Finite Group Databasebeta

← #29581574

Subgroups — #29581574

Order 15616.

454 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 87 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#21457210C211yesmaximal, characteristic
2#21456960C211yesmaximal, characteristic
3#21456365C211yesmaximal, characteristic
4#21446808C211yesmaximal, characteristic
5#21444241C211yesmaximal, characteristic
6#21443709C211yesmaximal, characteristic
7C61×#1526C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9C2×#14519777C2211yescharacteristic
10#16840171—22no—
11#16840167—21no—
12#16840160—21no—
13#16840158—21no—
14#16840112—21no—
15#16840090—22no—
16#16840047—21no—
17#16840047—22no—
18#16840044—21no—
19#16840018—22no—
20#16839972—21no—
21#16839969—21no—
22#16839815C2211yescharacteristic
23#16839804—22no—
24#16839786C2211yescharacteristic
25#16839530—22no—
26#16839494—22no—
27C61×#533C2211yescharacteristic
28C4×M16×C61C2211yescharacteristic
29C4×M16×C61C2211yescharacteristic
30C61×#330—22no—
31C22×Dic122—21no—
32C2×#11758202—22no—
33C2×#11758194D412yes—
34#14519777—21no—
35#14519777—21no—
36#14519777—21no—
37#14519777—21no—
38#14519777D411yescharacteristic
39#14519777D411yescharacteristic
40#14519769—21no—
41#14519756—21no—
42#14519756—21no—
43C4×Dic122—21no—
44C4×Dic122—21no—
45#14519736—21no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61D412yes—
48#14519711—24no—
49#14519709—24no—
50#14519708—22no—
51#14519708—24no—
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C4⋊#92418D412yes—
55#14519701—22no—
56#14519701—24no—
57#14519700—22no—
58C22⋊#92418—21no—
59C4×#92418—21no—
60C2×M16×C61—21no—
61C2×M16×C61—21no—
62C4⋊Q8×C61D411yescharacteristic
63C4⋊Q8×C61D411yescharacteristic
64#125×C61—21no—
65C22⋊Q8×C61—21no—
66C22⋊Q8×C61—21no—
67C4×Q8×C61—21no—
68C4×Q8×C61—21no—
69#117×C61—21no—
70C2×C42×C61C2311yescharacteristic
71#97×C61—21no—
72#97×C61D412yes—
73C4×C8×C61—21no—
74C4×C8×C61—21no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61#4111yescharacteristic
80C22×Dic61—22no—
81C2×Dic122—21no—
82C2×Dic122—21no—
83C2×Dic122—41no—
84C2×Dic122—41no—
85C2×Dic122—82no—
86M16×C61—24no—
87M16×C61—22no—
88M16×C61—41no—
89C2×C8×C61—22no—
90C2×C8×C61—41no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C22⋊C4×C61—41no—
98C42×C61—41no—
99C42×C61C2×D411yescharacteristic
100C42×C61C2×D411yescharacteristic
101C42×C61C2×D411yescharacteristic
102C42×C61C2×D411yescharacteristic
103C22⋊Dic61—41no—
104#11758202—82no—
105C4⋊Dic61—22no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—22no—
109C4⋊Dic61—21no—
110C4⋊Dic61—21no—
111C4⋊Dic61—41no—
112C4⋊Dic61—41no—
113#11758196—41no—
114#11758196—41no—
115C4×Dic61—41no—
116#11758194—42no—
117C2×#92418—22no—
118C2×#92418—22no—
119C23×C61#12111yescharacteristic
120Q8×C61—41no—
121Q8×C61—42no—
122Q8×C61—41no—
123C2×C4×C61—21no—
124C2×C4×C61—22no—
125C2×C4×C61—21no—
126C2×C4×C61—22no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61#12011yescharacteristic
130C2×C4×C61#12011yescharacteristic
131C2×C4×C61#12011yescharacteristic
132C2×C4×C61#12111yescharacteristic
133C2×C4×C61#12111yescharacteristic
134C2×C4×C61#12711yescharacteristic
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—42no—
138C2×Dic61—82no—
139Dic122—41no—
140Dic122—41no—
141Dic122—42no—
142Dic122—82no—
143Dic122—82no—
144C8×C61—24no—
145C8×C61—42no—
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#45511yescharacteristic
159C4×C61#45612yes—
160C4×C61#45611yescharacteristic
161C4×C61#47112yes—
162Dic61—81no—
163Dic61—81no—
164Dic61—82no—
165#3255—611no—
166#3229—611no—
167#3165—611no—
168#1997—611no—
169#1526D6111yescharacteristic
170#1485—611no—
171#1299—611no—
172C2×C61—41no—
173C2×C61#200511yescharacteristic
174C2×C61#200711yescharacteristic
175C2×C61#201911yescharacteristic
176#533—611no—
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—1222no—
186#475—1221no—
187#475—1221no—
188#470—1222no—
189#462—1221no—
190#462—1221no—
191#431—611no—
192#428—1222no—
193#423—611no—
194C4×M16C2×D6111yescharacteristic
195C4×M16C2×D6111yescharacteristic
196#343—1222no—
197#330—22no—
198Q8⋊C8—1222no—
199C61#1011911yescharacteristic, complemented, Sylow
200C22×Q8—1221no—
201C2×Q16—1222no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—612no—
205C4⋊Q8—611no—
206C4⋊Q8—611no—
207C4⋊Q8—1221no—
208C4⋊Q8—1221no—
209C4⋊Q8—1221no—
210C4⋊Q8—1221no—
211C4⋊Q8D24411yescharacteristic
212C4⋊Q8D24411yescharacteristic
213#125—21no—
214#125—1221no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—1221no—
218C22⋊Q8—1221no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—1221no—
222C4×Q8—1221no—
223#117—21no—
224#117—1221no—
225C2×C4⋊C4—612no—
226C2×C4⋊C4—1221no—
227C2×C42C22×D6111yescharacteristic
228#107—1224no—
229#106—1222no—
230#106—1224no—
231#106—1224no—
232C4⋊C8—612no—
233C4⋊C8—1221no—
234C4⋊C8—1221no—
235#103—1222no—
236#103—1222no—
237#103—1224no—
238C22⋊C8—1221no—
239#97—21no—
240#97C22⋊D6112yes—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—1221no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—1221no—
247C2×Q8—1221no—
248C2×Q8—4882no—
249C2×Q8—2441no—
250C2×Q8—2441no—
251C22×C4—1222no—
252C22×C4#1175822211yescharacteristic
253C22×C4D4×D6111yescharacteristic
254C22×C4D4×D6111yescharacteristic
255Q16—4882no—
256M16—24no—
257M16—22no—
258M16—41no—
259M16—2442no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—1222no—
263C2×C8—1222no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—1221no—
271C4⋊C4—1222no—
272C4⋊C4—1221no—
273C4⋊C4—1222no—
274C4⋊C4—1221no—
275C4⋊C4—1221no—
276C4⋊C4—2441no—
277C4⋊C4—2441no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C22⋊C4—41no—
281C22⋊C4—2441no—
282C42—41no—
283C42—2441no—
284C42C2×D24411yescharacteristic
285C42D4×D6111yescharacteristic
286C42D4×D6111yescharacteristic
287C42C2×C22⋊D6111yescharacteristic
288C23#1451975011yescharacteristic
289Q8—41no—
290Q8—42no—
291Q8—41no—
292Q8—4882no—
293Q8—4882no—
294Q8—2441no—
295Q8—2441no—
296Q8—2442no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—4882no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2442no—
307C2×C4#1451974511yescharacteristic
308C2×C4#1451974711yescharacteristic
309C2×C4#1451974711yescharacteristic
310C2×C4#1451975011yescharacteristic
311C2×C4#1451975111yescharacteristic
312C2×C4#1451977311yescharacteristic
313C8—24no—
314C8—42no—
315C8—2442no—
316C8—2442no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22#1683967111yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—81no—
325C4—81no—
326C4—4882no—
327C4—4881no—
328C4—4881no—
329C4#1683994111yescharacteristic
330C4#1683994612yes—
331C4#1683994611yescharacteristic
332C4#1684002812yes—
333C2—41no—
334C2#2144685411yescharacteristic
335C2#2144686711yescharacteristic
336C2#2144693211yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1526C2
socleC22×C61#392
prime radicalC23×C61#121
hypercenter#1526D61
hypocenterC61#10119
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1526

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
#214572102
#214569602
#214563652
#214468082
#214442412
#214437092
C61×#15262
#1011961