Finite Group Databasebeta

← #29581556

Subgroups — #29581556

Order 15616.

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

normal subgroups only

This group has 470 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#21456967C211yesmaximal, characteristic
2#21456366C211yesmaximal, characteristic
3#21446662C211yesmaximal, characteristic
4#21444414C211yesmaximal, characteristic
5#21444240C211yesmaximal, characteristic
6#21443708C211yesmaximal, characteristic
7C61×#3255C211yesmaximal, characteristic
8C2×#14519777C2211yescharacteristic
9#16840172—22no—
10#16840165—21no—
11#16840165—21no—
12#16840158—21no—
13#16840112—21no—
14#16840095—22no—
15#16840057—22no—
16#16839815C2211yescharacteristic
17#16839805—22no—
18C2×#14519695—22no—
19C4×#11758194C2211yescharacteristic
20C4×#11758194C2211yescharacteristic
21#16839504—22no—
22Q8⋊#92418—22no—
23C61×#533C2211yescharacteristic
24C61×#533C2211yescharacteristic
25C61×#475—21no—
26C61×#475—21no—
27C61×#470—22no—
28C61×#462—21no—
29C61×#462—21no—
30C61×#423C2211yescharacteristic
31C22×Dic122—21no—
32C2×#11758194—21no—
33C2×#11758194—21no—
34#14519777—21no—
35#14519777—21no—
36#14519777—21no—
37#14519777—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61D412yes—
40#14519710—24no—
41#14519710—24no—
42#14519708—24no—
43C4⋊#92418D412yes—
44#14519701—24no—
45#14519695—44no—
46C22⋊#92418—21no—
47#14519688—21no—
48#14519688D412yes—
49C4×#92418—21no—
50C4×#92418—21no—
51C4×#92418—21no—
52C2×Q16×C61—22no—
53C2×M16×C61D412yes—
54C4⋊Q8×C61D411yescharacteristic
55C4⋊Q8×C61D411yescharacteristic
56C4⋊Q8×C61D411yescharacteristic
57C4⋊Q8×C61D411yescharacteristic
58#125×C61—21no—
59#125×C61—21no—
60C22⋊Q8×C61—21no—
61C22⋊Q8×C61—21no—
62C22⋊Q8×C61—21no—
63C22⋊Q8×C61—21no—
64C4×Q8×C61—21no—
65C4×Q8×C61—21no—
66C4×Q8×C61—21no—
67C4×Q8×C61—21no—
68#117×C61—21no—
69#117×C61—21no—
70C2×C42×C61C2311yescharacteristic
71#106×C61—22no—
72C4⋊C8×C61—21no—
73C4⋊C8×C61—21no—
74#103×C61—22no—
75#103×C61—22no—
76C2×Q8×C61—21no—
77C2×Q8×C61—21no—
78C2×Q8×C61—21no—
79C2×Q8×C61—21no—
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61#4111yescharacteristic
83C22×Dic61—22no—
84C2×Dic122—42no—
85C2×Dic122—42no—
86C2×Dic122—41no—
87C2×Dic122—41no—
88Q16×C61—82no—
89M16×C61—42no—
90C2×C8×C61—22no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—41no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—41no—
103C22⋊C4×C61—41no—
104C22⋊C4×C61—41no—
105C42×C61—41no—
106C42×C61—41no—
107C42×C61C2×D411yescharacteristic
108C42×C61C2×D411yescharacteristic
109C42×C61C2×D411yescharacteristic
110C42×C61C2×D411yescharacteristic
111C4⋊Dic61—22no—
112C4⋊Dic61—22no—
113C4⋊Dic61—41no—
114C4⋊Dic61—41no—
115#11758194—24no—
116#11758194—22no—
117#11758194—41no—
118C2×#92418—22no—
119C2×#92418—22no—
120C2×#92418—41no—
121C23×C61#12111yescharacteristic
122Q8×C61—41no—
123Q8×C61—41no—
124Q8×C61—41no—
125Q8×C61—41no—
126Q8×C61—42no—
127Q8×C61—42no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61#12011yescharacteristic
138C2×C4×C61#12011yescharacteristic
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12711yescharacteristic
142C2×Dic61—44no—
143C2×Dic61—42no—
144Dic122—82no—
145Dic122—81no—
146Dic122—81no—
147C8×C61—42no—
148#92418—24no—
149#92418—42no—
150#92418—42no—
151#10117—611nomaximal, Sylow
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—21no—
155C22×C61#39211yescharacteristic
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—81no—
160C4×C61—81no—
161C4×C61—81no—
162C4×C61—81no—
163C4×C61#45311yescharacteristic
164C4×C61#45411yescharacteristic
165C4×C61#45612yes—
166C4×C61#47312yes—
167Dic61—82no—
168#3255D6111yescharacteristic
169#3230—611no—
170#3165—611no—
171#1966—611no—
172#1526—611no—
173#1485—611no—
174#1299—611no—
175C2×C61—41no—
176C2×C61#200511yescharacteristic
177C2×C61#200611yescharacteristic
178C2×C61#202011yescharacteristic
179#533C2×D6111yescharacteristic
180#533C2×D6111yescharacteristic
181C2×C4⋊Q8—611no—
182#501—1222no—
183#500—1221no—
184#500—1221no—
185#498—1221no—
186#487—1221no—
187#484—1222no—
188#477—1222no—
189#475—21no—
190#475—21no—
191#470—22no—
192#462—21no—
193#462—21no—
194#431—611no—
195#428—1222no—
196#423C2×D6111yescharacteristic
197C2×#101—1222no—
198C4×M16—611no—
199C4×M16—611no—
200#330—1222no—
201Q8⋊C8—1222no—
202C61#1011711yescharacteristic, complemented, Sylow
203C22×Q8—1221no—
204C2×Q16—22no—
205C2×M16—1221no—
206C2×M16—1221no—
207C2×M16C22⋊D6112yes—
208C4⋊Q8—1221no—
209C4⋊Q8—1221no—
210C4⋊Q8—1221no—
211C4⋊Q8—1221no—
212C4⋊Q8C22⋊D6111yescharacteristic
213C4⋊Q8C22⋊D6111yescharacteristic
214C4⋊Q8C22⋊D6111yescharacteristic
215C4⋊Q8C22⋊D6111yescharacteristic
216#125—21no—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226#117—21no—
227#117—21no—
228C2×C4⋊C4—612no—
229C2×C4⋊C4—1221no—
230C2×C42C22×D6111yescharacteristic
231#107—1224no—
232#107—1224no—
233#106—22no—
234#106—1224no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—612no—
238#103—22no—
239#103—22no—
240#103—1224no—
241#101—2444no—
242C22⋊C8—1221no—
243#97—612no—
244#97—1221no—
245C4×C8—1221no—
246C4×C8—1221no—
247C4×C8—1221no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—2442no—
253C2×Q8—2442no—
254C2×Q8—2441no—
255C2×Q8—2441no—
256C22×C4—1222no—
257C22×C4D4×D6111yescharacteristic
258C22×C4#1175822411yescharacteristic
259C22×C4C2×C22⋊D6111yescharacteristic
260Q16—82no—
261M16—42no—
262M16—1224no—
263M16—1222no—
264M16—2441no—
265C2×C8—22no—
266C2×C8—1222no—
267C2×C8—1222no—
268C2×C8—2441no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—1222no—
282C4⋊C4—1222no—
283C4⋊C4—2441no—
284C4⋊C4—2441no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42—41no—
288C42—41no—
289C42D4×D6111yescharacteristic
290C42D4×D6111yescharacteristic
291C42C2×C22⋊D6111yescharacteristic
292C42C2×C22⋊D6111yescharacteristic
293C23#1451974811yescharacteristic
294Q8—42no—
295Q8—42no—
296Q8—41no—
297Q8—41no—
298Q8—41no—
299Q8—41no—
300Q8—4882no—
301Q8—4881no—
302Q8—4881no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—2444no—
312C2×C4—2442no—
313C2×C4#1451974511yescharacteristic
314C2×C4#1451974511yescharacteristic
315C2×C4#1451974611yescharacteristic
316C2×C4#1451974911yescharacteristic
317C2×C4#1451974911yescharacteristic
318C2×C4#1451977311yescharacteristic
319C8—42no—
320C8—1224no—
321C8—2442no—
322C8—2442no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22#1683967011yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—4882no—
335C4#1683993011yescharacteristic
336C4#1683993511yescharacteristic
337C4#1683994312yes—
338C4#1684003512yes—
339C2—41no—
340C2#2144685211yescharacteristic
341C2#2144686111yescharacteristic
342C2#2144693711yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3255C2
socleC22×C61#392
prime radicalC23×C61#121
hypercenter#3255D61
hypocenterC61#10117
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3255

Complements

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

NH ≅ G/Ntype
202. C61#10117semidirect

Maximal subgroups

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

subgroupindex
#214569672
#214563662
#214466622
#214444142
#214442402
#214437082
C61×#32552
#1011761