Finite Group Databasebeta

← #29598263

Subgroups — #29598263

Order 15616.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456243C211yesmaximal, characteristic
2C2×#16839573C211yesmaximal, characteristic, complemented
3C2×#16839572C211yesmaximal, characteristic, complemented
4C2×#16839571C211yesmaximal, characteristic, complemented
5M32×Dic61C211yesmaximal, characteristic, complemented
6#21445980C211yesmaximal, characteristic, complemented
7C2×C61×#360C211yesmaximal, characteristic, complemented
8C2×#14519799C2211yescharacteristic
9C2×#14519797—21no—
10#16840308—21no—
11C2×#14519717C2211yescharacteristic
12#16840147C411yescharacteristic, complemented
13#16840145C411yescharacteristic, complemented
14#16840143—21no—
15#16840140C411yescharacteristic, complemented
16#16840137C411yescharacteristic, complemented
17#16840133—21no—
18C2×#14519710—21no—
19C2×#14519709C2211yescharacteristic
20C2×C8×Dic61C2211yescharacteristic
21#16839627—21no—
22#16839573—22no—
23#16839572—22no—
24#16839571—22no—
25#16839537—21no—
26C16×Dic61—21no—
27C2×#135×C61C2211yescharacteristic
28C2×M32×C61C2211yescharacteristic
29C2×#106×C61C2211yescharacteristic
30C61×#360—22no—
31C2×#11758222—21no—
32#14519799—21no—
33#14519799—21no—
34#14519799—21no—
35#14519799—21no—
36#14519797—42no—
37#14519797—42no—
38C2×Dic244C2×C411yescharacteristic
39C2×#11758202—21no—
40C2×#11758190—21no—
41C2×#11758200—21no—
42C2×#11758201—21no—
43C2×D488C2×C411yescharacteristic
44C2×#11758199—21no—
45C22×#92418D411yescharacteristic
46#14519775—21no—
47#14519773—21no—
48#14519766—21no—
49#14519764—21no—
50#14519749—42no—
51C4×C22⋊D61—42no—
52C2×C4⋊Dic61—21no—
53C2×C4×Dic61D411yescharacteristic
54#14519717—21no—
55#14519717D412yescomplemented
56C2×#11758185—21no—
57#14519710—42no—
58#14519709—21no—
59#14519709—21no—
60C8×Dic61C2×C411yescharacteristic
61C8×Dic61C2×C411yescharacteristic
62C8×Dic61D411yescharacteristic
63C8×Dic61D411yescharacteristic
64#14519684—42no—
65C2×#41×C61—21no—
66#135×C61—21no—
67#135×C61—21no—
68#135×C61—21no—
69#135×C61—21no—
70C2×Q16×C61C2×C411yescharacteristic
71C2×SD16×C61—21no—
72C2×D8×C61C2×C411yescharacteristic
73C22×C8×C61C2311yescharacteristic
74C2×C4⋊C4×C61—21no—
75M32×C61—21no—
76M32×C61D412yescomplemented
77C2×C16×C61—21no—
78#106×C61—21no—
79#106×C61—21no—
80#41×C61—41no—
81#41×C61—41no—
82#41×C61—41no—
83#41×C61—41no—
84C2×Q8×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—41no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61C2×D411yescharacteristic
90C2×C22⋊D61—41no—
91C2×C22⋊D61—42no—
92C22×Dic61—41no—
93C22×Dic61D812yes—
94#11758222—41no—
95#11758222—41no—
96#11758222—41no—
97#11758222—41no—
98C2×D244—21no—
99C2×C4×D61—41no—
100C2×Dic122—21no—
101Q16×C61—21no—
102Q16×C61—21no—
103SD16×C61—41no—
104SD16×C61—41no—
105D8×C61—21no—
106D8×C61—21no—
107C2×C8×C61C22×C411yescharacteristic
108C2×C8×C61C2×D411yescharacteristic
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61#4111yescharacteristic
112C2×C8×C61#4111yescharacteristic
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C22⋊Dic61—42no—
116#11758202—42no—
117#11758201—42no—
118#11758200—42no—
119#11758199—42no—
120#11758198—42no—
121C4⋊Dic61—41no—
122C4⋊Dic61—41no—
123#11758196—42no—
124C4×Dic61C22⋊C411yescharacteristic
125C4×Dic61C22⋊C411yescharacteristic
126C4×Dic61SD1612yes—
127C2×#92418C22⋊C411yescharacteristic
128C2×#92418C22⋊C411yescharacteristic
129C2×#92418D812yes—
130C2×#92418SD1612yes—
131Dic244—21no—
132Dic244—21no—
133D488—21no—
134D488—21no—
135#11758190—41no—
136#11758190—41no—
137C16×C61—22no—
138#11758185—22no—
139C23×C61—41no—
140C23×C61C2×D811yescharacteristic
141C22×D61—41no—
142Q8×C61—41no—
143Q8×C61—41no—
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—81no—
147D4×C61—81no—
148C2×C4×C61—41no—
149C2×C4×C61—81no—
150C2×C4×C61—81no—
151C2×C4×C61—82no—
152C2×C4×C61—81no—
153C2×C4×C61C2×C22⋊C411yescharacteristic
154C2×C4×C61#12411yescharacteristic
155C2×C4×C61#12711yescharacteristic
156C2×C4×C61C2×D811yescharacteristic
157C2×C4×C61C2×SD1611yescharacteristic
158C2×C4×C61C2×SD1611yescharacteristic
159C22⋊D61—84no—
160C22⋊D61—81no—
161C22⋊D61—81no—
162C2×Dic61—42no—
163C2×Dic61—41no—
164C2×Dic61—82no—
165C2×Dic61—81no—
166C2×Dic61#10212yes—
167C2×Dic61#10212yes—
168D244—41no—
169D244—41no—
170C4×D61—81no—
171C4×D61—81no—
172Dic122—41no—
173Dic122—41no—
174C8×C61C4×D411yescharacteristic
175C8×C61C4×D411yescharacteristic
176C8×C61#12111yescharacteristic
177C8×C61#12111yescharacteristic
178#92418#10212yes—
179#92418#10212yes—
180#13211—611nomaximal, Sylow
181C22×C61—22no—
182C22×C61—82no—
183C22×C61—81no—
184C22×C61C2×#10211yescharacteristic
185C22×C61#48611yescharacteristic
186C22×C61#49311yescharacteristic
187C2×D61—82no—
188C2×D61—81no—
189C4×C61—81no—
190C4×C61—161no—
191C4×C61#39011yescharacteristic
192C4×C61C2×#10211yescharacteristic
193C4×C61#48611yescharacteristic
194C4×C61#49211yescharacteristic
195Dic61—22no—
196Dic61—22no—
197Dic61—81no—
198Dic61—161no—
199#3147—611no—
200C2×#360—611no—
201C2×#360—611no—
202C2×#360—611no—
203C2×#360D6111yescharacteristic, complemented
204C4×M32—611no—
205#1846—611no—
206C2×C61—21no—
207C2×C61—21no—
208C2×C61—161no—
209C2×C61#195711yescharacteristic
210C2×C61#214512yes—
211D61—161no—
212C2×#135—611no—
213C2×#135—1221no—
214C2×#135C2×D6111yescharacteristic
215#532—1221no—
216C2×M32—611no—
217C2×M32C2×D6111yescharacteristic
218#495—611no—
219#495—611no—
220#495—1221no—
221#494—611no—
222#493—611no—
223#492—1221no—
224C2×#107—1221no—
225C2×#106—611no—
226C2×#106C2×D6111yescharacteristic
227C2×C4×C8—611no—
228#382—1221no—
229#360—22no—
230#360—1222no—
231#360—1222no—
232#360—1222no—
233#346—1221no—
234C4×C16—1221no—
235C61#1321111yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41—1221no—
238#135—21no—
239#135—21no—
240#135—21no—
241#135—21no—
242#135—1221no—
243#135—1221no—
244#135—1221no—
245#135—1221no—
246#135—2442no—
247#135—2442no—
248C2×Q16—611no—
249C2×Q16—1221no—
250C2×Q16C4×D6111yescharacteristic
251C2×SD16—21no—
252C2×SD16—1221no—
253C2×SD16—1221no—
254C2×SD16—1221no—
255C2×D8—611no—
256C2×D8—1221no—
257C2×D8C4×D6111yescharacteristic
258C22×C8—611no—
259C22×C8C22×D6111yescharacteristic
260C4⋊Q8—1221no—
261#127—1221no—
262#124—1221no—
263#124—1221no—
264#121—2442no—
265C4×D4—2442no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—1221no—
268C2×C42—611no—
269M32—21no—
270M32—612no—
271M32—1221no—
272M32C22⋊D6112yescomplemented
273C2×C16—21no—
274C2×C16—1221no—
275#107—2442no—
276#106—21no—
277#106—21no—
278#106—1221no—
279#106—1221no—
280C4×C8—611no—
281C4×C8—611no—
282C4×C8—611no—
283C4×C8—611no—
284#95—2442no—
285#41—41no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—2441no—
290#41—2441no—
291#41—2441no—
292#41—2441no—
293C2×Q8—21no—
294C2×Q8—1221no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1221no—
298C2×D4—2442no—
299C2×D4—2441no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—612no—
303C22×C4—2441no—
304C22×C4—2441no—
305C22×C4D4×D6111yescharacteristic
306Q16—21no—
307Q16—21no—
308Q16—1221no—
309Q16—1221no—
310Q16—2442no—
311SD16—41no—
312SD16—41no—
313SD16—2442no—
314SD16—2442no—
315SD16—2441no—
316SD16—2441no—
317D8—21no—
318D8—21no—
319D8—1221no—
320D8—1221no—
321D8—2442no—
322C2×C8—611no—
323C2×C8—611no—
324C2×C8—612no—
325C2×C8—612no—
326C2×C8C2×C4×D6111yescharacteristic
327C2×C8#1175822211yescharacteristic
328C2×C8D4×D6111yescharacteristic
329C2×C8D4×D6111yescharacteristic
330C2×C8#1175822411yescharacteristic
331C2×C8C2×C22⋊D6111yescharacteristic
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—2442no—
335C4⋊C4—2441no—
336C4⋊C4—2441no—
337C22⋊C4—2442no—
338C22⋊C4—2442no—
339C42—611no—
340C42—611no—
341C42—612no—
342C16—22no—
343C16—1222no—
344C23—41no—
345C23—2441no—
346C23D8×D6111yescharacteristic
347Q8—41no—
348Q8—41no—
349Q8—2441no—
350Q8—2441no—
351D4—41no—
352D4—41no—
353D4—81no—
354D4—81no—
355D4—4884no—
356D4—4881no—
357D4—4881no—
358D4—2441no—
359D4—2441no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—612no—
366C2×C4—612no—
367C2×C4—4882no—
368C2×C4—4881no—
369C2×C4—4881no—
370C2×C4—4881no—
371C2×C4—2442no—
372C2×C4—2441no—
373C2×C4C22⋊C4×D6111yescharacteristic
374C2×C4#1451976311yescharacteristic
375C2×C4#1451977311yescharacteristic
376C2×C4D8×D6111yescharacteristic
377C2×C4SD16×D6111yescharacteristic
378C2×C4SD16×D6111yescharacteristic
379C8—612no—
380C8—612no—
381C8#1451973711yescharacteristic
382C8C4×C22⋊D6111yescharacteristic
383C8#1451974911yescharacteristic
384C8#1451975011yescharacteristic
385C22—22no—
386C22—82no—
387C22—81no—
388C22—4882no—
389C22—4881no—
390C22#102×D6111yescharacteristic
391C22#1684010711yescharacteristic
392C22#1684013711yescharacteristic
393C4—81no—
394C4—161no—
395C4—1222no—
396C4—1222no—
397C4—9761no—
398C4—4881no—
399C4#1683966211yescharacteristic
400C4#102×D6111yescharacteristic
401C4#1684010711yescharacteristic
402C4#1684013311yescharacteristic
403C2—21no—
404C2—21no—
405C2—161no—
406C2—9761no—
407C2#2144662311yescharacteristic
408C2#2144755312yes—
subgroupG/H
centerC22#102×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC2×C61×#360C2
socleC22×C61C2×#102
prime radical#21456243C2
cosocleC22×C8×C61C23
hypercenterC2×#360D61
hypocenterC61#13211
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#360

Complements

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

NH ≅ G/Ntype
2. C2×#16839573C2semidirect
3. C2×#16839572C2semidirect
4. C2×#16839571C2semidirect
5. M32×Dic61C2semidirect
6. #21445980C2semidirect
7. C2×C61×#360C2semidirect
12. #16840147C4semidirect
13. #16840145C4semidirect
15. #16840140C4semidirect
16. #16840137C4semidirect
55. #14519717D4semidirect
76. M32×C61D4semidirect
203. C2×#360D61semidirect
235. C61#13211semidirect
272. M32C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214562432
C2×#168395732
C2×#168395722
C2×#168395712
M32×Dic612
#214459802
C2×C61×#3602
#1321161