Finite Group Databasebeta

← #29571570

Subgroups — #29571570

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#21445811C211yesmaximal, characteristic
3#21444507C212yesmaximal, complemented
4#21444275C212yesmaximal, complemented
5C61×#1954C211yesmaximal, characteristic
6#16840471—24no—
7#16840469—22no—
8D4×Dic122—22no—
9#16840393—21no—
10#16840388C414yes—
11#16840319C2212yes—
12C2×#14519756—22no—
13#16840237—21no—
14#16839786C2212yes—
15C2×#14519694—21no—
16#16839531—21no—
17#16839510—22no—
18#16839509—22no—
19#16839494—24no—
20C2×C4×D4×C61C2211yescharacteristic
21C2×C4⋊C8×C61C2211yescharacteristic
22C2×C22⋊C8×C61—22no—
23C2×C4×C8×C61C2211yescharacteristic
24C22×Dic122—22no—
25C2×#11758194—22no—
26#14519777D412yes—
27#14519777D412yes—
28#14519776—24no—
29#14519769—22no—
30#14519768—24no—
31#14519756—24no—
32#14519756—24no—
33#14519756—24no—
34#14519756—22no—
35#14519756C2×C414yes—
36#14519754—28no—
37#14519754—28no—
38C4×Dic122—22no—
39C4×Dic122D414yes—
40D4×Dic61—24no—
41D4×Dic61—24no—
42#14519736—22no—
43C2×C4⋊Dic61—22no—
44C2×#11758196—24no—
45C2×C22⋊Dic61—24no—
46C4⋊#92418—22no—
47C4⋊#92418—22no—
48#14519694—42no—
49C22×D4×C61D411yescharacteristic
50C23×C4×C61D412yes—
51C22×C8×C61—21no—
52C22×C8×C61—21no—
53C4×D4×C61—22no—
54C4×D4×C61—22no—
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61C2×C412yes—
57C2×C42×C61C2311yescharacteristic
58C4⋊C8×C61—22no—
59C22⋊C8×C61—44no—
60C4×C8×C61—22no—
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—
81C2×C8×C61—22no—
82C2×C8×C61—41no—
83C2×C8×C61—41no—
84C2×C8×C61—42no—
85C2×C8×C61—41no—
86C4⋊C4×C61—22no—
87C22⋊C4×C61—24no—
88C42×C61C2×D412yes—
89C42×C61C2×D412yes—
90C22⋊Dic61—28no—
91C22⋊Dic61—24no—
92C4⋊Dic61—24no—
93C4⋊Dic61—22no—
94C4⋊Dic61—22no—
95C4⋊Dic61—22no—
96C4⋊Dic61C22⋊C414yes—
97#11758196—28no—
98#11758196—24no—
99#11758196—24no—
100C4×Dic61—24no—
101#11758194—42no—
102#11758194—42no—
103C2×#92418—42no—
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—42no—
136C8×C61—81no—
137#92418—82no—
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#1954D6111yescharacteristic
160#1800—611no—
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#423—612no—
179C2×C4⋊C8C2×D6111yescharacteristic
180C2×#100—1221no—
181C2×C22⋊C8—22no—
182C2×C4×C8C2×D6111yescharacteristic
183#343—1221no—
184#333—1222no—
185#332—1222no—
186Q8⋊C8—1224no—
187C61#835511yescharacteristic, complemented, Sylow
188C22×Q8—1222no—
189C22×D4D24411yescharacteristic
190C23×C4C22⋊D6112yes—
191C2×M16—1222no—
192C22×C8—21no—
193C22×C8—21no—
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⋊C4D24411yescharacteristic
216C2×C22⋊C4—1224no—
217C2×C22⋊C4C4×D6112yes—
218C2×C42C22×D6111yescharacteristic
219C4⋊C8—22no—
220C4⋊C8—1222no—
221C4⋊C8—1222no—
222#100—2442no—
223C22⋊C8—44no—
224C4×C8—22no—
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#1175819012yes—
234C2×D4Dic24412yes—
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—2442no—
246M16—2442no—
247C2×C8—22no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—41no—
252C2×C8—2442no—
253C4⋊C4—22no—
254C4⋊C4—614no—
255C4⋊C4—1228no—
256C4⋊C4—1224no—
257C4⋊C4—1224no—
258C4⋊C4—1224no—
259C4⋊C4—1222no—
260C4⋊C4—1222no—
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#1451970214yes—
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#1451974511yescharacteristic
294C2×C4#1451974511yescharacteristic
295C2×C4#1451974712yes—
296C2×C4C2×#1175819011yescharacteristic
297C2×C4C2×Dic24411yescharacteristic
298C2×C4#1451981012yes—
299C8—42no—
300C8—81no—
301C8—4882no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—44no—
306C22—42no—
307C22—42no—
308C22—82no—
309C22#1683961711yescharacteristic
310C22C2×#1451970211yescharacteristic
311C22#1683977211yescharacteristic
312C4—22no—
313C4—21no—
314C4—42no—
315C4—2444no—
316C4—2444no—
317C4#1683990312yes—
318C4#1683992012yes—
319C4#1683992911yescharacteristic
320C4#1683993911yescharacteristic
321C2—21no—
322C2—21no—
323C2—44no—
324C2#2144575511yescharacteristic
325C2#2144576911yescharacteristic
326C2#2144577211yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1954C2
socleC22×C61#379
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#1954D61
hypocenterC61#8355
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1954

Complements

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

NH ≅ G/Ntype
3. #21444507C2semidirect
4. #21444275C2semidirect
187. C61#8355semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214458112
#214445072
#214442752
C61×#19542
#835561