Finite Group Databasebeta

← #29596906

Subgroups — #29596906

Order 15616.

508 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C16×D61C211yesmaximal, characteristic, complemented
2#21454515C211yesmaximal, characteristic
3C2×#16839547C211yesmaximal, characteristic, complemented
4C2×#16839545C211yesmaximal, characteristic, complemented
5#21447429C211yesmaximal, characteristic, complemented
6#21443908C211yesmaximal, characteristic, complemented
7C61×#1379C211yesmaximal, characteristic, complemented
8C2×#14519783C2211yescharacteristic
9C22×C8×D61C2211yescharacteristic
10#16840274C411yescharacteristic
11C2×#14519717C2211yescharacteristic
12C22×#11758185C2211yescharacteristic
13C2×C16×D61—22no—
14C2×C16×D61—22no—
15C2×C16×D61—22no—
16#16839835—22no—
17C8×D244—22no—
18#16839789C411yescharacteristic
19#16839730C411yescharacteristic
20#16839725C411yescharacteristic
21#16839547—22no—
22#16839545—22no—
23#16839541—22no—
24C2×M32×C61C2211yescharacteristic
25C22×C16×C61C2211yescharacteristic
26C61×#405C2211yescharacteristic
27C2×#11758222C2×C411yescharacteristic
28C22×C4×D61C2×C411yescharacteristic
29#14519783—21no—
30#14519783—21no—
31#14519783—21no—
32#14519783—21no—
33C2×#11758194C2×C411yescharacteristic
34C2×#11758189—21no—
35C22×#92418C2×C411yescharacteristic
36C2×C8×D61—21no—
37C2×C8×D61—21no—
38C2×C8×D61—21no—
39C2×C8×D61—21no—
40C2×C8×D61—21no—
41C2×C8×D61D412yes—
42C2×C8×D61D412yes—
43#14519761—21no—
44#14519757C812yes—
45#14519752C812yes—
46#14519747—21no—
47C4×D244—22no—
48#14519717—21no—
49#14519717—21no—
50C2×#11758185—21no—
51C2×#11758185—21no—
52C2×#11758185—21no—
53C2×#11758185D412yescomplemented
54C16×D61—28no—
55C4⋊#92418—22no—
56#14519689—22no—
57#14519689—22no—
58C2×M16×C61C2×C411yescharacteristic
59C22×C8×C61C2311yescharacteristic
60#117×C61C2×C411yescharacteristic
61M32×C61—21no—
62M32×C61—21no—
63C2×C16×C61—21no—
64C2×C16×C61—21no—
65C2×C16×C61—21no—
66C2×C16×C61D412yescomplemented
67#97×C61—21no—
68C4×C8×C61—21no—
69C23×D61M1611yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C2×C22⋊D61—21no—
72C22×Dic61M1611yescharacteristic
73#11758222—21no—
74#11758222—21no—
75#11758222—21no—
76#11758222—21no—
77C2×D244—22no—
78C2×D244C2×C811yescharacteristic
79C2×C4×D61—21no—
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61C22⋊C412yes—
85C2×C4×D61C2×C812yes—
86C2×Dic122C2×C811yescharacteristic
87M16×C61—21no—
88M16×C61—21no—
89C2×C8×C61—21no—
90C2×C8×C61C2×D411yescharacteristic
91C2×C8×C61C2×D411yescharacteristic
92C2×C8×C61C2×D411yescharacteristic
93C2×C8×C61C2×D411yescharacteristic
94C2×C8×C61#4111yescharacteristic
95C2×C8×C61#4111yescharacteristic
96C4⋊C4×C61C2×C812yes—
97C22⋊C4×C61—21no—
98C42×C61—21no—
99#11758198—22no—
100#11758198—22no—
101C4⋊Dic61—22no—
102#11758194—21no—
103#11758194—21no—
104C2×#92418—21no—
105C2×#92418—21no—
106C2×#92418—21no—
107C2×#92418C22⋊C412yes—
108#11758189—41no—
109#11758189—41no—
110C8×D61—22no—
111C8×D61—22no—
112C8×D61—22no—
113C8×D61—22no—
114C8×D61—41no—
115C8×D61—41no—
116C16×C61—22no—
117C16×C61—41no—
118#11758185—22no—
119#11758185—41no—
120C23×C61C2×M1611yescharacteristic
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—24no—
124C22×D61—21no—
125C22×D61C22⋊C812yes—
126C2×C4×C61—22no—
127C2×C4×C61C2×C22⋊C411yescharacteristic
128C2×C4×C61C4×D411yescharacteristic
129C2×C4×C61C4×D411yescharacteristic
130C2×C4×C61C22×C811yescharacteristic
131C2×C4×C61#13111yescharacteristic
132C2×C4×C61#13111yescharacteristic
133C22⋊D61—41no—
134C22⋊D61—41no—
135C2×Dic61—21no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61C22⋊C812yes—
139D244—21no—
140D244—21no—
141D244—44no—
142C4×D61—22no—
143C4×D61—22no—
144C4×D61—22no—
145C4×D61—22no—
146C4×D61—41no—
147C4×D61—41no—
148Dic122—21no—
149Dic122—21no—
150C8×C61—41no—
151C8×C61#12011yescharacteristic
152C8×C61#12111yescharacteristic
153C8×C61#12111yescharacteristic
154C8×C61#12311yescharacteristic
155#92418—22no—
156#92418—41no—
157#12983—611nomaximal, Sylow
158C22×C61—22no—
159C22×C61C2×C22⋊C811yescharacteristic
160C22×C61#43511yescharacteristic
161C22×C61#43511yescharacteristic
162C2×D61—24no—
163C2×D61—24no—
164C2×D61—22no—
165C2×D61—22no—
166C2×D61—42no—
167C2×D61—41no—
168C2×D61#35014yescomplemented
169C4×C61—42no—
170C4×C61#38611yescharacteristic
171C4×C61#40811yescharacteristic
172C4×C61C8×D411yescharacteristic
173C4×C61C8×D411yescharacteristic
174Dic61—22no—
175Dic61—41no—
176C23×C16—611no—
177#2929—611no—
178C2×#350—611no—
179C2×#350—611no—
180#2112—611no—
181#1379—611no—
182#1379D6111yescharacteristic, complemented
183C2×C61—21no—
184C2×C61—21no—
185C2×C61#186911yescharacteristic
186C2×C61C2×#35012yes—
187D61—24no—
188D61—81no—
189C2×#131—611no—
190C23×C8—611no—
191#525—611no—
192C2×M32—611no—
193C2×M32C2×D6111yescharacteristic
194C22×C16—611no—
195C22×C16—1222no—
196C22×C16—1222no—
197C22×C16—1222no—
198C22×C16C2×D6111yescharacteristic
199#435—1222no—
200C8×D4—1222no—
201#424—611no—
202#408—611no—
203#407—611no—
204#405C2×D6111yescharacteristic
205#350—1222no—
206#350—1222no—
207C22⋊C16—1222no—
208C61#1298311yescharacteristic, complemented, Sylow
209C2×#41—611no—
210C23×C4—611no—
211#131—1221no—
212#131—1221no—
213#131—1221no—
214#131—1221no—
215C2×M16—611no—
216C2×M16—1221no—
217C2×M16C4×D6111yescharacteristic
218C22×C8—611no—
219C22×C8—612no—
220C22×C8—612no—
221C22×C8—1221no—
222C22×C8—1221no—
223C22×C8—1221no—
224C22×C8—1221no—
225C22×C8—1221no—
226C22×C8C22×D6111yescharacteristic
227#123—1221no—
228C22⋊Q8—612no—
229#121—612no—
230#120—1221no—
231C4×D4—1222no—
232#117C4×D6111yescharacteristic
233M32—21no—
234M32—21no—
235M32—1221no—
236M32—1221no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—612no—
241C2×C16—1228no—
242C2×C16—1221no—
243C2×C16—1221no—
244C2×C16—1221no—
245C2×C16D24412yescomplemented
246C4⋊C8—1222no—
247C22⋊C8—1222no—
248C22⋊C8—1222no—
249#97—21no—
250C4×C8—21no—
251C24—611no—
252#41—1221no—
253#41—1221no—
254#41—1221no—
255#41—1221no—
256C2×Q8—611no—
257C2×D4—611no—
258C2×D4—1222no—
259C2×D4—1221no—
260C22×C4—611no—
261C22×C4—612no—
262C22×C4—612no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4C2×C4×D6111yescharacteristic
269M16—21no—
270M16—21no—
271M16—1221no—
272M16—1221no—
273M16—2441no—
274M16—2441no—
275C2×C8—21no—
276C2×C8—612no—
277C2×C8—1221no—
278C2×C8—1221no—
279C2×C8—1222no—
280C2×C8—1222no—
281C2×C8—1222no—
282C2×C8—1222no—
283C2×C8—1221no—
284C2×C8—2441no—
285C2×C8—2441no—
286C2×C8C2×D24411yescharacteristic
287C2×C8#1175822211yescharacteristic
288C2×C8D4×D6111yescharacteristic
289C2×C8D4×D6111yescharacteristic
290C2×C8D4×D6111yescharacteristic
291C2×C8#1175822611yescharacteristic
292C4⋊C4—1222no—
293C4⋊C4C8×D6112yes—
294C22⋊C4—21no—
295C22⋊C4—1222no—
296C22⋊C4—1222no—
297C42—21no—
298C16—22no—
299C16—41no—
300C16—1222no—
301C16—2441no—
302C23—612no—
303C23—1224no—
304C23—1221no—
305C23—1221no—
306C23—1221no—
307C23M16×D6111yescharacteristic
308Q8—1221no—
309Q8—1221no—
310D4—1221no—
311D4—1221no—
312D4—2444no—
313D4—2441no—
314D4—2441no—
315C2×C4—22no—
316C2×C4—612no—
317C2×C4—1221no—
318C2×C4—1221no—
319C2×C4—1222no—
320C2×C4—1222no—
321C2×C4—1222no—
322C2×C4—1222no—
323C2×C4—1221no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4C22⋊C4×D6111yescharacteristic
327C2×C4#1451973911yescharacteristic
328C2×C4C4×D24411yescharacteristic
329C2×C4C2×C8×D6111yescharacteristic
330C2×C4#1451978311yescharacteristic
331C2×C4#1451978511yescharacteristic
332C8—41no—
333C8—1222no—
334C8—2441no—
335C8#1451974711yescharacteristic
336C8#1451975011yescharacteristic
337C8#1451975211yescharacteristic
338C8#1451976211yescharacteristic
339C22—22no—
340C22—614no—
341C22—1224no—
342C22—1224no—
343C22—1222no—
344C22—1222no—
345C22—2442no—
346C22—2441no—
347C22C22⋊C8×D6111yescharacteristic
348C22#1683983211yescharacteristic
349C22#1683983511yescharacteristic
350C4—42no—
351C4—1222no—
352C4—2441no—
353C4#1683964011yescharacteristic
354C4#1683972711yescharacteristic
355C4#1683982711yescharacteristic
356C4C8×D24411yescharacteristic
357C2—21no—
358C2—21no—
359C2—1224no—
360C2—4881no—
361C2#2144611011yescharacteristic
362C2#350×D6112yes—
subgroupG/H
centerC2×C8D4×D61
commutatorC2×C4×C61C22×C8
FrattiniC22×C8C22×D61
FittingC61×#1379C2
socleC22×C61C2×C22⋊C8
prime radical#16840274C4
cosocleC22×C8×C61C23
hypercenter#1379D61
hypocenterC61#12983
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ #1379

Complements

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

NH ≅ G/Ntype
1. C22×C16×D61C2semidirect
3. C2×#16839547C2semidirect
4. C2×#16839545C2semidirect
5. #21447429C2semidirect
6. #21443908C2semidirect
7. C61×#1379C2semidirect
53. C2×#11758185D4semidirect
66. C2×C16×C61D4semidirect
168. C2×D61#350semidirect
182. #1379D61semidirect
208. C61#12983semidirect
245. C2×C16D244semidirect

Maximal subgroups

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

subgroupindex
C22×C16×D612
#214545152
C2×#168395472
C2×#168395452
#214474292
#214439082
C61×#13792
#1298361