Finite Group Databasebeta

← #29583245

Subgroups — #29583245

Order 15616.

602 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 75 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#21451484C211yesmaximal, characteristic, complemented
2#21445763C212yesmaximal
3#21443727C211yesmaximal, characteristic, complemented
4#21443671C212yesmaximal, complemented
5C61×#1782C211yesmaximal, characteristic
6C2×#14519704—22no—
7C2×#14519704—22no—
8#16839726C2212yes—
9#16839698—21no—
10#16839674—21no—
11#16839666—22no—
12#16839649—22no—
13C4×C22⋊Dic61C2212yes—
14C22×#41×C61C2211yescharacteristic
15C2×#99×C61—21no—
16C2×#99×C61—21no—
17C61×#381C2211yescharacteristic
18C61×#381C2211yescharacteristic
19C22×Dic122—21no—
20C2×#11758194—22no—
21C2×C4⋊Dic61—21no—
22C2×#11758196—22no—
23C2×C22⋊Dic61—22no—
24C2×C4×Dic61D412yes—
25C2×C4×Dic61D412yes—
26#14519704—44no—
27#14519704—44no—
28C22⋊#92418—22no—
29C22⋊#92418—22no—
30#14519684—21no—
31#14519684—21no—
32#14519684—22no—
33#14519684—22no—
34C2×#41×C61—22no—
35C2×#41×C61—22no—
36C2×#41×C61—42no—
37C2×#41×C61—42no—
38C22×Q8×C61Q811yescharacteristic, complemented
39C22×D4×C61—21no—
40C22×D4×C61Q811yescharacteristic, complemented
41C23×C4×C61—21no—
42C23×C4×C61C2311yescharacteristic
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61D411yescharacteristic
45C2×C22⋊C4×C61D411yescharacteristic
46#99×C61—42no—
47#99×C61—42no—
48#95×C61—22no—
49#95×C61—22no—
50C24×C61—21no—
51C24×C61C2×D411yescharacteristic
52#41×C61—24no—
53#41×C61—24no—
54#41×C61—42no—
55#41×C61—42no—
56#41×C61—44no—
57#41×C61—44no—
58C2×Q8×C61—22no—
59C2×Q8×C61—42no—
60C2×D4×C61—22no—
61C2×D4×C61—22no—
62C2×D4×C61—22no—
63C2×D4×C61—42no—
64C2×D4×C61—41no—
65C2×D4×C61—41no—
66C2×D4×C61—41no—
67C2×D4×C61—41no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—41no—
72C22×C4×C61—41no—
73C22×C4×C61—42no—
74C22×C4×C61—42no—
75C22×C4×C61—42no—
76C22×C4×C61—41no—
77C22×C4×C61C2×D412yes—
78C22×C4×C61C2×Q811yescharacteristic
79C22×C4×C61#4112yes—
80C22×C4×C61#4111yescharacteristic
81C22×Dic61—22no—
82C22×Dic61—22no—
83C2×Dic122—44no—
84C2×Dic122—81no—
85C22⋊C4×C61—41no—
86C22⋊C4×C61—41no—
87C22⋊C4×C61—41no—
88C22⋊C4×C61—41no—
89C22⋊Dic61—44no—
90C4⋊Dic61—81no—
91#11758196—44no—
92C4×Dic61—42no—
93C4×Dic61—42no—
94#11758194—42no—
95#11758194—42no—
96C2×#92418—42no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—22no—
100C23×C61—42no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61—82no—
104C23×C61#12011yescharacteristic
105C23×C61C22⋊Q811yescharacteristic
106C23×C61#12411yescharacteristic
107Q8×C61—24no—
108Q8×C61—42no—
109D4×C61—24no—
110D4×C61—42no—
111D4×C61—42no—
112D4×C61—42no—
113D4×C61—42no—
114D4×C61—42no—
115C2×C4×C61—22no—
116C2×C4×C61—22no—
117C2×C4×C61—22no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—42no—
121C2×C4×C61—44no—
122C2×C4×C61—44no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—81no—
130C2×C4×C61—82no—
131C2×C4×C61C22⋊Q812yes—
132C2×C4×C61#12412yes—
133C2×Dic61—24no—
134C2×Dic61—42no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61—42no—
139Dic122—84no—
140#92418—82no—
141#10386—611nomaximal, Sylow
142C22×C61—22no—
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—22no—
146C22×C61—22no—
147C22×C61—42no—
148C22×C61—81no—
149C22×C61—81no—
150C22×C61—82no—
151C22×C61—82no—
152C22×C61—82no—
153C22×C61#39311yescharacteristic
154C22×C61B(4,𝔽2)11yescharacteristic
155C22×C61B(4,𝔽2)11yescharacteristic
156C4×C61—22no—
157C4×C61—22no—
158C4×C61—42no—
159C4×C61—81no—
160C4×C61—81no—
161C4×C61—161no—
162Dic61—44no—
163Dic61—82no—
164#2607—611no—
165#1789—612no—
166#1782D6111yescharacteristic
167#1306—611no—
168#1284—612no—
169C2×C61—21no—
170C2×C61—21no—
171C2×C61—41no—
172C2×C61—41no—
173C2×C61—82no—
174C2×C61#202911yescharacteristic
175C2×C61#203112yes—
176C22×#41C2×D6111yescharacteristic
177C2×#104—1222no—
178C2×#104—1222no—
179C2×#99—21no—
180C2×#99—21no—
181#407—612no—
182#399—1221no—
183#393—1221no—
184#391—1222no—
185#387—1222no—
186#381C2×D6111yescharacteristic
187#381C2×D6111yescharacteristic
188C4×C22⋊C4—612no—
189C61#1038611yescharacteristic, complemented, Sylow
190C2×#41—22no—
191C2×#41—22no—
192C2×#41—42no—
193C2×#41—42no—
194C22×Q8—1221no—
195C22×Q8Dic12211yescharacteristic, complemented
196C22×D4—21no—
197C22×D4Dic12211yescharacteristic, complemented
198C23×C4—21no—
199C23×C4C22×D6111yescharacteristic
200C2×M16—1222no—
201C2×C4⋊C4—1221no—
202C2×C4⋊C4—1222no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—1222no—
205C2×C22⋊C4C22⋊D6111yescharacteristic
206C2×C22⋊C4C22⋊D6111yescharacteristic
207C2×C42—612no—
208C2×C42—612no—
209#104—2444no—
210#104—2444no—
211#99—42no—
212#99—42no—
213C22⋊C8—1222no—
214C22⋊C8—1222no—
215#95—22no—
216#95—22no—
217#95—1221no—
218#95—1222no—
219#95—1221no—
220#95—1222no—
221C24—21no—
222C24C2×C22⋊D6111yescharacteristic
223#41—24no—
224#41—24no—
225#41—42no—
226#41—42no—
227#41—44no—
228#41—44no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×Q8—4881no—
232C2×Q8—2444no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—1222no—
251C22×C4—1222no—
252C22×C4C2×Dic12211yescharacteristic
253C22×C4#1175822211yescharacteristic
254C22×C4D4×D6112yes—
255C22×C4#1175822412yes—
256M16—2442no—
257M16—2442no—
258C2×C8—2442no—
259C4⋊C4—4881no—
260C4⋊C4—2444no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—2444no—
266C42—2442no—
267C42—2442no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—82no—
275C23#1451974511yescharacteristic
276C23#1451975611yescharacteristic
277C23#1451976411yescharacteristic
278Q8—24no—
279Q8—42no—
280Q8—4884no—
281D4—24no—
282D4—42no—
283D4—42no—
284D4—42no—
285D4—42no—
286D4—42no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—1224no—
304C2×C4—2442no—
305C2×C4—2442no—
306C2×C4—2442no—
307C2×C4—2442no—
308C2×C4—2442no—
309C2×C4#1451975412yes—
310C2×C4#1451976312yes—
311C8—4882no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—81no—
319C22—81no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22#1683967411yescharacteristic
324C22#1683994811yescharacteristic
325C22#1683994811yescharacteristic
326C4—22no—
327C4—22no—
328C4—42no—
329C4—81no—
330C4—81no—
331C4—161no—
332C4—4882no—
333C4—2444no—
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—82no—
339C2#2144698311yescharacteristic
340C2#2144699312yes—
subgroupG/H
centerC22#16839674
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1782C2
socleC22×C61#393
prime radicalC22×#41×C61C22
hypercenter#1782D61
hypocenterC61#10386
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #21451484C2semidirect
3. #21443727C2semidirect
4. #21443671C2semidirect
38. C22×Q8×C61Q8semidirect
40. C22×D4×C61Q8semidirect
189. C61#10386semidirect
195. C22×Q8Dic122semidirect
197. C22×D4Dic122semidirect

Maximal subgroups

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

subgroupindex
#214514842
#214457632
#214437272
#214436712
C61×#17822
#1038661