107—1222no—
237C4⋊C8—612no—
238C4⋊C8—1221no—
239#103—1222no—
240#102—1222no—
241C22⋊C8—1221no—
242C22⋊C8—1221no—
243#97—1221no—
244#97—1221no—
245#97—1221no—
246C4×C8—1221no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95C22⋊D6111yescharacteristic
251#95C22⋊D6111yescharacteristic
252C24—21no—
253C2×Q8—1221no—
254C2×Q8—1221no—
255C2×D4—21no—
256C2×D4—42no—
257C2×D4#1175819812yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—82no—
265C22×C4C2×C22⋊D6111yescharacteristic
266C22×C4C2×C22⋊D6111yescharacteristic
267C22×C4C2×C22⋊D6111yescharacteristic
268SD16—2444no—
269M16—1222no—
270M16—2441no—
271C2×C8—1222no—
272C2×C8—2441no—
273C2×C8—2441no—
274C4⋊C4—21no—
275C4⋊C4—42no—
276C4⋊C4—82no—
277C4⋊C4—1221no—
278C4⋊C4—1221no—
279C4⋊C4—1221no—
280C4⋊C4—1221no—
281C4⋊C4—2441no—
282C4⋊C4—2441no—
283C4⋊C4#1175819812yes—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—82no—
287C22⋊C4—82no—
288C22⋊C4—2441no—
289C42—42no—
290C42—42no—
291C42—2441no—
292C42C2×C4×D6111yescharacteristic
293C42C2×D24411yescharacteristic
294C42#1175822211yescharacteristic
295C42#1175822211yescharacteristic
296C23—22no—
297C23—42no—
298C23—42no—
299C23—41no—
300C23—41no—
301C23#1451974611yescharacteristic
302Q8—4881no—
303Q8—2442no—
304D4—22no—
305D4—22no—
306D4—42no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4C2×#1175819811yescharacteristic
327C2×C4C4×C22⋊D6111yescharacteristic
328C2×C4C4×C22⋊D6111yescharacteristic
329C2×C4#1451975111yescharacteristic
330C2×C4#1451975111yescharacteristic
331C2×C4#1451976011yescharacteristic
332C8—4881no—
333C8—2442no—
334C8—2442no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—82no—
341C22—82no—
342C22#1683964111yescharacteristic
343C22#1683994011yescharacteristic
344C22#1683994511yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—42no—
351C4—81no—
352C4—82no—
353C4—81no—
354C4—4881no—
355C4—4881no—
356C4#1683978012yes—
357C2—21no—
358C2—21no—
359C2—82no—
360C2#2144618811yescharacteristic
361C2#2144626711yescharacteristic
362C2#2144630611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2319C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#2319D61
hypocenterC61#9327
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. #21446762C2semidirect
3. #21444330C2semidirect
5. #21443712C2semidirect
6. #21443675C2semidirect
10. #16840042C4semidirect
11. #16839967C4semidirect
210. C61#9327semidirect

Maximal subgroups

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

subgroupindex
#214559182
#214467622
#214443302
#214442622
#214437122
#214436752
C61×#23192
#932761