Finite Group Databasebeta

← #29598217

Subgroups — #29598217

Order 15616.

438 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454511C211yesmaximal, characteristic
2#21447584C211yesmaximal, characteristic, complemented
3#21447447C211yesmaximal, characteristic, complemented
4#21447429C211yesmaximal, characteristic, complemented
5#21447429C211yesmaximal, characteristic, complemented
6#21447423C211yesmaximal, characteristic, complemented
7C61×#2117C211yesmaximal, characteristic, complemented
8C22×C8×D61C2211yescharacteristic
9#16840269C411yescharacteristic
10C2×#14519717C2211yescharacteristic
11C2×#14519717C2211yescharacteristic
12#16839831—22no—
13C8×C22⋊D61—22no—
14#16839790C411yescharacteristic
15#16839729C411yescharacteristic
16#16839725C411yescharacteristic
17#16839717C2211yescharacteristic
18#16839582—22no—
19#16839546—22no—
20#16839541—22no—
21#16839541—22no—
22#16839539—22no—
23C2×#131×C61C2211yescharacteristic
24C2×M32×C61C2211yescharacteristic
25C2×M32×C61C2211yescharacteristic
26C61×#350—22no—
27C22×C4×D61C2×C411yescharacteristic
28C2×#11758194C2×C411yescharacteristic
29C22×#92418C2×C411yescharacteristic
30C2×C8×D61—21no—
31C2×C8×D61—21no—
32C2×C8×D61—21no—
33C2×C8×D61—21no—
34C2×C8×D61—21no—
35C2×C8×D61—21no—
36#14519758—21no—
37#14519753—21no—
38#14519748—21no—
39#14519745—21no—
40C4×C22⋊D61—22no—
41#14519736C2×C411yescharacteristic
42#14519717—21no—
43#14519717—21no—
44#14519717—21no—
45#14519717—21no—
46C2×#11758185—21no—
47C2×#11758185D412yescomplemented
48#14519706—22no—
49C22⋊#92418—22no—
50#14519689—22no—
51#14519687D412yes—
52C8×Dic61D412yes—
53C2×#41×C61C2×C411yescharacteristic
54#131×C61—21no—
55#131×C61—21no—
56#131×C61—21no—
57#131×C61—21no—
58C2×M16×C61—21no—
59C2×M16×C61C2×C411yescharacteristic
60C22×C8×C61—21no—
61C22×C8×C61C2311yescharacteristic
62M32×C61—21no—
63M32×C61—21no—
64M32×C61—21no—
65M32×C61—21no—
66C2×C16×C61—21no—
67C2×C16×C61D412yescomplemented
68C23×D61M1611yescharacteristic
69#41×C61—21no—
70#41×C61—21no—
71#41×C61—21no—
72#41×C61—21no—
73C2×Q8×C61M1611yescharacteristic
74C2×D4×C61—21no—
75C2×D4×C61M1611yescharacteristic
76C22×C4×C61—21no—
77C22×C4×C61C22×C411yescharacteristic
78C2×C22⋊D61—22no—
79C22×Dic61M1611yescharacteristic
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86M16×C61—21no—
87M16×C61—21no—
88M16×C61—41no—
89M16×C61—41no—
90C2×C8×C61—21no—
91C2×C8×C61—41no—
92C2×C8×C61—41no—
93C2×C8×C61C2×D411yescharacteristic
94C2×C8×C61C2×D411yescharacteristic
95C2×C8×C61C2×D411yescharacteristic
96C2×C8×C61C2×D411yescharacteristic
97C2×C8×C61#4111yescharacteristic
98C2×C8×C61#4111yescharacteristic
99C22⋊Dic61—22no—
100C22⋊Dic61—21no—
101#11758198—22no—
102C4⋊Dic61—21no—
103#11758196—22no—
104C4×Dic61C22⋊C412yes—
105#11758194—21no—
106#11758194—21no—
107C2×#92418—21no—
108C2×#92418—21no—
109C2×#92418—21no—
110C2×#92418C22⋊C412yes—
111C8×D61—41no—
112C8×D61—41no—
113C8×D61—41no—
114C8×D61—41no—
115C16×C61—22no—
116C16×C61—41no—
117#11758185—22no—
118#11758185—41no—
119C23×C61—21no—
120C23×C61C2×M1611yescharacteristic
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—42no—
125Q8×C61—21no—
126Q8×C61—21no—
127D4×C61—21no—
128D4×C61—21no—
129D4×C61—41no—
130D4×C61—41no—
131C2×C4×C61—21no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61C2×C22⋊C411yescharacteristic
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61C2×M1611yescharacteristic
138C2×C4×C61#13111yescharacteristic
139C2×C4×C61#13111yescharacteristic
140C22⋊D61—44no—
141C2×Dic61—22no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C4×D61—41no—
146C4×D61—41no—
147C4×D61—41no—
148C4×D61—41no—
149C8×C61—41no—
150C8×C61#12111yescharacteristic
151C8×C61#12111yescharacteristic
152C8×C61#12411yescharacteristic
153C8×C61#12711yescharacteristic
154#92418—22no—
155#92418—41no—
156#13200—611nomaximal, Sylow
157C22×C61—41no—
158C22×C61—42no—
159C22×C61—41no—
160C22×C61#40711yescharacteristic
161C22×C61#43511yescharacteristic
162C22×C61#43511yescharacteristic
163C2×D61—22no—
164C2×D61—22no—
165C2×D61—22no—
166C2×D61—41no—
167C2×D61—41no—
168C2×D61—41no—
169C2×D61—41no—
170C4×C61—41no—
171C4×C61#39011yescharacteristic
172C4×C61#40811yescharacteristic
173C4×C61#43511yescharacteristic
174C4×C61#43511yescharacteristic
175Dic61—42no—
176Dic61—41no—
177#2929—611no—
178#2153—611no—
179#2117—611no—
180#2117D6111yescharacteristic, complemented
181#2112—611no—
182#2112—611no—
183#2110—611no—
184C2×C61—21no—
185C2×C61—21no—
186C2×C61—21no—
187C2×C61—81no—
188C2×C61#195511yescharacteristic
189D61—42no—
190C2×#131C2×D6111yescharacteristic
191C23×C8—611no—
192#525—611no—
193C2×M32—611no—
194C2×M32—611no—
195C2×M32C2×D6111yescharacteristic
196C2×M32C2×D6111yescharacteristic
197#435—1222no—
198C8×D4—1222no—
199#424—611no—
200#408—611no—
201#407—611no—
202#405—611no—
203#364—1222no—
204#350—22no—
205#350—1222no—
206C22⋊C16—1222no—
207C22⋊C16—1222no—
208#347—1222no—
209C61#1320011yescharacteristic, complemented, Sylow
210C2×#41C4×D6111yescharacteristic
211C23×C4—611no—
212#131—21no—
213#131—21no—
214#131—21no—
215#131—21no—
216C2×M16—21no—
217C2×M16—611no—
218C2×M16C4×D6111yescharacteristic
219C22×C8—21no—
220C22×C8—611no—
221C22×C8—1221no—
222C22×C8—1221no—
223C22×C8—1221no—
224C22×C8—1221no—
225C22×C8—1221no—
226C22×C8—1221no—
227C22×C8C22×D6111yescharacteristic
228#123—1221no—
229C22⋊Q8—1221no—
230#121—1221no—
231#120—1221no—
232C4×D4—1222no—
233#117—611no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—1221no—
239M32—1221no—
240M32—1221no—
241M32—1221no—
242C2×C16—21no—
243C2×C16—612no—
244C2×C16—1221no—
245C2×C16C22⋊D6112yescomplemented
246C4⋊C8—1222no—
247C22⋊C8—1222no—
248C22⋊C8—1222no—
249#97—612no—
250C4×C8—612no—
251C24—611no—
252#41—21no—
253#41—21no—
254#41—21no—
255#41—21no—
256C2×Q8#1175818911yescharacteristic
257C2×D4—21no—
258C2×D4—1222no—
259C2×D4#1175818911yescharacteristic
260C22×C4—21no—
261C22×C4—611no—
262C22×C4—1221no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4C2×C4×D6111yescharacteristic
269M16—21no—
270M16—21no—
271M16—41no—
272M16—41no—
273M16—1221no—
274M16—1221no—
275C2×C8—21no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—612no—
279C2×C8—1221no—
280C2×C8—1221no—
281C2×C8—1221no—
282C2×C8—2441no—
283C2×C8—2441no—
284C2×C8—2441no—
285C2×C8—2441no—
286C2×C8#1175822211yescharacteristic
287C2×C8D4×D6111yescharacteristic
288C2×C8D4×D6111yescharacteristic
289C2×C8D4×D6111yescharacteristic
290C2×C8#1175822411yescharacteristic
291C2×C8C2×C22⋊D6111yescharacteristic
292C4⋊C4—1222no—
293C4⋊C4—1221no—
294C22⋊C4—1222no—
295C22⋊C4—1222no—
296C22⋊C4—1221no—
297C42—612no—
298C16—22no—
299C16—41no—
300C16—1222no—
301C16—2441no—
302C23—21no—
303C23—1221no—
304C23—1221no—
305C23—1221no—
306C23—2442no—
307C23M16×D6111yescharacteristic
308Q8—21no—
309Q8—21no—
310D4—21no—
311D4—21no—
312D4—41no—
313D4—41no—
314D4—2444no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—1222no—
319C2×C4—1221no—
320C2×C4—1221no—
321C2×C4—1221no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4C22⋊C4×D6111yescharacteristic
327C2×C4#1451973711yescharacteristic
328C2×C4C4×C22⋊D6111yescharacteristic
329C2×C4C2×#1175818911yescharacteristic
330C2×C4#1451978311yescharacteristic
331C2×C4#1451978511yescharacteristic
332C8—41no—
333C8—1222no—
334C8—2441no—
335C8#1451974911yescharacteristic
336C8#1451975011yescharacteristic
337C8#1451976311yescharacteristic
338C8#1451977311yescharacteristic
339C22—41no—
340C22—42no—
341C22—41no—
342C22—1222no—
343C22—1222no—
344C22—1222no—
345C22—2441no—
346C22—2441no—
347C22—2441no—
348C22—2441no—
349C22#1683972411yescharacteristic
350C22#1683983111yescharacteristic
351C22#1683983611yescharacteristic
352C4—41no—
353C4—2442no—
354C4—2441no—
355C4#1683966211yescharacteristic
356C4#1683972711yescharacteristic
357C4#1683983011yescharacteristic
358C4#1683983311yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—81no—
363C2—2442no—
364C2#2144661511yescharacteristic
subgroupG/H
centerC8#14519773
commutatorC22×C4×C61C22×C4
FrattiniC22×C8C22×D61
FittingC61×#2117C2
socleC2×C61#1955
prime radical#16840269C4
cosocleC22×C8×C61C23
hypercenter#2117D61
hypocenterC61#13200
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
2. #21447584C2semidirect
3. #21447447C2semidirect
4. #21447429C2semidirect
5. #21447429C2semidirect
6. #21447423C2semidirect
7. C61×#2117C2semidirect
47. C2×#11758185D4semidirect
67. C2×C16×C61D4semidirect
180. #2117D61semidirect
209. C61#13200semidirect
245. C2×C16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214545112
#214475842
#214474472
#214474292
#214474232
C61×#21172
#1320061