Finite Group Databasebeta

← #29581570

Subgroups — #29581570

Order 15616.

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

normal subgroups only

This group has 454 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#21456960C211yesmaximal, characteristic
2#21456366C211yesmaximal, characteristic
3#21446807C211yesmaximal, characteristic
4#21444414C211yesmaximal, characteristic
5#21444240C211yesmaximal, characteristic
6#21443708C211yesmaximal, characteristic
7C61×#3255C211yesmaximal, characteristic
8C2×#14519777C2211yescharacteristic
9#16840172—22no—
10#16840165—21no—
11#16840158—21no—
12#16840158—21no—
13#16840112—21no—
14#16840090—22no—
15#16840047—22no—
16#16839815C2211yescharacteristic
17#16839805—22no—
18C4×#11758194C2211yescharacteristic
19C4×#11758194C2211yescharacteristic
20#16839530—22no—
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#14519708—24no—
42#14519708—24no—
43C4⋊#92418D412yes—
44#14519701—24no—
45C22⋊#92418—21no—
46#14519688—21no—
47#14519688D412yes—
48C4×#92418—21no—
49C4×#92418—21no—
50C4×#92418—21no—
51C2×Q16×C61—22no—
52C2×M16×C61D412yes—
53C4⋊Q8×C61D411yescharacteristic
54C4⋊Q8×C61D411yescharacteristic
55C4⋊Q8×C61D411yescharacteristic
56C4⋊Q8×C61D411yescharacteristic
57#125×C61—21no—
58#125×C61—21no—
59C22⋊Q8×C61—21no—
60C22⋊Q8×C61—21no—
61C22⋊Q8×C61—21no—
62C22⋊Q8×C61—21no—
63C4×Q8×C61—21no—
64C4×Q8×C61—21no—
65C4×Q8×C61—21no—
66C4×Q8×C61—21no—
67#117×C61—21no—
68#117×C61—21no—
69C2×C42×C61C2311yescharacteristic
70#106×C61—22no—
71C4⋊C8×C61—21no—
72C4⋊C8×C61—21no—
73#103×C61—22no—
74#103×C61—22no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77C2×Q8×C61—21no—
78C2×Q8×C61—21no—
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61#4111yescharacteristic
82C22×Dic61—22no—
83C2×Dic122—41no—
84C2×Dic122—41no—
85C2×Dic122—82no—
86Q16×C61—82no—
87M16×C61—42no—
88C2×C8×C61—22no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61—21no—
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—41no—
98C4⋊C4×C61—41no—
99C4⋊C4×C61—41no—
100C4⋊C4×C61—41no—
101C22⋊C4×C61—41no—
102C22⋊C4×C61—41no—
103C42×C61—41no—
104C42×C61—41no—
105C42×C61C2×D411yescharacteristic
106C42×C61C2×D411yescharacteristic
107C42×C61C2×D411yescharacteristic
108C42×C61C2×D411yescharacteristic
109C4⋊Dic61—22no—
110C4⋊Dic61—22no—
111C4⋊Dic61—41no—
112C4⋊Dic61—41no—
113#11758194—24no—
114#11758194—22no—
115#11758194—41no—
116C2×#92418—22no—
117C2×#92418—22no—
118C2×#92418—41no—
119C23×C61#12111yescharacteristic
120Q8×C61—41no—
121Q8×C61—41no—
122Q8×C61—42no—
123Q8×C61—41no—
124Q8×C61—41no—
125Q8×C61—42no—
126C2×C4×C61—21no—
127C2×C4×C61—22no—
128C2×C4×C61—21no—
129C2×C4×C61—22no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61#12011yescharacteristic
135C2×C4×C61#12011yescharacteristic
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61#12711yescharacteristic
140C2×Dic61—42no—
141C2×Dic61—82no—
142Dic122—82no—
143Dic122—82no—
144C8×C61—42no—
145#92418—24no—
146#92418—42no—
147#92418—42no—
148#10119—611nomaximal, Sylow
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—21no—
152C22×C61#39211yescharacteristic
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—81no—
157C4×C61—81no—
158C4×C61—81no—
159C4×C61—81no—
160C4×C61#45511yescharacteristic
161C4×C61#45612yes—
162C4×C61#45611yescharacteristic
163C4×C61#47112yes—
164Dic61—82no—
165#3255D6111yescharacteristic
166#3229—611no—
167#3165—611no—
168#1997—611no—
169#1526—611no—
170#1485—611no—
171#1299—611no—
172C2×C61—41no—
173C2×C61#200511yescharacteristic
174C2×C61#200711yescharacteristic
175C2×C61#201911yescharacteristic
176#533C2×D6111yescharacteristic
177#533C2×D6111yescharacteristic
178C2×C4⋊Q8—611no—
179#501—1222no—
180#500—1221no—
181#498—1221no—
182#498—1221no—
183#487—1221no—
184#483—1222no—
185#475—21no—
186#475—21no—
187#475—1222no—
188#470—22no—
189#462—21no—
190#462—21no—
191#431—611no—
192#428—1222no—
193#423C2×D6111yescharacteristic
194C4×M16—611no—
195C4×M16—611no—
196#343—1222no—
197#330—1222no—
198Q8⋊C8—1222no—
199C61#1011911yescharacteristic, complemented, Sylow
200C22×Q8—1221no—
201C2×Q16—22no—
202C2×M16—1221no—
203C2×M16—1221no—
204C2×M16C22⋊D6112yes—
205C4⋊Q8—1221no—
206C4⋊Q8—1221no—
207C4⋊Q8—1221no—
208C4⋊Q8—1221no—
209C4⋊Q8C22⋊D6111yescharacteristic
210C4⋊Q8C22⋊D6111yescharacteristic
211C4⋊Q8C22⋊D6111yescharacteristic
212C4⋊Q8C22⋊D6111yescharacteristic
213#125—21no—
214#125—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223#117—21no—
224#117—21no—
225C2×C4⋊C4—612no—
226C2×C4⋊C4—1221no—
227C2×C42C22×D6111yescharacteristic
228#107—1224no—
229#106—22no—
230#106—1224no—
231#106—1224no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—612no—
235#103—22no—
236#103—22no—
237#103—1224no—
238C22⋊C8—1221no—
239#97—612no—
240#97—1221no—
241C4×C8—1221no—
242C4×C8—1221no—
243C4×C8—1221no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—4882no—
249C2×Q8—2441no—
250C2×Q8—2441no—
251C22×C4—1222no—
252C22×C4D4×D6111yescharacteristic
253C22×C4#1175822411yescharacteristic
254C22×C4C2×C22⋊D6111yescharacteristic
255Q16—82no—
256M16—42no—
257M16—1224no—
258M16—1222no—
259M16—2441no—
260C2×C8—22no—
261C2×C8—1222no—
262C2×C8—1222no—
263C2×C8—2441no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—1222no—
277C4⋊C4—1222no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C42—41no—
283C42—41no—
284C42D4×D6111yescharacteristic
285C42D4×D6111yescharacteristic
286C42C2×C22⋊D6111yescharacteristic
287C42C2×C22⋊D6111yescharacteristic
288C23#1451974811yescharacteristic
289Q8—41no—
290Q8—41no—
291Q8—42no—
292Q8—41no—
293Q8—41no—
294Q8—42no—
295Q8—4882no—
296Q8—4882no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—4882no—
306C2×C4—2442no—
307C2×C4#1451974511yescharacteristic
308C2×C4#1451974511yescharacteristic
309C2×C4#1451974611yescharacteristic
310C2×C4#1451974911yescharacteristic
311C2×C4#1451974911yescharacteristic
312C2×C4#1451977311yescharacteristic
313C8—42no—
314C8—1224no—
315C8—2442no—
316C8—2442no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22#1683967011yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—4882no—
329C4#1683994011yescharacteristic
330C4#1683994312yes—
331C4#1683994511yescharacteristic
332C4#1684002712yes—
333C2—41no—
334C2#2144685211yescharacteristic
335C2#2144686611yescharacteristic
336C2#2144693011yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3255C2
socleC22×C61#392
prime radicalC23×C61#121
hypercenter#3255D61
hypocenterC61#10119
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
199. C61#10119semidirect

Maximal subgroups

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

subgroupindex
#214569602
#214563662
#214468072
#214444142
#214442402
#214437082
C61×#32552
#1011961