Finite Group Databasebeta

← #29592066

Subgroups — #29592066

Order 15616.

570 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 63 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#21446950C211yesmaximal, characteristic, complemented
2#21444736C212yesmaximal, complemented
3#21443816C212yesmaximal, complemented
4#21443749C211yesmaximal, characteristic, complemented
5C61×#2815C211yesmaximal, characteristic, complemented
6#16840324C2211yescharacteristic
7C2×#14519760—21no—
8#16839896—22no—
9C2×#14519691—21no—
10#16839529C2211yescharacteristic
11#16839495—22no—
12#16839486C2212yes—
13C2×#121×C61—21no—
14C61×#397—22no—
15C61×#392—22no—
16C61×#392—21no—
17C61×#392C2212yes—
18C61×#381C2211yescharacteristic
19C22×C22⋊D61—21no—
20C2×#11758199—42no—
21#14519761—41no—
22#14519760—42no—
23#14519760—42no—
24#14519751—22no—
25#14519747—41no—
26C2×#11758196—41no—
27C2×C22⋊Dic61—21no—
28C2×#11758198—41no—
29#14519698—42no—
30#14519691—42no—
31C22⋊#92418—22no—
32C22×D4×C61—21no—
33C22×D4×C61—22no—
34C22×D4×C61—21no—
35C22×D4×C61D412yes—
36C23×C4×C61D411yescharacteristic
37#121×C61—42no—
38#121×C61—42no—
39C2×C4⋊C4×C61—21no—
40C2×C22⋊C4×C61—22no—
41C2×C22⋊C4×C61—22no—
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61D411yescharacteristic
44C2×C22⋊C4×C61C2311yescharacteristic
45#95×C61—21no—
46#95×C61—22no—
47#95×C61D412yes—
48C24×C61—22no—
49C24×C61C2×D411yescharacteristic
50C23×D61—21no—
51C2×D4×C61—42no—
52C2×D4×C61—42no—
53C2×D4×C61—42no—
54C2×D4×C61—42no—
55C2×D4×C61—82no—
56C2×D4×C61—81no—
57C2×D4×C61—81no—
58C2×D4×C61—82no—
59C2×D4×C61—82no—
60C2×D4×C61—82no—
61C2×D4×C61—81no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—22no—
66C22×C4×C61—41no—
67C22×C4×C61—41no—
68C22×C4×C61D812yescomplemented
69C22×C4×C61C2×D412yes—
70C2×C22⋊D61—41no—
71C2×C22⋊D61—41no—
72C2×C22⋊D61—41no—
73C2×C22⋊D61—41no—
74C2×C22⋊D61—41no—
75C2×C22⋊D61—41no—
76C22×Dic61—21no—
77C22×Dic61—41no—
78C2×D244—22no—
79C4⋊C4×C61—42no—
80C22⋊C4×C61—41no—
81C22⋊C4×C61—44no—
82C22⋊C4×C61—41no—
83C22⋊C4×C61—82no—
84C22⋊C4×C61—82no—
85C22⋊Dic61—22no—
86C22⋊Dic61—41no—
87#11758199—84no—
88#11758198—42no—
89#11758198—44no—
90C4⋊Dic61—22no—
91#11758196—44no—
92C2×#92418—42no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—42no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61#12011yescharacteristic
104C23×C61C2×D811yescharacteristic
105C23×C61AGL(1,ℤ/8)11yescharacteristic
106C22×D61—41no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61—82no—
110D4×C61—44no—
111D4×C61—82no—
112D4×C61—84no—
113D4×C61—84no—
114D4×C61—84no—
115D4×C61—84no—
116C2×C4×C61—22no—
117C2×C4×C61—22no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61—81no—
128C22⋊D61—81no—
129C22⋊D61—81no—
130C22⋊D61—81no—
131C22⋊D61—81no—
132C2×Dic61—41no—
133C2×Dic61—42no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—82no—
137D244—82no—
138#92418—82no—
139#12094—611nomaximal, Sylow
140C22×C61—22no—
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61—42no—
146C22×C61—81no—
147C22×C61—82no—
148C22×C61—82no—
149C22×C61—82no—
150C22×C61—82no—
151C22×C61—82no—
152C22×C61—82no—
153C22×C61—82no—
154C22×C61#44711yescharacteristic
155C22×C61#45311yescharacteristic
156C22×C61B(4,𝔽2)11yescharacteristic
157C2×D61—82no—
158C2×D61—81no—
159C2×D61—82no—
160C2×D61—161no—
161C4×C61—42no—
162C4×C61—42no—
163C4×C61—82no—
164C4×C61—81no—
165Dic61—82no—
166Dic61—81no—
167#2815D6111yescharacteristic, complemented
168#2022—611no—
169#1595—612no—
170#1343—612no—
171#1317—611no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61—41no—
175C2×C61—41no—
176C2×C61—162no—
177C2×C61#159511yescharacteristic
178C2×C61#220012yes—
179D61—161no—
180#534—611no—
181C2×#123—1221no—
182C2×#121—21no—
183#447—1222no—
184C2×#99—1221no—
185#397—22no—
186#392—22no—
187#392—21no—
188#392C2×D6112yes—
189#381C2×D6111yescharacteristic
190#342—611no—
191#327—1222no—
192#323—612no—
193C61#1209411yescharacteristic, complemented, Sylow
194C22×D4—21no—
195C22×D4—22no—
196C22×D4—21no—
197C22×D4—1221no—
198C22×D4C22⋊D6112yes—
199C23×C4C22⋊D6111yescharacteristic
200C2×D8—2442no—
201#123—2441no—
202#123—2442no—
203#123—2442no—
204#121—42no—
205#121—42no—
206#121—1222no—
207#120—2441no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—2441no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—1221no—
214C2×C22⋊C4—2441no—
215C2×C22⋊C4C22⋊D6111yescharacteristic
216C2×C22⋊C4C22×D6111yescharacteristic
217#102—2442no—
218#99—2442no—
219C22⋊C8—1222no—
220#95—21no—
221#95—22no—
222#95C22⋊D6112yes—
223C24—22no—
224C24—1221no—
225C24C2×C22⋊D6111yescharacteristic
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—82no—
231C2×D4—81no—
232C2×D4—81no—
233C2×D4—82no—
234C2×D4—82no—
235C2×D4—82no—
236C2×D4—81no—
237C2×D4—1222no—
238C2×D4—2441no—
239C2×D4—2441no—
240C2×D4—2441no—
241C2×D4—2441no—
242C2×D4—2441no—
243C2×D4—2441no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—1221no—
251C22×C4—2441no—
252C22×C4#1175819912yescomplemented
253C22×C4C2×C22⋊D6112yes—
254D8—4884no—
255C2×C8—2442no—
256C4⋊C4—42no—
257C4⋊C4—1222no—
258C4⋊C4—2444no—
259C22⋊C4—41no—
260C22⋊C4—44no—
261C22⋊C4—41no—
262C22⋊C4—82no—
263C22⋊C4—82no—
264C22⋊C4—1222no—
265C22⋊C4—2442no—
266C22⋊C4—2444no—
267C22⋊C4—2441no—
268C23—21no—
269C23—21no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—4882no—
279C23—2441no—
280C23—2441no—
281C23—2441no—
282C23#1451974611yescharacteristic
283C23C2×#1175819911yescharacteristic
284C23#1451980311yescharacteristic
285D4—44no—
286D4—82no—
287D4—84no—
288D4—84no—
289D4—84no—
290D4—84no—
291D4—4882no—
292D4—4881no—
293D4—4881no—
294D4—4881no—
295D4—4881no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—4882no—
309C2×C4—2442no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C8—4882no—
314C22—22no—
315C22—21no—
316C22—21no—
317C22—41no—
318C22—41no—
319C22—42no—
320C22—81no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—9761no—
329C22—4882no—
330C22—4881no—
331C22—4882no—
332C22#1683989611yescharacteristic
333C22#1683993011yescharacteristic
334C22#1683994911yescharacteristic
335C4—42no—
336C4—42no—
337C4—82no—
338C4—81no—
339C4—4882no—
340C4—4881no—
341C2—21no—
342C2—21no—
343C2—41no—
344C2—41no—
345C2—162no—
346C2—9761no—
347C2#2144473711yescharacteristic
348C2#2144782612yes—
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C22⋊C4×C61C23
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2815C2
socleC22×C61B(4,𝔽2)
hypercenter#2815D61
hypocenterC61#12094
LCSG ▷ C2×C22⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2815

Complements

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

NH ≅ G/Ntype
1. #21446950C2semidirect
2. #21444736C2semidirect
3. #21443816C2semidirect
4. #21443749C2semidirect
5. C61×#2815C2semidirect
68. C22×C4×C61D8semidirect
167. #2815D61semidirect
193. C61#12094semidirect
252. C22×C4#11758199semidirect

Maximal subgroups

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

subgroupindex
#214469502
#214447362
#214438162
#214437492
C61×#28152
#1209461