Finite Group Databasebeta

← #29581541

Subgroups — #29581541

Order 15616.

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

normal subgroups only

This group has 466 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#21457210C211yesmaximal, characteristic
2#21456767C211yesmaximal, characteristic
3#21446809C211yesmaximal, characteristic
4#21444413C211yesmaximal, characteristic
5#21444240C211yesmaximal, characteristic
6#21443709C211yesmaximal, characteristic
7C61×#3139C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840141—21no—
10#16840112—21no—
11#16840047—21no—
12#16840044—21no—
13#16840018—22no—
14#16839972—21no—
15#16839969—22no—
16#16839969—21no—
17#16839918—22no—
18#16839815C2211yescharacteristic
19#16839786C2211yescharacteristic
20C4×#11758194C2211yescharacteristic
21#16839531—22no—
22#16839503—22no—
23Q8⋊#92418—22no—
24C61×#533C2211yescharacteristic
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#491—22no—
27C61×#487—21no—
28C61×#487—21no—
29C61×#419—22no—
30C4×M16×C61C2211yescharacteristic
31C2×#11758202—22no—
32C2×#11758202—24no—
33C2×#11758194—21no—
34C2×#11758194D412yes—
35#14519777D411yescharacteristic
36#14519777D411yescharacteristic
37#14519769—21no—
38#14519756—21no—
39#14519756—21no—
40C4×Dic122—21no—
41C4×Dic122—21no—
42#14519736—21no—
43#14519708—22no—
44C4⋊#92418—21no—
45C4⋊#92418—21no—
46C4⋊#92418D412yes—
47#14519701—22no—
48#14519701—24no—
49#14519700—22no—
50C22⋊#92418—21no—
51#14519688—21no—
52C4×#92418—21no—
53C4×#92418—21no—
54C22×Q8×C61D412yes—
55C2×M16×C61—21no—
56C4⋊Q8×C61—21no—
57C4⋊Q8×C61—21no—
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61D411yescharacteristic
61C4⋊Q8×C61D411yescharacteristic
62#125×C61—21no—
63C22⋊Q8×C61—21no—
64C22⋊Q8×C61—21no—
65C4×Q8×C61—21no—
66C4×Q8×C61—21no—
67#117×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C42×C61C2311yescharacteristic
71#103×C61—24no—
72#103×C61—24no—
73#97×C61D412yes—
74C4×C8×C61—21no—
75C2×Q8×C61—22no—
76C2×Q8×C61—21no—
77C2×Q8×C61—22no—
78C2×Q8×C61—21no—
79C2×Q8×C61—44no—
80C22×C4×C61—22no—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61#4111yescharacteristic
84C2×Dic122—21no—
85C2×Dic122—21no—
86M16×C61—42no—
87C2×C8×C61—22no—
88C4⋊C4×C61—21no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61—21no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61—41no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—41no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C22⋊C4×C61—41no—
99C42×C61—41no—
100C42×C61C2×D411yescharacteristic
101C42×C61C2×D411yescharacteristic
102C42×C61C2×D411yescharacteristic
103C42×C61C2×D411yescharacteristic
104C22⋊Dic61—41no—
105#11758202—82no—
106#11758202—84no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110C4⋊Dic61—21no—
111#11758196—41no—
112#11758196—41no—
113C4×Dic61—41no—
114#11758194—22no—
115#11758194—41no—
116#11758194—42no—
117C2×#92418—22no—
118C2×#92418—22no—
119C2×#92418—41no—
120C23×C61#12111yescharacteristic
121Q8×C61—44no—
122Q8×C61—41no—
123Q8×C61—44no—
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—42no—
133C2×C4×C61—82no—
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—41no—
141C2×Dic61—41no—
142Dic122—42no—
143Dic122—41no—
144Dic122—41no—
145C8×C61—42no—
146#92418—42no—
147#92418—42no—
148#92418—42no—
149#10115—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61#39211yescharacteristic
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—82no—
158C4×C61—81no—
159C4×C61—81no—
160C4×C61#45511yescharacteristic
161C4×C61#45512yes—
162C4×C61#45611yescharacteristic
163C4×C61#47312yes—
164Dic61—81no—
165Dic61—81no—
166#3255—611no—
167#3209—611no—
168#3139D6111yescharacteristic
169#1997—611no—
170#1526—611no—
171#1485—611no—
172#1299—611no—
173C2×C61—41no—
174C2×C61#200411yescharacteristic
175C2×C61#200711yescharacteristic
176C2×C61#202011yescharacteristic
177#533—611no—
178#533C2×D6111yescharacteristic
179C2×C4⋊Q8C2×D6111yescharacteristic
180#494—1221no—
181#491—22no—
182#487—21no—
183#487—21no—
184#487—1221no—
185#475—1221no—
186#475—1221no—
187#470—1222no—
188#462—1222no—
189#462—1221no—
190#462—1221no—
191#451—1222no—
192#431—611no—
193#423—611no—
194#419—22no—
195C4×M16—611no—
196C4×M16C2×D6111yescharacteristic
197#343—1222no—
198#330—1222no—
199Q8⋊C8—1222no—
200C61#1011511yescharacteristic, complemented, Sylow
201C22×Q8C22⋊D6112yes—
202C2×Q16—1224no—
203C2×Q16—1222no—
204C2×M16—21no—
205C2×M16—612no—
206C2×M16—1221no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—611no—
212C4⋊Q8—611no—
213C4⋊Q8C22⋊D6111yescharacteristic
214C4⋊Q8C22⋊D6111yescharacteristic
215#125—21no—
216#125—1221no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C22⋊Q8—1221no—
220C22⋊Q8—1221no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—1221no—
224C4×Q8—1221no—
225#117—21no—
226#117—1221no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C42C22×D6111yescharacteristic
230#106—1222no—
231C4⋊C8—612no—
232C4⋊C8—1221no—
233C4⋊C8—1221no—
234#103—24no—
235#103—24no—
236#103—1222no—
237#103—1224no—
238#103—1222no—
239C22⋊C8—1221no—
240#97—1221no—
241#97C22⋊D6112yes—
242C4×C8—21no—
243C4×C8—1221no—
244C4×C8—1221no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—21no—
249C2×Q8—44no—
250C2×Q8—1221no—
251C2×Q8—1221no—
252C22×C4—22no—
253C22×C4D4×D6111yescharacteristic
254C22×C4#1175822411yescharacteristic
255C22×C4C2×C22⋊D6111yescharacteristic
256Q16—4882no—
257Q16—4884no—
258M16—42no—
259M16—1222no—
260M16—2441no—
261M16—2442no—
262C2×C8—22no—
263C2×C8—1222no—
264C2×C8—1222no—
265C2×C8—2441no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—1221no—
277C4⋊C4—1221no—
278C4⋊C4—1221no—
279C4⋊C4—1221no—
280C4⋊C4—2441no—
281C4⋊C4—2441no—
282C22⋊C4—41no—
283C22⋊C4—2441no—
284C42—41no—
285C42—2441no—
286C42D4×D6111yescharacteristic
287C42D4×D6111yescharacteristic
288C42C2×C22⋊D6111yescharacteristic
289C42C2×C22⋊D6111yescharacteristic
290C23#1451974911yescharacteristic
291Q8—44no—
292Q8—41no—
293Q8—44no—
294Q8—41no—
295Q8—42no—
296Q8—2441no—
297Q8—2442no—
298Q8—2441no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4#1451974511yescharacteristic
310C2×C4#1451974511yescharacteristic
311C2×C4#1451974611yescharacteristic
312C2×C4#1451974811yescharacteristic
313C2×C4#1451974911yescharacteristic
314C2×C4#1451977311yescharacteristic
315C8—42no—
316C8—2442no—
317C8—2442no—
318C8—2442no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22#1683967011yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—81no—
327C4—82no—
328C4—81no—
329C4—4881no—
330C4—4881no—
331C4#1683993811yescharacteristic
332C4#1683993912yes—
333C4#1683994311yescharacteristic
334C4#1684003512yes—
335C2—41no—
336C2#2144684811yescharacteristic
337C2#2144686411yescharacteristic
338C2#2144693711yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3139C2
socleC22×C61#392
prime radicalC23×C61#121
hypercenter#3139D61
hypocenterC61#10115
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
200. C61#10115semidirect

Maximal subgroups

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

subgroupindex
#214572102
#214567672
#214468092
#214444132
#214442402
#214437092
C61×#31392
#1011561