Finite Group Databasebeta

← #29663970

Subgroups — #29663970

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840110C211yesmaximal, characteristic, complemented
2#21455628C211yesmaximal, characteristic, complemented
3#21455300C211yesmaximal, characteristic
4#21454775C211yesmaximal, characteristic, complemented
5#21445238C214yesmaximal, complemented
6#21445231C214yesmaximal, complemented
7#21444574C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic, complemented
9C61×#1548C211yesmaximal, characteristic, complemented
10C2×#14519805—21no—
11C2×#14519774C2211yescharacteristic
12C4×#11758222C2211yescharacteristic
13#16840110—21no—
14#16840110—21no—
15#16840110C2214yes—
16#16840054C2212yescomplemented
17#16840049C2212yescomplemented
18#16840044—21no—
19#16840039—21no—
20#16839963—21no—
21#16839963C2212yes—
22#16839958—21no—
23#16839958C2212yes—
24#16839886C2212yescomplemented
25#16839827C2212yescomplemented
26#16839825—22no—
27#16839813—21no—
28#16839802—21no—
29#16839786C2211yescharacteristic
30#16839768—21no—
31#16839761—21no—
32C2×#14519697—22no—
33C2×C4×#92418C2211yescharacteristic
34C4×C8×D61—21no—
35#16839512C2214yescomplemented
36#16839505C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#333C2214yescomplemented
41C2×#11758222D411yescharacteristic
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—21no—
45#14519805—44no—
46C2×#11758190—22no—
47C2×D488—22no—
48C2×#11758194—21no—
49C22×#92418D412yes—
50C2×C8×D61—22no—
51#14519774—21no—
52#14519774—21no—
53#14519774—21no—
54#14519774C2312yes—
55C4×Dic122D411yescharacteristic
56C4×Dic122C2311yescharacteristic
57C4×D244D411yescharacteristic
58C4×D244C2311yescharacteristic
59C4×C22⋊D61—21no—
60C4×C22⋊D61—21no—
61#14519736D411yescharacteristic
62#14519734—21no—
63C42×D61—21no—
64#14519710—22no—
65#14519708—22no—
66C4⋊#92418—21no—
67C4⋊#92418C2312yes—
68#14519706—22no—
69#14519700—22no—
70#14519699—22no—
71#14519699—22no—
72#14519697—22no—
73#14519697—24no—
74#14519697—24no—
75#14519689—22no—
76C4×#92418D411yescharacteristic
77C4×#92418D411yescharacteristic
78C4×#92418C2312yes—
79C8×Dic61—22no—
80C22×Q8×C61—21no—
81C2×M16×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61C2312yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C42×C61C2311yescharacteristic
89C4⋊C8×C61—21no—
90C4⋊C8×C61C2312yes—
91C4⋊C8×C61C2312yes—
92C22⋊C8×C61—21no—
93C4×C8×C61—21no—
94C23×D61—41no—
95C2×Q8×C61—22no—
96C2×Q8×C61—41no—
97C2×Q8×C61—82no—
98C22×C4×C61—41no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61#4111yescharacteristic
102C2×C22⋊D61—21no—
103#11758222—22no—
104#11758222SD1614yes—
105C2×D244—22no—
106C2×D244—22no—
107C2×D244—22no—
108C2×D244—44no—
109C2×D244—42no—
110C2×D244—42no—
111C2×D244—41no—
112C2×D244—41no—
113C2×D244—41no—
114C2×D244—82no—
115C2×D244C2×D411yescharacteristic
116C2×C4×D61—21no—
117C2×C4×D61—21no—
118C2×C4×D61—21no—
119C2×Dic122C2×D411yescharacteristic
120M16×C61—42no—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—41no—
126C4⋊C4×C61—41no—
127C42×C61C2×D411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61#4111yescharacteristic
130C42×C61C2411yescharacteristic
131C22⋊Dic61—21no—
132#11758198—21no—
133#11758198—21no—
134C4⋊Dic61C2×D411yescharacteristic
135C4⋊Dic61C2×D411yescharacteristic
136#11758196—21no—
137#11758196—21no—
138C4×Dic61—21no—
139C4×Dic61—21no—
140C4×Dic61—21no—
141#11758194—42no—
142C2×#92418—22no—
143C2×#92418—42no—
144C2×#92418C2×D412yes—
145C2×#92418C2×D412yes—
146D488—44no—
147#11758190—44no—
148C8×D61—44no—
149C23×C61#12111yescharacteristic
150C22×D61—21no—
151C22×D61—42no—
152C22×D61—82no—
153C22×D61—82no—
154C22×D61—82no—
155Q8×C61—82no—
156Q8×C61—82no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—42no—
160C2×C4×C61—82no—
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61C2×SD1612yes—
165C2×C4×C61#13512yes—
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C2×#4111yescharacteristic
169C22⋊D61—24no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173D244—21no—
174D244—42no—
175D244—42no—
176D244—44no—
177D244—82no—
178D244—82no—
179D244—82no—
180D244—82no—
181D244C2×SD1612yes—
182C4×D61—24no—
183C4×D61—42no—
184C4×D61—42no—
185Dic122—21no—
186Dic122C2×SD1612yes—
187C8×C61—42no—
188C8×C61—42no—
189#92418—22no—
190#92418—22no—
191#92418—42no—
192#19274—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61#46512yes—
196C22×C61C2×#12111yescharacteristic
197C2×D61—22no—
198C2×D61—41no—
199C2×D61—41no—
200C2×D61—82no—
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—164no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—82no—
207C4×C61#46512yes—
208C4×C61#47212yes—
209C4×C61C22×SD1611yescharacteristic
210C4×C61C2×#13511yescharacteristic
211C4×C61#57512yes—
212Dic61—22no—
213Dic61—41no—
214Dic61—41no—
215C2×#486—611no—
216#3079—611no—
217#3038—611no—
218C8×#41—611no—
219#1678—614no—
220#1677—614no—
221#1566—611no—
222#1548D6111yescharacteristic, complemented
223#1545—611no—
224C2×C61—22no—
225C2×C61#304011yescharacteristic
226C2×C61C2×#46511yescharacteristic
227C2×C61C2×#47211yescharacteristic
228D61—42no—
229D61—162no—
230C2×AGL(1,ℤ/8)—1221no—
231C2×C4⋊Q8C2×D6111yescharacteristic
232C2×#127—611no—
233C4×#41—611no—
234#486—614no—
235#486—1221no—
236#486—1221no—
237#477—612no—
238#476—612no—
239#475—1221no—
240#474—1221no—
241#460—612no—
242#460—1221no—
243#459—612no—
244#459—1221no—
245C8×Q8—612no—
246C8×D4—612no—
247C8×D4—1222no—
248#431—1221no—
249#431C2×D6111yescharacteristic
250#428—1221no—
251#423—611no—
252#423C2×D6111yescharacteristic
253#418—1221no—
254#417—1221no—
255C2×#102—1222no—
256C2×C4×C8—611no—
257C2×C4×C8—1221no—
258#334—614no—
259#333C2×D6114yescomplemented
260#331—614no—
261C61#1927411yescharacteristic, complemented, Sylow
262C2×#41—611no—
263C22×Q8—21no—
264C22×D4—1221no—
265C22×D4—1221no—
266C22×D4—1221no—
267AGL(1,ℤ/8)—2444no—
268C2×SD16—1222no—
269C2×D8—1222no—
270C2×M16—21no—
271C2×M16—1221no—
272C22×C8—612no—
273C22×C8—1222no—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8C22×D6112yes—
278#127—612no—
279#127—1221no—
280#127—1221no—
281#127—1221no—
282C4×Q8—611no—
283C4×Q8—611no—
284C4×D4—611no—
285C4×D4—611no—
286C4×D4—1221no—
287C4×D4—1221no—
288#117—611no—
289#117—1221no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C42—1221no—
293C2×C42C22×D6111yescharacteristic
294#107—1222no—
295#106—1222no—
296C4⋊C8—21no—
297C4⋊C8—612no—
298C4⋊C8—1222no—
299C4⋊C8—1221no—
300C4⋊C8C22×D6112yes—
301C4⋊C8C22×D6112yes—
302#103—1222no—
303#102—1224no—
304#102—1224no—
305#102—1222no—
306#102—1222no—
307#102—1222no—
308C22⋊C8—21no—
309C22⋊C8—1222no—
310C4×C8—21no—
311C4×C8—611no—
312C4×C8—611no—
313C4×C8—612no—
314C4×C8—1222no—
315C24—2441no—
316#41—614no—
317#41—1222no—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×Q8—82no—
321C2×Q8—611no—
322C2×D4—611no—
323C2×D4—1221no—
324C2×D4—1222no—
325C2×D4—1222no—
326C2×D4—1222no—
327C2×D4—4882no—
328C2×D4—2444no—
329C2×D4—2441no—
330C2×D4—2441no—
331C2×D4—2441no—
332C2×D4—2442no—
333C2×D4—2442no—
334C22×C4—41no—
335C22×C4—1221no—
336C22×C4—1221no—
337C22×C4—1221no—
338C22×C4#1175822211yescharacteristic
339C22×C4D4×D6111yescharacteristic
340C22×C4D4×D6111yescharacteristic
341SD16—2444no—
342D8—2444no—
343M16—42no—
344M16—2442no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—612no—
348C2×C8—612no—
349C2×C8—1222no—
350C2×C8—2444no—
351C2×C8—2442no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—611no—
357C4⋊C4—611no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C22⋊C4—1221no—
361C22⋊C4—1221no—
362C22⋊C4—1221no—
363C42—1221no—
364C42—1221no—
365C42—1221no—
366C42#1175822211yescharacteristic
367C42D4×D6111yescharacteristic
368C42D4×D6111yescharacteristic
369C42C23×D6111yescharacteristic
370C23—1221no—
371C23—4882no—
372C23—4882no—
373C23—4882no—
374C23—2442no—
375C23#1451975011yescharacteristic
376Q8—82no—
377Q8—82no—
378Q8—612no—
379Q8—1221no—
380D4—612no—
381D4—1224no—
382D4—1221no—
383D4—4882no—
384D4—4882no—
385D4—4882no—
386D4—4882no—
387D4—2442no—
388D4—2442no—
389D4—2444no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—42no—
393C2×C4—82no—
394C2×C4—1221no—
395C2×C4—1224no—
396C2×C4—1221no—
397C2×C4—1221no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4#1451975011yescharacteristic
401C2×C4#1451975011yescharacteristic
402C2×C4#1451975011yescharacteristic
403C2×C4SD16×D6112yes—
404C2×C4#1451980012yes—
405C2×C4C2×D4×D6111yescharacteristic
406C2×C4C2×D4×D6111yescharacteristic
407C2×C4C2×#1175822211yescharacteristic
408C8—42no—
409C8—42no—
410C8—1222no—
411C8—1222no—
412C8—2442no—
413C22—21no—
414C22—21no—
415C22—1222no—
416C22—9764no—
417C22—4882no—
418C22—4882no—
419C22—4882no—
420C22—2441no—
421C22—2441no—
422C22#1683999412yes—
423C22C2×#1451975011yescharacteristic
424C4—21no—
425C4—21no—
426C4—82no—
427C4—1222no—
428C4—2441no—
429C4—2441no—
430C4#1683999412yes—
431C4#1684003112yes—
432C4C2×SD16×D6111yescharacteristic
433C4C2×#1451980011yescharacteristic
434C4#1684056112yes—
435C2—22no—
436C2—9762no—
437C2—2442no—
438C2#2145531611yescharacteristic
439C2C2×#1683999411yescharacteristic
440C2C2×#1684003111yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1548C2
socleC22×C61C2×#121
prime radical#21455300C2
hypercenter#1548D61
hypocenterC61#19274
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. C2×#16840110C2semidirect
2. #21455628C2semidirect
4. #21454775C2semidirect
5. #21445238C2semidirect
6. #21445231C2semidirect
7. #21444574C2semidirect
8. #21444486C2semidirect
9. C61×#1548C2semidirect
16. #16840054C22semidirect
17. #16840049C22semidirect
24. #16839886C22semidirect
25. #16839827C22semidirect
35. #16839512C22semidirect
40. C61×#333C22semidirect
222. #1548D61semidirect
259. #333C2×D61semidirect
261. C61#19274semidirect

Maximal subgroups

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

subgroupindex
C2×#168401102
#214556282
#214553002
#214547752
#214452382
#214452312
#214445742
#214444862
C61×#15482
#1927461