Finite Group Databasebeta

← #29571573

Subgroups — #29571573

Order 15616.

774 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 143 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#21459622C211yesmaximal, characteristic
2#21446614C211yesmaximal, characteristic
3#21444510C212yesmaximal, complemented
4#21444273C212yesmaximal, complemented
5C61×#1800C211yesmaximal, characteristic
6#16840471—24no—
7#16840469—22no—
8D4×Dic122—22no—
9#16840393—21no—
10#16840388C414yes—
11#16840319C2212yes—
12C2×#14519756—22no—
13#16840237—21no—
14C2×C4⋊#92418C2211yescharacteristic
15C2×C22⋊#92418—22no—
16C2×C4×#92418C2211yescharacteristic
17#16839510—22no—
18#16839507—22no—
19#16839492—24no—
20C2×C4×D4×C61C2211yescharacteristic
21C61×#423C2212yes—
22C2×#100×C61—21no—
23C61×#343—21no—
24C22×Dic122—22no—
25C22×#92418—21no—
26C22×#92418—21no—
27#14519777D412yes—
28#14519777D412yes—
29#14519776—24no—
30#14519769—22no—
31#14519768—24no—
32#14519756—24no—
33#14519756—24no—
34#14519756—24no—
35#14519756—22no—
36#14519756C2×C414yes—
37#14519754—28no—
38#14519754—28no—
39C4×Dic122—22no—
40C4×Dic122D414yes—
41D4×Dic61—24no—
42D4×Dic61—24no—
43#14519736—22no—
44C2×C4⋊Dic61—22no—
45C2×#11758196—24no—
46C2×C22⋊Dic61—24no—
47C4⋊#92418—22no—
48C22⋊#92418—44no—
49C4×#92418—22no—
50C22×D4×C61D411yescharacteristic
51C23×C4×C61D412yes—
52C2×M16×C61—22no—
53C4×D4×C61—22no—
54C4×D4×C61—22no—
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61C2×C412yes—
57C2×C42×C61C2311yescharacteristic
58C4⋊C8×C61—22no—
59C4⋊C8×C61—22no—
60#100×C61—42no—
61C24×C61C22⋊C412yes—
62C2×D4×C61—22no—
63C2×D4×C61—22no—
64C2×D4×C61SD1612yes—
65C2×D4×C61Q1612yes—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—42no—
69C22×C4×C61—42no—
70C22×C4×C61C22⋊C412yes—
71C22×C4×C61C22×C411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×Dic61—24no—
75C22×Dic61—24no—
76C2×Dic122—22no—
77C2×Dic122—24no—
78C2×Dic122—82no—
79C2×Dic122—82no—
80C2×Dic122C22⋊C414yes—
81M16×C61—42no—
82M16×C61—42no—
83C2×C8×C61—42no—
84C4⋊C4×C61—22no—
85C22⋊C4×C61—24no—
86C42×C61C2×D412yes—
87C42×C61C2×D412yes—
88C22⋊Dic61—28no—
89C22⋊Dic61—24no—
90C4⋊Dic61—24no—
91C4⋊Dic61—22no—
92C4⋊Dic61—22no—
93C4⋊Dic61—22no—
94C4⋊Dic61C22⋊C414yes—
95#11758196—28no—
96#11758196—24no—
97#11758196—24no—
98C4×Dic61—24no—
99C2×#92418—22no—
100C2×#92418—41no—
101C2×#92418—41no—
102C2×#92418—42no—
103C2×#92418—41no—
104C23×C61—22no—
105C23×C61—22no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61#10314yes—
109C23×C61C2×C22⋊C411yescharacteristic
110D4×C61—24no—
111D4×C61—24no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—22no—
115C2×C4×C61—41no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—82no—
120C2×C4×C61C2×C22⋊C412yes—
121C2×C4×C61#12011yescharacteristic
122C2×C4×C61#12012yes—
123C2×C4×C61#12011yescharacteristic
124C2×C4×C61C2×SD1611yescharacteristic
125C2×C4×C61C2×Q1611yescharacteristic
126C2×C4×C61#13712yes—
127C2×Dic61—24no—
128C2×Dic61—24no—
129C2×Dic61—84no—
130C2×Dic61—84no—
131Dic122—42no—
132Dic122—42no—
133Dic122—44no—
134Dic122—84no—
135C8×C61—82no—
136#92418—42no—
137#92418—81no—
138#8355—611nomaximal, Sylow
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—24no—
142C22×C61—44no—
143C22×C61—42no—
144C22×C61—42no—
145C22×C61—82no—
146C22×C61#37911yescharacteristic
147C22×C61C2×#10311yescharacteristic
148C22×C61#41911yescharacteristic
149C4×C61—21no—
150C4×C61—22no—
151C4×C61—42no—
152C4×C61#44812yes—
153C4×C61#45112yes—
154C4×C61#45311yescharacteristic
155C4×C61#45511yescharacteristic
156Dic61—44no—
157Dic61—44no—
158#3501—611no—
159#1954—611no—
160#1800D6111yescharacteristic
161#1550—612no—
162#1493—612no—
163C2×C61—21no—
164C2×C61—21no—
165C2×C61—44no—
166C2×C61#178711yescharacteristic
167C2×C61#179111yescharacteristic
168C2×C61#179211yescharacteristic
169#554—1224no—
170#554—1222no—
171D4×Q8—1222no—
172#544—1221no—
173#543—614no—
174#533—612no—
175C2×C22⋊Q8—1222no—
176#518—1221no—
177C2×C4×D4C2×D6111yescharacteristic
178#423C2×D6112yes—
179C2×C4⋊C8—611no—
180C2×#100—21no—
181C2×C22⋊C8—1222no—
182C2×C4×C8—611no—
183#343—21no—
184#333—1222no—
185#332—1222no—
186Q8⋊C8—1224no—
187C61#835511yescharacteristic, complemented, Sylow
188C22×Q8—1222no—
189C22×D4C22⋊D6111yescharacteristic
190C23×C4D24412yes—
191C2×M16—22no—
192C22×C8—1221no—
193C22×C8—1221no—
194C4⋊Q8—612no—
195C4⋊Q8—612no—
196C4⋊Q8—1224no—
197#125—1224no—
198#125—1222no—
199C22⋊Q8—614no—
200C22⋊Q8—1228no—
201C22⋊Q8—1228no—
202C22⋊Q8—1224no—
203C22⋊Q8—1224no—
204C22⋊Q8—1224no—
205C22⋊Q8—1222no—
206C4×Q8—614no—
207C4×Q8—1222no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—1224no—
211C4×D4—1224no—
212#117—1222no—
213C2×C4⋊C4—1224no—
214C2×C4⋊C4—1222no—
215C2×C4⋊C4C22⋊D6111yescharacteristic
216C2×C22⋊C4—1224no—
217C2×C22⋊C4C4×D6112yes—
218C2×C42C22×D6111yescharacteristic
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—1222no—
222#100—42no—
223C22⋊C8—2444no—
224C4×C8—1222no—
225C24#1175819812yes—
226C2×Q8—614no—
227C2×Q8—1224no—
228C2×Q8—1222no—
229C2×Q8—4882no—
230C2×Q8—4882no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4#1175820012yes—
234C2×D4#1175820212yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—1224no—
240C22×C4—1224no—
241C22×C4#1175819812yes—
242C22×C4C2×C4×D6111yescharacteristic
243C22×C4C2×D24411yescharacteristic
244C22×C4C2×C22⋊D6111yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—42no—
248C2×C8—1222no—
249C2×C8—2442no—
250C2×C8—2441no—
251C2×C8—2441no—
252C2×C8—2441no—
253C4⋊C4—22no—
254C4⋊C4—614no—
255C4⋊C4—1228no—
256C4⋊C4—1224no—
257C4⋊C4—1224no—
258C4⋊C4—1222no—
259C4⋊C4—1222no—
260C4⋊C4—1224no—
261C4⋊C4—1222no—
262C22⋊C4—24no—
263C22⋊C4—1228no—
264C22⋊C4—1224no—
265C42—1224no—
266C42D4×D6112yes—
267C42D4×D6112yes—
268C23—22no—
269C23—22no—
270C23—42no—
271C23—42no—
272C23#1451970014yes—
273C23C2×#1175819811yescharacteristic
274Q8—4884no—
275Q8—2444no—
276Q8—2442no—
277Q8—2442no—
278D4—24no—
279D4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—82no—
288C2×C4—1224no—
289C2×C4—1224no—
290C2×C4—4884no—
291C2×C4—4884no—
292C2×C4C22⋊C4×D6112yes—
293C2×C4#1451974512yes—
294C2×C4#1451974711yescharacteristic
295C2×C4#1451974711yescharacteristic
296C2×C4C2×#1175820011yescharacteristic
297C2×C4C2×#1175820211yescharacteristic
298C2×C4#1451980712yes—
299C8—82no—
300C8—4881no—
301C8—2442no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—44no—
306C22—42no—
307C22—42no—
308C22—82no—
309C22#1683961711yescharacteristic
310C22C2×#1451970011yescharacteristic
311C22#1683977011yescharacteristic
312C4—22no—
313C4—21no—
314C4—42no—
315C4—2444no—
316C4—2444no—
317C4#1683990012yes—
318C4#1683991712yes—
319C4#1683993211yescharacteristic
320C4#1683994211yescharacteristic
321C2—21no—
322C2—21no—
323C2—44no—
324C2#2144575311yescharacteristic
325C2#2144576811yescharacteristic
326C2#2144577511yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1800C2
socleC22×C61#379
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#1800D61
hypocenterC61#8355
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #21444510C2semidirect
4. #21444273C2semidirect
187. C61#8355semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214466142
#214445102
#214442732
C61×#18002
#835561