Finite Group Databasebeta

← #29578108

Subgroups — #29578108

Order 15616.

428 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 89 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#21454916C211yesmaximal, characteristic, complemented
3#21444547C211yesmaximal, characteristic, complemented
4#21444243C211yesmaximal, characteristic
5#21444159C211yesmaximal, characteristic
6#21443682C211yesmaximal, characteristic, complemented
7C61×#1307C211yesmaximal, characteristic, complemented
8#16840319C2211yescharacteristic
9C2×C4×D244C2211yescharacteristic
10#16840098C412yes—
11#16840043—21no—
12#16839968—21no—
13#16839915C412yes—
14#16839890—22no—
15#16839889—21no—
16#16839886—21no—
17#16839864—22no—
18#16839820—22no—
19#16839819C2211yescharacteristic
20#16839815C2211yescharacteristic
21#16839510—21no—
22#16839509—21no—
23#16839489—21no—
24#16839489—21no—
25#16839485—22no—
26C61×#432C2211yescharacteristic
27C61×#432C2211yescharacteristic
28C61×#423C2211yescharacteristic
29C22×D244C2×C411yescharacteristic
30C22×C4×D61—21no—
31C2×#11758190—22no—
32#14519777D411yescharacteristic
33#14519777D411yescharacteristic
34#14519769—21no—
35#14519756—21no—
36#14519756C2×C412yes—
37C4×Dic122—21no—
38C4×Dic122D412yes—
39C4×D244—21no—
40C4×D244—21no—
41C4×D244—21no—
42C4×D244—21no—
43#14519736—21no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×#11758198—21no—
46#14519711—22no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49#14519706—22no—
50#14519706—22no—
51#14519702—22no—
52#14519699—22no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55#14519688—21no—
56#14519687—22no—
57C4×#92418D411yescharacteristic
58C4×#92418D411yescharacteristic
59C8×Dic61—22no—
60C2×M16×C61—21no—
61C2×C42×C61C2311yescharacteristic
62C4⋊C8×C61—21no—
63C4⋊C8×C61—21no—
64C4⋊C8×C61—21no—
65C4⋊C8×C61—21no—
66C22⋊C8×C61C2×C412yes—
67C22⋊C8×C61D412yes—
68#97×C61—21no—
69#97×C61—21no—
70C23×D61—21no—
71C22×C4×C61C22×C411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61#4111yescharacteristic
74C22×Dic61—21no—
75C2×D244—21no—
76C2×D244—21no—
77C2×D244—42no—
78C2×C4×D61—41no—
79C2×C4×D61—41no—
80C2×C4×D61—81no—
81C2×C4×D61—81no—
82C2×Dic122—21no—
83C2×Dic122C22⋊C412yes—
84M16×C61—81no—
85C2×C8×C61—22no—
86C2×C8×C61—22no—
87C2×C8×C61—41no—
88C42×C61C2×D411yescharacteristic
89C42×C61C2×D411yescharacteristic
90C42×C61C2×D411yescharacteristic
91C42×C61#4111yescharacteristic
92C22⋊Dic61—22no—
93#11758198—41no—
94#11758198—41no—
95C4⋊Dic61—21no—
96C4⋊Dic61—21no—
97C4⋊Dic61—21no—
98C4⋊Dic61—21no—
99C4⋊Dic61—21no—
100C4⋊Dic61C22⋊C412yes—
101#11758196—22no—
102#11758196—22no—
103C4×Dic61—22no—
104C2×#92418—21no—
105C2×#92418—21no—
106C2×#92418—41no—
107#11758190—44no—
108C23×C61C4×D411yescharacteristic
109C22×D61—22no—
110C22×D61—41no—
111C22×D61—41no—
112C22×D61—81no—
113C22×D61—81no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61C2×C22⋊C411yescharacteristic
119C2×C4×C61C4×D411yescharacteristic
120C2×C4×C61#12011yescharacteristic
121C2×C4×C61#12111yescharacteristic
122C2×C4×C61#12111yescharacteristic
123C2×C4×C61#12311yescharacteristic
124C2×Dic61—22no—
125C2×Dic61—22no—
126C2×Dic61—41no—
127C2×Dic61—81no—
128D244—22no—
129D244—22no—
130D244—41no—
131D244—41no—
132C4×D61—82no—
133C4×D61—81no—
134C4×D61—81no—
135Dic122—41no—
136Dic122—41no—
137Dic122—42no—
138C8×C61—42no—
139C8×C61—42no—
140C8×C61—81no—
141#92418—41no—
142#92418—41no—
143#92418—81no—
144#9530—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—21no—
148C22×C61#38611yescharacteristic
149C2×D61—42no—
150C2×D61—82no—
151C2×D61—81no—
152C2×D61—81no—
153C2×D61—82no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61#47111yescharacteristic
160C4×C61#47211yescharacteristic
161Dic61—42no—
162Dic61—42no—
163Dic61—81no—
164#3110—611no—
165#2985—611no—
166#1559—611no—
167#1486—611no—
168#1466—611no—
169#1307D6111yescharacteristic, complemented
170#1287—611no—
171C2×C61—41no—
172C2×C61#189111yescharacteristic
173C2×C61#190511yescharacteristic
174C2×C61#190711yescharacteristic
175D61—82no—
176#533—611no—
177C2×C4×D4—611no—
178#484—612no—
179#474—1221no—
180#461—1221no—
181#450—612no—
182#446—1221no—
183#446—1222no—
184C8×Q8—1221no—
185#440—1222no—
186#433—1222no—
187#432—611no—
188#432C2×D6111yescharacteristic
189#432C2×D6111yescharacteristic
190#431—611no—
191#423C2×D6111yescharacteristic
192#333—1221no—
193#332—1221no—
194D4⋊C8—1221no—
195D4⋊C8—1221no—
196#323—1222no—
197C61#953011yescharacteristic, complemented, Sylow
198C22×D4—611no—
199C23×C4—1221no—
200C2×SD16—1222no—
201C2×M16—21no—
202C4⋊Q8—611no—
203C4⋊Q8—611no—
204#125—1221no—
205C22⋊Q8—612no—
206C22⋊Q8—1221no—
207C4×Q8—612no—
208C4×Q8—1221no—
209C4×D4—1221no—
210C4×D4—1221no—
211C4×D4—1221no—
212C4×D4—1221no—
213#117—1221no—
214C2×C4⋊C4—611no—
215C2×C22⋊C4—1221no—
216C2×C42C22×D6111yescharacteristic
217#107—1222no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—1222no—
223C4⋊C8—1222no—
224C4⋊C8—1221no—
225C4⋊C8—1221no—
226#103—1222no—
227#102—1222no—
228C22⋊C8—1221no—
229C22⋊C8—1221no—
230C22⋊C8C4×D6112yes—
231C22⋊C8D24412yes—
232#97—21no—
233#97—21no—
234#97—1222no—
235#97—1221no—
236C4×C8—611no—
237C4×C8—611no—
238C4×C8—1222no—
239C24—1221no—
240C2×Q8—612no—
241C2×Q8—1221no—
242C2×D4—1221no—
243C2×D4—1221no—
244C2×D4—2442no—
245C22×C4—1221no—
246C22×C4—4881no—
247C22×C4—4881no—
248C22×C4—2441no—
249C22×C4—2441no—
250C22×C4C2×C4×D6111yescharacteristic
251C22×C4C2×D24411yescharacteristic
252C22×C4#1175822211yescharacteristic
253SD16—2444no—
254M16—81no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—1221no—
259C2×C8—1221no—
260C2×C8—2441no—
261C4⋊C4—612no—
262C4⋊C4—1222no—
263C4⋊C4—1221no—
264C4⋊C4—1221no—
265C4⋊C4—1221no—
266C4⋊C4—1222no—
267C4⋊C4—1221no—
268C4⋊C4—1221no—
269C22⋊C4—1222no—
270C22⋊C4—2441no—
271C22⋊C4—2441no—
272C42—1222no—
273C42D4×D6111yescharacteristic
274C42D4×D6111yescharacteristic
275C42D4×D6111yescharacteristic
276C42#1175822611yescharacteristic
277C23—1222no—
278C23—4881no—
279C23—4881no—
280C23—2441no—
281C23—2441no—
282C23C4×D24411yescharacteristic
283Q8—2442no—
284Q8—2441no—
285Q8—2441no—
286D4—1222no—
287D4—1222no—
288D4—2441no—
289D4—2441no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—1222no—
295C2×C4—1222no—
296C2×C4—4882no—
297C2×C4—4881no—
298C2×C4—4881no—
299C2×C4—4881no—
300C2×C4—2441no—
301C2×C4C22⋊C4×D6111yescharacteristic
302C2×C4#1451973911yescharacteristic
303C2×C4#1451974711yescharacteristic
304C2×C4#1451975011yescharacteristic
305C2×C4#1451975211yescharacteristic
306C2×C4#1451976211yescharacteristic
307C8—42no—
308C8—42no—
309C8—81no—
310C8—4881no—
311C8—2441no—
312C8—2441no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—4882no—
317C22—4882no—
318C22—4881no—
319C22—4881no—
320C22—2442no—
321C22#1683964011yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—4881no—
328C4—2442no—
329C4—2442no—
330C4#1684002811yescharacteristic
331C4#1684003111yescharacteristic
332C2—41no—
333C2—4882no—
334C2#2144623611yescharacteristic
335C2#2144631211yescharacteristic
336C2#2144632811yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1307C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1307D61
hypocenterC61#9530
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
2. #21454916C2semidirect
3. #21444547C2semidirect
6. #21443682C2semidirect
7. C61×#1307C2semidirect
169. #1307D61semidirect
197. C61#9530semidirect

Maximal subgroups

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

subgroupindex
#214559192
#214549162
#214445472
#214442432
#214441592
#214436822
C61×#13072
#953061