Finite Group Databasebeta

← #29665245

Subgroups — #29665245

Order 15616.

728 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#21455279C211yesmaximal, characteristic
2C2×#16839964C211yesmaximal, characteristic, complemented
3#21454842C211yesmaximal, characteristic, complemented
4#21445253C214yesmaximal, complemented
5#21445205C214yesmaximal, complemented
6#21444594C211yesmaximal, characteristic, complemented
7#21444491C211yesmaximal, characteristic, complemented
8#21444468C211yesmaximal, characteristic, complemented
9C61×#3226C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C22×#11758201—21no—
12C2×C4×D244C2211yescharacteristic
13#16839974—22no—
14#16839964—22no—
15#16839964C2212yes—
16#16839964C2214yes—
17#16839954C2212yes—
18C8×D244C2214yes—
19#16839827—22no—
20#16839815C2211yescharacteristic
21#16839786C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23#16839763—21no—
24C2×#14519701—21no—
25C22⋊C8×D61—22no—
26#16839515C2214yescomplemented
27#16839508C2212yescomplemented
28#16839508C2214yescomplemented
29#16839506C2212yescomplemented
30C61×#532C2211yescharacteristic
31C2×C4⋊Q8×C61C2211yescharacteristic
32C61×#500C2212yescomplemented
33C61×#486C2212yescomplemented
34C61×#480—22no—
35C61×#474—22no—
36C61×#431C2211yescharacteristic
37C22×D244D411yescharacteristic
38C22×C4×D61—21no—
39#14519802—44no—
40C2×#11758201—24no—
41C2×#11758201—22no—
42C2×#11758201—44no—
43C2×#11758199—22no—
44C2×#11758194—21no—
45C22×#92418—21no—
46C2×C8×D61—24no—
47C4×D244—21no—
48C4×D244—21no—
49C4×D244D412yes—
50C4×D244C2312yes—
51C2×C4⋊Dic61D411yescharacteristic
52C2×#11758198—21no—
53C4⋊#92418—21no—
54C4⋊#92418—21no—
55C4⋊#92418C2312yes—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58#14519701—24no—
59#14519701—22no—
60#14519698—22no—
61C22⋊#92418—21no—
62#14519689—24no—
63C4×#92418—21no—
64C2×#41×C61—21no—
65C22×Q8×C61—21no—
66C4⋊Q8×C61—21no—
67C4⋊Q8×C61—21no—
68C4⋊Q8×C61—21no—
69C4⋊Q8×C61C2312yes—
70C4⋊Q8×C61C2311yescharacteristic
71#127×C61C2311yescharacteristic
72#124×C61—21no—
73#121×C61—21no—
74#121×C61—21no—
75C4×D4×C61—21no—
76C4×D4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—21no—
79C2×C42×C61C2311yescharacteristic
80#107×C61—24no—
81C4⋊C8×C61D412yes—
82#102×C61—24no—
83C22⋊C8×C61D412yes—
84C4×C8×C61C2312yes—
85C23×D61—21no—
86#41×C61—42no—
87#41×C61—42no—
88C2×Q8×C61—21no—
89C2×Q8×C61—22no—
90C2×Q8×C61—41no—
91C2×Q8×C61—42no—
92C2×Q8×C61—42no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61#4111yescharacteristic
102C22×Dic61—21no—
103C2×D244—21no—
104C2×D244—22no—
105C2×D244D814yes—
106C2×D244C2×D412yes—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—82no—
110C2×C8×C61C2×D414yes—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61#4111yescharacteristic
122C42×C61C2411yescharacteristic
123#11758201—48no—
124#11758201—44no—
125#11758199—44no—
126#11758198—22no—
127#11758198—22no—
128C4⋊Dic61—21no—
129C4⋊Dic61C2×D412yes—
130#11758194—42no—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418—22no—
134C2×#92418—42no—
135C8×D61—84no—
136C23×C61—41no—
137C23×C61#12111yescharacteristic
138C22×D61—22no—
139C22×D61—24no—
140C22×D61—81no—
141C22×D61—81no—
142Q8×C61—42no—
143Q8×C61—41no—
144Q8×C61—42no—
145Q8×C61—41no—
146Q8×C61—82no—
147D4×C61—41no—
148D4×C61—41no—
149D4×C61—42no—
150D4×C61—81no—
151D4×C61—82no—
152C2×C4×C61—22no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—82no—
157C2×C4×C61—82no—
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61C2×D812yes—
162C2×C4×C61#13512yes—
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C2×Dic61—22no—
167C2×Dic61—81no—
168D244—22no—
169D244—24no—
170D244C2×D814yes—
171C4×D61—44no—
172C4×D61—82no—
173C8×C61—24no—
174#92418—42no—
175#92418—42no—
176#92418—42no—
177#19392—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61#46612yes—
184C22×C61C2×#12111yescharacteristic
185C2×D61—24no—
186C2×D61—42no—
187C2×D61—42no—
188C2×D61—84no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—81no—
193C4×C61—82no—
194C4×C61#46612yes—
195C4×C61C22×D811yescharacteristic
196C4×C61C2×#13511yescharacteristic
197C4×C61#57712yes—
198Dic61—42no—
199#3226D6111yescharacteristic, complemented
200#3036—611no—
201C2×#461—611no—
202#2972—611no—
203#1681—614no—
204#1673—614no—
205#1571—611no—
206#1546—611no—
207#1540—611no—
208C2×C61—22no—
209C2×C61—161no—
210C2×C61#304211yescharacteristic
211C2×C61C2×#46611yescharacteristic
212C2×C61#306911yescharacteristic
213D61—44no—
214C2×AGL(1,ℤ/8)—1221no—
215C22×SD16—1221no—
216#532C2×D6111yescharacteristic
217C2×C4⋊Q8C2×D6111yescharacteristic
218C2×C4×D4—611no—
219#500C2×D6112yescomplemented
220#486C2×D6112yescomplemented
221#480—22no—
222#474—22no—
223#463—1222no—
224#461—612no—
225#461—614no—
226#461—1222no—
227#459—612no—
228C8×D4—614no—
229C8×D4—1222no—
230#431—611no—
231#431C2×D6111yescharacteristic
232#423—611no—
233C2×C4⋊C8—611no—
234#417—1221no—
235C2×#103—1221no—
236C2×C22⋊C8—1222no—
237#335—614no—
238#332—612no—
239#332—614no—
240#331—612no—
241C61#1939211yescharacteristic, complemented, Sylow
242C2×#41—21no—
243C22×Q8—21no—
244C22×D4—611no—
245C23×C4—1221no—
246AGL(1,ℤ/8)—2444no—
247C2×SD16—1224no—
248C2×SD16—1222no—
249C2×SD16—2444no—
250C2×D8—1222no—
251C2×M16—1221no—
252C22×C8—1221no—
253C22×C8—1224no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8C22×D6112yes—
258C4⋊Q8C22×D6111yescharacteristic
259#127C22×D6111yescharacteristic
260#124—21no—
261#121—21no—
262#121—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—612no—
266C4×D4—612no—
267C4×D4—1221no—
268C4×D4—1221no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—611no—
272C2×C22⋊C4—1221no—
273C2×C42C22×D6111yescharacteristic
274#107—24no—
275C4⋊C8—612no—
276C4⋊C8—612no—
277C4⋊C8—612no—
278C4⋊C8—1221no—
279C4⋊C8—1221no—
280C4⋊C8C22⋊D6112yes—
281#103—1224no—
282#103—1222no—
283#102—24no—
284#102—1222no—
285C22⋊C8—1221no—
286C22⋊C8—1224no—
287C22⋊C8C22⋊D6112yes—
288C4×C8—1221no—
289C4×C8C22×D6112yes—
290C24—1221no—
291#41—42no—
292#41—42no—
293C2×Q8—21no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—612no—
303C2×D4—614no—
304C2×D4—1221no—
305C2×D4—1222no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—1221no—
309C22×C4—1222no—
310C22×C4—1222no—
311C22×C4—4882no—
312C22×C4D4×D6111yescharacteristic
313C22×C4#1175822411yescharacteristic
314C22×C4C2×C22⋊D6111yescharacteristic
315SD16—2448no—
316SD16—2444no—
317D8—2444no—
318M16—2442no—
319C2×C8—1222no—
320C2×C8—1222no—
321C2×C8—1222no—
322C2×C8—4884no—
323C2×C8—2442no—
324C2×C8C2×C22⋊D6114yes—
325C4⋊C4—22no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—612no—
332C4⋊C4—1221no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—1222no—
336C22⋊C4—1222no—
337C42D4×D6111yescharacteristic
338C42#1175822411yescharacteristic
339C42C2×C22⋊D6111yescharacteristic
340C42C23×D6111yescharacteristic
341C23—41no—
342C23—1224no—
343C23—1222no—
344C23—4881no—
345C23—4881no—
346C23#1451974811yescharacteristic
347Q8—42no—
348Q8—41no—
349Q8—42no—
350Q8—41no—
351Q8—82no—
352D4—41no—
353D4—41no—
354D4—42no—
355D4—81no—
356D4—82no—
357D4—614no—
358D4—1224no—
359D4—1222no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—1222no—
367C2×C4—4882no—
368C2×C4—4881no—
369C2×C4—2444no—
370C2×C4#1451974811yescharacteristic
371C2×C4#1451974811yescharacteristic
372C2×C4#1451974811yescharacteristic
373C2×C4D8×D6112yes—
374C2×C4#1451979612yes—
375C2×C4C2×D4×D6111yescharacteristic
376C2×C4C22×C22⋊D6111yescharacteristic
377C2×C4C2×#1175822411yescharacteristic
378C8—24no—
379C8—2442no—
380C8—2442no—
381C8—2442no—
382C22—21no—
383C22—21no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—1224no—
388C22—4884no—
389C22—2442no—
390C22—2442no—
391C22#1683999512yes—
392C22C2×#1451974811yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—81no—
397C4—82no—
398C4—2442no—
399C4#1683999512yes—
400C4C2×D8×D6111yescharacteristic
401C4C2×#1451979611yescharacteristic
402C4#1684057012yes—
403C2—22no—
404C2—161no—
405C2—2444no—
406C2#2145532711yescharacteristic
407C2C2×#1683999511yescharacteristic
408C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3226C2
socleC22×C61C2×#121
prime radical#21455279C2
hypercenter#3226D61
hypocenterC61#19392
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3226

Complements

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

NH ≅ G/Ntype
2. C2×#16839964C2semidirect
3. #21454842C2semidirect
4. #21445253C2semidirect
5. #21445205C2semidirect
6. #21444594C2semidirect
7. #21444491C2semidirect
8. #21444468C2semidirect
9. C61×#3226C2semidirect
26. #16839515C22semidirect
27. #16839508C22semidirect
28. #16839508C22semidirect
29. #16839506C22semidirect
32. C61×#500C22semidirect
33. C61×#486C22semidirect
199. #3226D61semidirect
219. #500C2×D61semidirect
220. #486C2×D61semidirect
241. C61#19392semidirect

Maximal subgroups

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

subgroupindex
#214552792
C2×#168399642
#214548422
#214452532
#214452052
#214445942
#214444912
#214444682
C61×#32262
#1939261