Finite Group Databasebeta

← #29576937

Subgroups — #29576937

Order 15616.

492 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 93 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#21455919C211yesmaximal, characteristic
2#21448463C211yesmaximal, characteristic
3#21446763C211yesmaximal, characteristic, complemented
4#21444330C211yesmaximal, characteristic, complemented
5#21444263C211yesmaximal, characteristic
6#21443676C211yesmaximal, characteristic, complemented
7C61×#1300C211yesmaximal, characteristic, complemented
8#16840319C2211yescharacteristic
9C2×C4×D244C2211yescharacteristic
10#16840098—21no—
11#16840043C412yescomplemented
12#16839968C412yescomplemented
13#16839915—21no—
14#16839864—22no—
15#16839819C2211yescharacteristic
16#16839664—21no—
17#16839660—21no—
18#16839651—22no—
19#16839624—21no—
20#16839621—21no—
21C4×C4⋊Dic61C2211yescharacteristic
22#16839523—22no—
23#16839498—21no—
24#16839498—21no—
25#16839491—22no—
26C61×#432C2211yescharacteristic
27C4×M16×C61C2211yescharacteristic
28C61×#376C2211yescharacteristic
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31C2×#11758190—22no—
32#14519777C2×C411yescharacteristic
33#14519777C2×C411yescharacteristic
34#14519769—21no—
35#14519756—21no—
36#14519756—21no—
37C4×Dic122—21no—
38C4×Dic122—21no—
39C4×D244—21no—
40C4×D244—21no—
41C4×D244C2×C412yes—
42C4×D244D412yes—
43#14519736—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—22no—
47C2×#11758198—21no—
48C2×#11758198—22no—
49C2×#11758198—22no—
50C2×C4×Dic61D412yes—
51#14519711—22no—
52C4⋊#92418—21no—
53#14519702—22no—
54#14519699—22no—
55C22⋊#92418—21no—
56#14519688—21no—
57#14519684—22no—
58#14519684—21no—
59C2×M16×C61—21no—
60C2×C42×C61—21no—
61C2×C42×C61C2311yescharacteristic
62C4⋊C8×C61C2×C412yes—
63C22⋊C8×C61—21no—
64#97×C61—21no—
65#97×C61—21no—
66C4×C8×C61—21no—
67#95×C61—21no—
68#95×C61D411yescharacteristic
69#95×C61D411yescharacteristic
70C23×D61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×Dic61—21no—
77C22×Dic61—22no—
78C2×D244—21no—
79C2×D244—42no—
80C2×D244C22⋊C412yes—
81C2×C4×D61—22no—
82C2×C4×D61—22no—
83C2×C4×D61—82no—
84C2×Dic122—21no—
85C2×Dic122—21no—
86M16×C61—22no—
87M16×C61—41no—
88C2×C8×C61—22no—
89C2×C8×C61—41no—
90C42×C61—42no—
91C42×C61C22×C411yescharacteristic
92C42×C61C2×D411yescharacteristic
93C42×C61#4111yescharacteristic
94C42×C61#4111yescharacteristic
95C22⋊Dic61—41no—
96#11758198—22no—
97#11758198—22no—
98#11758198—82no—
99#11758198—82no—
100C4⋊Dic61—21no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105C4⋊Dic61—42no—
106C4⋊Dic61C22⋊C412yes—
107#11758196—41no—
108#11758196—41no—
109#11758196—82no—
110C4×Dic61—41no—
111C4×Dic61—42no—
112C2×#92418—41no—
113#11758190—44no—
114C23×C61#12011yescharacteristic
115C22×D61—22no—
116C22×D61—42no—
117C22×D61—42no—
118C22×D61—41no—
119C22×D61—41no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—22no—
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×C61C2×C22⋊C411yescharacteristic
131C2×C4×C61C4×D411yescharacteristic
132C2×C4×C61C4×D411yescharacteristic
133C2×C4×C61#12111yescharacteristic
134C2×C4×C61#12111yescharacteristic
135C2×C4×C61#12311yescharacteristic
136C2×Dic61—22no—
137C2×Dic61—42no—
138C2×Dic61—42no—
139C2×Dic61—42no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—42no—
143D244—22no—
144D244—22no—
145D244—42no—
146C4×D61—82no—
147C4×D61—82no—
148Dic122—42no—
149Dic122—81no—
150C8×C61—42no—
151C8×C61—42no—
152#92418—81no—
153#9327—611nomaximal, Sylow
154C22×C61—21no—
155C22×C61—21no—
156C22×C61#38611yescharacteristic
157C22×C61#45511yescharacteristic
158C22×C61#45611yescharacteristic
159C2×D61—42no—
160C2×D61—42no—
161C2×D61—42no—
162C2×D61—82no—
163C2×D61—82no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—21no—
169C4×C61—81no—
170C4×C61—81no—
171C4×C61#42112yes—
172Dic61—42no—
173Dic61—82no—
174Dic61—81no—
175Dic61—81no—
176#3110—611no—
177#2319—611no—
178#1986—611no—
179#1506—611no—
180#1490—611no—
181#1300D6111yescharacteristic, complemented
182#1285—611no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61#188211yescharacteristic
186C2×C61#189711yescharacteristic
187C2×C61#190411yescharacteristic
188D61—82no—
189#533—611no—
190C2×C4×D4—611no—
191#484—1221no—
192#474—612no—
193#461—612no—
194#450—1221no—
195#440—1222no—
196#432—611no—
197#432C2×D6111yescharacteristic
198C4×M16C2×D6111yescharacteristic
199#390—1221no—
200#389—1221no—
201#388—1222no—
202#381—1221no—
203#380—1221no—
204C4×C4⋊C4—611no—
205#376C2×D6111yescharacteristic
206#339—1222no—
207#328—1221no—
208#328—1221no—
209D4⋊C8—1222no—
210C61#932711yescharacteristic, complemented, Sylow
211C22×D4—611no—
212C23×C4—1221no—
213C2×SD16—1222no—
214C2×M16—21no—
215C4⋊Q8—611no—
216C4⋊Q8—611no—
217#125—1221no—
218C22⋊Q8—1221no—
219C22⋊Q8—1221no—
220C4×Q8—1221no—
221C4×Q8—1221no—
222C4×D4—612no—
223C4×D4—612no—
224C4×D4—1221no—
225C4×D4—1221no—
226#117—1221no—
227C2×C4⋊C4—611no—
228C2×C4⋊C4—1222no—
229C2×C4⋊C4—1221no—
230C2×C22⋊C4—1221no—
231C2×C22⋊C4—1222no—
232C2×C22⋊C4—1222no—
233C2×C42—21no—
234C2×C42—612no—
235C2×C42C22×D6111yescharacteristic
236#107—1222no—
237C4⋊C8—1221no—
238C4⋊C8C4×D6112yes—
239#103—1222no—
240#102—1222no—
241C22⋊C8—21no—
242C22⋊C8—1221no—
243#97—21no—
244#97—21no—
245#97—1221no—
246C4×C8—21no—
247#95—21no—
248#95—1222no—
249#95—1221no—
250#95D24411yescharacteristic
251#95D24411yescharacteristic
252C24—1221no—
253C2×Q8—1221no—
254C2×Q8—1221no—
255C2×D4—612no—
256C2×D4—1221no—
257C2×D4—2442no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1221no—
261C22×C4—1222no—
262C22×C4—1222no—
263C22×C4—1222no—
264C22×C4—4882no—
265C22×C4C2×D24411yescharacteristic
266C22×C4D4×D6111yescharacteristic
267C22×C4D4×D6111yescharacteristic
268SD16—2444no—
269M16—22no—
270M16—41no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—2441no—
274C4⋊C4—612no—
275C4⋊C4—1221no—
276C4⋊C4—1221no—
277C4⋊C4—1221no—
278C4⋊C4—1221no—
279C4⋊C4—1221no—
280C4⋊C4—4882no—
281C4⋊C4—2442no—
282C4⋊C4—2441no—
283C4⋊C4—2441no—
284C22⋊C4—1222no—
285C22⋊C4—1222no—
286C22⋊C4—4882no—
287C22⋊C4—4882no—
288C22⋊C4—2441no—
289C42—42no—
290C42—2442no—
291C42—2441no—
292C42C2×C4×D6111yescharacteristic
293C42#1175822211yescharacteristic
294C42D4×D6111yescharacteristic
295C42#1175822611yescharacteristic
296C23—1222no—
297C23—2442no—
298C23—2442no—
299C23—2441no—
300C23—2441no—
301C23#1451974711yescharacteristic
302Q8—4881no—
303Q8—2442no—
304D4—1222no—
305D4—1222no—
306D4—2442no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—1222no—
318C2×C4—4882no—
319C2×C4—4882no—
320C2×C4—2442no—
321C2×C4—2442no—
322C2×C4—2442no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2442no—
326C2×C4C22⋊C4×D6111yescharacteristic
327C2×C4#1451973911yescharacteristic
328C2×C4C4×D24411yescharacteristic
329C2×C4#1451975011yescharacteristic
330C2×C4#1451975211yescharacteristic
331C2×C4#1451976211yescharacteristic
332C8—42no—
333C8—42no—
334C8—4881no—
335C22—21no—
336C22—21no—
337C22—4882no—
338C22—4882no—
339C22—2442no—
340C22—2442no—
341C22—2442no—
342C22#1683964011yescharacteristic
343C22#1683994111yescharacteristic
344C22#1683994611yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—81no—
351C4—81no—
352C4—4882no—
353C4—4881no—
354C4—4881no—
355C4—2442no—
356C4#1683977712yes—
357C2—21no—
358C2—21no—
359C2—4882no—
360C2#2144618911yescharacteristic
361C2#2144626611yescharacteristic
362C2#2144630511yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1300C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1300D61
hypocenterC61#9327
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
3. #21446763C2semidirect
4. #21444330C2semidirect
6. #21443676C2semidirect
7. C61×#1300C2semidirect
11. #16840043C4semidirect
12. #16839968C4semidirect
181. #1300D61semidirect
210. C61#9327semidirect

Maximal subgroups

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

subgroupindex
#214559192
#214484632
#214467632
#214443302
#214442632
#214436762
C61×#13002
#932761