Finite Group Databasebeta

← #29665178

Subgroups — #29665178

Order 15616.

708 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840039C211yesmaximal, characteristic, complemented
2#21455279C211yesmaximal, characteristic
3#21454842C211yesmaximal, characteristic, complemented
4#21445239C214yesmaximal, complemented
5#21445157C214yesmaximal, complemented
6#21444590C211yesmaximal, characteristic, complemented
7#21444504C211yesmaximal, characteristic, complemented
8#21444468C211yesmaximal, characteristic, complemented
9C61×#3206C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C2×C4×D244C2211yescharacteristic
12#16840039—22no—
13#16840039C2214yes—
14#16839974—22no—
15#16839964C2212yes—
16#16839954C2212yes—
17C8×D244C2214yes—
18#16839827—22no—
19#16839815C2211yescharacteristic
20C2×#14519710—21no—
21#16839786C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23#16839763—21no—
24C2×#14519697—21no—
25C22⋊C8×D61—22no—
26#16839513C2214yescomplemented
27#16839508C2214yescomplemented
28#16839508C2212yescomplemented
29#16839506C2212yescomplemented
30C61×#532C2211yescharacteristic
31C2×C4⋊Q8×C61C2211yescharacteristic
32C61×#492C2212yescomplemented
33C61×#487C2212yescomplemented
34C61×#461—22no—
35C61×#448—22no—
36C61×#431C2211yescharacteristic
37C22×D244D411yescharacteristic
38C22×C4×D61—21no—
39#14519802—44no—
40C2×#11758201—22no—
41C2×#11758199—22no—
42C2×#11758194—21no—
43C22×#92418—21no—
44C2×C8×D61—24no—
45C4×D244—21no—
46C4×D244—21no—
47C4×D244D412yes—
48C4×D244C2312yes—
49C2×C4⋊Dic61D411yescharacteristic
50C2×#11758198—21no—
51#14519710—24no—
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C4⋊#92418C2312yes—
55C4⋊#92418C2312yes—
56C4⋊#92418C2312yes—
57#14519701—22no—
58#14519698—22no—
59#14519697—24no—
60C22⋊#92418—21no—
61#14519689—24no—
62C4×#92418—21no—
63C2×#41×C61—21no—
64C22×Q8×C61—21no—
65C2×SD16×C61—24no—
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
80C4⋊C8×C61D412yes—
81#103×C61—24no—
82C22⋊C8×C61D412yes—
83C4×C8×C61C2312yes—
84C23×D61—21no—
85#41×C61—42no—
86#41×C61—42no—
87C2×Q8×C61—22no—
88C2×Q8×C61—21no—
89C2×Q8×C61—41no—
90C2×Q8×C61—82no—
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61#4111yescharacteristic
100C22×Dic61—21no—
101C2×D244—21no—
102C2×D244—22no—
103C2×D244SD1614yescomplemented
104C2×D244C2×D412yes—
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—82no—
108SD16×C61—84no—
109C2×C8×C61C2×D414yes—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61#4111yescharacteristic
121C42×C61C2411yescharacteristic
122#11758201—44no—
123#11758199—44no—
124#11758198—22no—
125#11758198—22no—
126C4⋊Dic61—21no—
127C4⋊Dic61C2×D412yes—
128#11758194—42no—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418—22no—
132C2×#92418—42no—
133C8×D61—84no—
134C23×C61—41no—
135C23×C61#12111yescharacteristic
136C22×D61—22no—
137C22×D61—24no—
138C22×D61—81no—
139C22×D61—81no—
140Q8×C61—41no—
141Q8×C61—41no—
142Q8×C61—44no—
143Q8×C61—82no—
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—42no—
147D4×C61—81no—
148D4×C61—82no—
149C2×C4×C61—22no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—82no—
154C2×C4×C61—82no—
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61C2×SD1612yes—
159C2×C4×C61#13512yes—
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C2×#4111yescharacteristic
163C2×Dic61—22no—
164C2×Dic61—81no—
165D244—22no—
166D244—24no—
167D244C2×SD1614yes—
168C4×D61—44no—
169C4×D61—82no—
170C8×C61—24no—
171#92418—42no—
172#92418—42no—
173#92418—42no—
174#19385—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—81no—
180C22×C61#46512yes—
181C22×C61C2×#12111yescharacteristic
182C2×D61—24no—
183C2×D61—42no—
184C2×D61—42no—
185C2×D61—84no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—82no—
190C4×C61—81no—
191C4×C61#46512yes—
192C4×C61C22×SD1611yescharacteristic
193C4×C61C2×#13511yescharacteristic
194C4×C61#57612yes—
195Dic61—42no—
196#3206D6111yescharacteristic, complemented
197C2×#474—611no—
198#3036—611no—
199#2972—611no—
200#1679—614no—
201#1665—614no—
202#1570—611no—
203#1549—611no—
204#1540—611no—
205C2×C61—22no—
206C2×C61—161no—
207C2×C61#304111yescharacteristic
208C2×C61C2×#46511yescharacteristic
209C2×C61#306911yescharacteristic
210D61—44no—
211C2×AGL(1,ℤ/8)—1221no—
212#532C2×D6111yescharacteristic
213C2×C4⋊Q8C2×D6111yescharacteristic
214C2×C4×D4—611no—
215#492C2×D6112yescomplemented
216#487C2×D6112yescomplemented
217#474—614no—
218#474—1222no—
219#463—1222no—
220#461—22no—
221#461—612no—
222#459—612no—
223#448—22no—
224C8×D4—614no—
225C8×D4—1222no—
226#431—611no—
227#431C2×D6111yescharacteristic
228C2×#107—1221no—
229#423—611no—
230C2×C4⋊C8—611no—
231#417—1221no—
232C2×#102—1221no—
233C2×C22⋊C8—1222no—
234#334—614no—
235#332—612no—
236#332—614no—
237#331—612no—
238C61#1938511yescharacteristic, complemented, Sylow
239C2×#41—21no—
240C22×Q8—21no—
241C22×D4—611no—
242C23×C4—1221no—
243AGL(1,ℤ/8)—2444no—
244C2×SD16—24no—
245C2×SD16—1222no—
246C2×D8—1222no—
247C2×M16—1221no—
248C22×C8—1221no—
249C22×C8—1224no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8C22×D6112yes—
254C4⋊Q8C22×D6111yescharacteristic
255#127C22×D6111yescharacteristic
256#124—21no—
257#121—21no—
258#121—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—612no—
262C4×D4—612no—
263C4×D4—1221no—
264C4×D4—1221no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—611no—
268C2×C22⋊C4—1221no—
269C2×C42C22×D6111yescharacteristic
270#107—1224no—
271C4⋊C8—612no—
272C4⋊C8—612no—
273C4⋊C8—612no—
274C4⋊C8—1221no—
275C4⋊C8—1221no—
276C4⋊C8C22⋊D6112yes—
277#103—24no—
278#103—1222no—
279#102—1224no—
280#102—1222no—
281C22⋊C8—1221no—
282C22⋊C8—1224no—
283C22⋊C8C22⋊D6112yes—
284C4×C8—1221no—
285C4×C8C22×D6112yes—
286C24—1221no—
287#41—42no—
288#41—42no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—41no—
292C2×Q8—82no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—612no—
298C2×D4—614no—
299C2×D4—1221no—
300C2×D4—1222no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—1221no—
304C22×C4—1222no—
305C22×C4—1222no—
306C22×C4—4882no—
307C22×C4D4×D6111yescharacteristic
308C22×C4#1175822411yescharacteristic
309C22×C4C2×C22⋊D6111yescharacteristic
310SD16—84no—
311SD16—2444no—
312D8—2444no—
313M16—2442no—
314C2×C8—1222no—
315C2×C8—1222no—
316C2×C8—1222no—
317C2×C8—4884no—
318C2×C8—2442no—
319C2×C8C2×C22⋊D6114yes—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—612no—
327C4⋊C4—1221no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—1222no—
331C22⋊C4—1222no—
332C42D4×D6111yescharacteristic
333C42#1175822411yescharacteristic
334C42C2×C22⋊D6111yescharacteristic
335C42C23×D6111yescharacteristic
336C23—41no—
337C23—1224no—
338C23—1222no—
339C23—4881no—
340C23—4881no—
341C23#1451974811yescharacteristic
342Q8—41no—
343Q8—41no—
344Q8—44no—
345Q8—82no—
346D4—41no—
347D4—41no—
348D4—42no—
349D4—81no—
350D4—82no—
351D4—614no—
352D4—1224no—
353D4—1222no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—1222no—
361C2×C4—4882no—
362C2×C4—4881no—
363C2×C4—2444no—
364C2×C4#1451974811yescharacteristic
365C2×C4#1451974811yescharacteristic
366C2×C4#1451974811yescharacteristic
367C2×C4SD16×D6112yes—
368C2×C4#1451979812yes—
369C2×C4C2×D4×D6111yescharacteristic
370C2×C4C22×C22⋊D6111yescharacteristic
371C2×C4C2×#1175822411yescharacteristic
372C8—24no—
373C8—2442no—
374C8—2442no—
375C8—2442no—
376C22—21no—
377C22—21no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—1224no—
382C22—4884no—
383C22—2442no—
384C22—2442no—
385C22#1683998812yes—
386C22C2×#1451974811yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—81no—
391C4—82no—
392C4—2442no—
393C4#1683998812yes—
394C4C2×SD16×D6111yescharacteristic
395C4C2×#1451979811yescharacteristic
396C4#1684056512yes—
397C2—22no—
398C2—161no—
399C2—2444no—
400C2#2145531811yescharacteristic
401C2C2×#1683998811yescharacteristic
402C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3206C2
socleC22×C61C2×#121
prime radical#21455279C2
hypercenter#3206D61
hypocenterC61#19385
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
1. C2×#16840039C2semidirect
3. #21454842C2semidirect
4. #21445239C2semidirect
5. #21445157C2semidirect
6. #21444590C2semidirect
7. #21444504C2semidirect
8. #21444468C2semidirect
9. C61×#3206C2semidirect
26. #16839513C22semidirect
27. #16839508C22semidirect
28. #16839508C22semidirect
29. #16839506C22semidirect
32. C61×#492C22semidirect
33. C61×#487C22semidirect
103. C2×D244SD16semidirect
196. #3206D61semidirect
215. #492C2×D61semidirect
216. #487C2×D61semidirect
238. C61#19385semidirect

Maximal subgroups

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

subgroupindex
C2×#168400392
#214552792
#214548422
#214452392
#214451572
#214445902
#214445042
#214444682
C61×#32062
#1938561