Finite Group Databasebeta

← #29577359

Subgroups — #29577359

Order 15616.

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

normal subgroups only

This group has 496 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#21454457C211yesmaximal, characteristic, complemented
2#21444417C212yesmaximal, complemented
3#21444153C211yesmaximal, characteristic, complemented
4#21443712C212yesmaximal, complemented
5C61×#2610C211yesmaximal, characteristic
6#16839823—24no—
7#16839819C2212yes—
8#16839786—21no—
9#16839726—21no—
10C4×#11758194C2212yes—
11#16839502—24no—
12#16839488—21no—
13#16839486—21no—
14C2×#124×C61C2211yescharacteristic
15C2×#117×C61C2211yescharacteristic
16C61×#395C411yescharacteristic
17C61×#394C411yescharacteristic
18C61×#393C411yescharacteristic
19C61×#392C411yescharacteristic
20C61×#390—21no—
21C61×#388—21no—
22C61×#388—21no—
23C61×#386—21no—
24C61×#376C2211yescharacteristic
25C2×#11758194—22no—
26C4⋊#92418—24no—
27C4⋊#92418—22no—
28C22⋊#92418—24no—
29C22⋊#92418—22no—
30#14519688—22no—
31#14519688D414yescomplemented
32C4×#92418—22no—
33C22×Q8×C61C2×C411yescharacteristic
34C22×D4×C61—21no—
35C22×D4×C61C2×C411yescharacteristic
36C23×C4×C61D411yescharacteristic
37#124×C61—22no—
38#124×C61—22no—
39#117×C61—22no—
40#117×C61—22no—
41C2×C4⋊C4×C61—21no—
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C4⋊C4×C61D411yescharacteristic
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—21no—
46C2×C22⋊C4×C61—21no—
47C2×C22⋊C4×C61—21no—
48C2×C22⋊C4×C61—21no—
49C2×C22⋊C4×C61—21no—
50C2×C22⋊C4×C61C2×C411yescharacteristic
51C2×C22⋊C4×C61D411yescharacteristic
52C2×C42×C61—21no—
53C2×C42×C61D411yescharacteristic
54C2×C42×C61C2311yescharacteristic
55#95×C61—21no—
56#95×C61—21no—
57#95×C61—21no—
58#95×C61C2×C411yescharacteristic
59#95×C61C2×C411yescharacteristic
60C24×C61—21no—
61C24×C61C22⋊C411yescharacteristic
62C2×Q8×C61—22no—
63C2×Q8×C61—81no—
64C2×D4×C61—22no—
65C2×D4×C61—42no—
66C2×D4×C61—42no—
67C2×D4×C61—81no—
68C2×D4×C61—81no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—41no—
75C22×C4×C61—41no—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C4⋊C4×C61—22no—
83C4⋊C4×C61—22no—
84C4⋊C4×C61—81no—
85C22⋊C4×C61—22no—
86C22⋊C4×C61—22no—
87C22⋊C4×C61—42no—
88C22⋊C4×C61—42no—
89C22⋊C4×C61—41no—
90C22⋊C4×C61—42no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—81no—
93C22⋊C4×C61—81no—
94C42×C61—21no—
95C42×C61—21no—
96C42×C61—42no—
97C42×C61C2×D412yes—
98C42×C61#4112yes—
99#11758194—42no—
100#11758194—42no—
101C2×#92418—24no—
102C2×#92418—42no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61C2×C22⋊C411yescharacteristic
116Q8×C61—82no—
117D4×C61—82no—
118D4×C61—84no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—22no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61C4×D412yes—
142C2×C4×C61#12011yescharacteristic
143C2×C4×C61#12112yes—
144C2×C4×C61#12311yescharacteristic
145#92418—44no—
146#92418—82no—
147#9389—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—42no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—41no—
154C22×C61—81no—
155C22×C61—81no—
156C22×C61—81no—
157C22×C61—81no—
158C22×C61—81no—
159C22×C61—81no—
160C22×C61—81no—
161C22×C61—81no—
162C22×C61#38611yescharacteristic
163C22×C61#42111yescharacteristic
164C22×C61#42111yescharacteristic
165C4×C61—22no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—22no—
169C4×C61—42no—
170C4×C61—41no—
171C4×C61—41no—
172C4×C61—41no—
173C4×C61—81no—
174C4×C61—81no—
175#2921—611no—
176#2610D6111yescharacteristic
177#1527—612no—
178#1464—611no—
179#1300—612no—
180C2×C61—21no—
181C2×C61—21no—
182C2×C61—81no—
183C2×C61—161no—
184C2×C61#188411yescharacteristic
185C2×C61#189711yescharacteristic
186C2×C61#191411yescharacteristic
187C2×#124C2×D6111yescharacteristic
188C2×#117C2×D6111yescharacteristic
189#433—1224no—
190#432—612no—
191#423—1221no—
192#407—1221no—
193C4×M16—612no—
194#395Dic6111yescharacteristic
195#394Dic6111yescharacteristic
196#393Dic6111yescharacteristic
197#392Dic6111yescharacteristic
198#390—21no—
199#388—21no—
200#388—21no—
201#386—21no—
202#376C2×D6111yescharacteristic
203#329—1224no—
204#324—1221no—
205#323—1221no—
206C61#938911yescharacteristic, complemented, Sylow
207C22×Q8C2×Dic6111yescharacteristic
208C22×D4—21no—
209C22×D4C2×Dic6111yescharacteristic
210C23×C4C22⋊D6111yescharacteristic
211C2×M16—1222no—
212#124—22no—
213#124—22no—
214#117—22no—
215#117—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×Dic6111yescharacteristic
218C2×C4⋊C4C22⋊D6111yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C2×Dic6111yescharacteristic
226C2×C22⋊C4C22⋊D6111yescharacteristic
227C2×C42—21no—
228C2×C42C22⋊D6111yescharacteristic
229C2×C42C22×D6111yescharacteristic
230C4⋊C8—1224no—
231C4⋊C8—1222no—
232C22⋊C8—1224no—
233C22⋊C8—1222no—
234#97—614no—
235#97—1222no—
236C4×C8—1222no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95C2×Dic6111yescharacteristic
241#95C2×Dic6111yescharacteristic
242C24—21no—
243C24C22⋊Dic6111yescharacteristic
244C2×Q8—22no—
245C2×Q8—81no—
246C2×D4—22no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—81no—
250C2×D4—81no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C22⋊Dic6111yescharacteristic
259C22×C4C22⋊Dic6111yescharacteristic
260C22×C4C22⋊Dic6111yescharacteristic
261C22×C4C22×Dic6111yescharacteristic
262C22×C4C2×C22⋊D6111yescharacteristic
263C22×C4C2×C22⋊D6111yescharacteristic
264M16—2442no—
265M16—2442no—
266C2×C8—1224no—
267C2×C8—2442no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—81no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—81no—
279C22⋊C4—81no—
280C42—21no—
281C42—21no—
282C42—42no—
283C42D4×D6112yes—
284C42#1175822412yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Dic6111yescharacteristic
298Q8—82no—
299D4—82no—
300D4—84no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4D4×Dic6112yes—
324C2×C4#1451974511yescharacteristic
325C2×C4#1451974912yes—
326C2×C4#1451975811yescharacteristic
327C8—4882no—
328C8—2444no—
329C22—21no—
330C22—21no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22#1683964211yescharacteristic
344C22#1683977911yescharacteristic
345C22#1683977911yescharacteristic
346C4—22no—
347C4—21no—
348C4—21no—
349C4—22no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C2—21no—
357C2—21no—
358C2—81no—
359C2—161no—
360C2#2144620311yescharacteristic
361C2#2144626811yescharacteristic
362C2#2144636711yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2610C2
socleC22×C61#386
prime radicalC61×#392C4
cosocleC2×C42×C61C23
hypercenter#2610D61
hypocenterC61#9389
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #21454457C2semidirect
2. #21444417C2semidirect
3. #21444153C2semidirect
4. #21443712C2semidirect
31. #14519688D4semidirect
206. C61#9389semidirect

Maximal subgroups

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

subgroupindex
#214544572
#214444172
#214441532
#214437122
C61×#26102
#938961