Finite Group Databasebeta

← #29663432

Subgroups — #29663432

Order 15616.

630 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456795C212yesmaximal, complemented
2#21454794C211yesmaximal, characteristic, complemented
3#21445334C211yesmaximal, characteristic, complemented
4#21445329C211yesmaximal, characteristic, complemented
5#21445300C211yesmaximal, characteristic, complemented
6#21445295C211yesmaximal, characteristic, complemented
7#21445260C212yesmaximal, complemented
8#21445253C212yesmaximal, complemented
9#21444567C211yesmaximal, characteristic, complemented
10#21444524C212yesmaximal, complemented
11C61×#3169C211yesmaximal, characteristic, complemented
12#16840296C2211yescharacteristic
13#16840144—22no—
14#16840116—22no—
15#16839969C2212yes—
16#16839964C2212yes—
17#16839959C2212yes—
18#16839954C2212yes—
19#16839924—22no—
20#16839907—22no—
21#16839892C2211yescharacteristic
22C8×Dic122C2211yescharacteristic
23#16839841C2211yescharacteristic
24#16839833—21no—
25C8×D244C2211yescharacteristic
26C8×C22⋊D61—21no—
27#16839821—21no—
28#16839819C2211yescharacteristic
29#16839815C2212yes—
30#16839716—21no—
31#16839515C2212yes—
32#16839513C2211yescharacteristic
33#16839513C2211yescharacteristic
34#16839508C2212yes—
35#16839508C2212yes—
36#16839506C2212yes—
37#16839506C2212yes—
38C61×#532C2212yes—
39C61×#500C2211yescharacteristic
40C61×#493C2211yescharacteristic
41C61×#486C2212yes—
42C61×#466—21no—
43C61×#466—21no—
44C4×D8×C61—21no—
45C4×D8×C61—21no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758202—22no—
48C2×#11758200—22no—
49C2×#11758201—22no—
50C2×#11758199—22no—
51C2×#11758189—21no—
52C2×C8×D61—21no—
53#14519771—21no—
54#14519769D411yescharacteristic
55#14519765D411yescharacteristic
56#14519760—21no—
57#14519756D411yescharacteristic
58#14519751D411yescharacteristic
59C4×Dic122C2311yescharacteristic
60C4×D244C2311yescharacteristic
61C4×C22⋊D61—21no—
62#14519734—21no—
63C4⋊#92418C2312yes—
64C4⋊#92418C2312yes—
65C4⋊#92418C2311yescharacteristic
66C4⋊#92418C2311yescharacteristic
67#14519706—21no—
68#14519706—21no—
69#14519701—22no—
70#14519701—22no—
71#14519698—22no—
72#14519698—22no—
73C22⋊#92418—22no—
74C22⋊#92418—21no—
75#14519689—21no—
76#14519689—21no—
77#14519688—21no—
78#14519687—21no—
79C4×#92418—22no—
80C8×Dic61—21no—
81C2×#41×C61—22no—
82C2×D8×C61—21no—
83C2×D8×C61—21no—
84C22×C8×C61D411yescharacteristic
85C22×C8×C61D411yescharacteristic
86C4⋊Q8×C61C2312yes—
87#127×C61C2312yes—
88#124×C61—22no—
89#121×C61—22no—
90#121×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C2×C42×C61C2311yescharacteristic
94#107×C61—21no—
95#107×C61—21no—
96#102×C61—22no—
97#102×C61—22no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102#41×C61—82no—
103#41×C61—82no—
104C2×Q8×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—42no—
109C22×C4×C61—42no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C2×C22⋊D61—21no—
114C2×D244C2×D411yescharacteristic
115C2×C4×D61—21no—
116C2×Dic122C2×D411yescharacteristic
117D8×C61—42no—
118D8×C61—42no—
119C2×C8×C61—22no—
120C2×C8×C61—41no—
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D411yescharacteristic
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2×D411yescharacteristic
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C22⋊C4×C61—42no—
128C22⋊C4×C61—42no—
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61#4111yescharacteristic
132C42×C61C2411yescharacteristic
133C22⋊Dic61—21no—
134#11758202—82no—
135#11758201—82no—
136#11758200—82no—
137#11758199—82no—
138#11758198—21no—
139#11758198—21no—
140#11758198—21no—
141#11758198—21no—
142C4⋊Dic61C2×D411yescharacteristic
143C4⋊Dic61C2×D411yescharacteristic
144#11758196—21no—
145#11758196—21no—
146#11758196—21no—
147#11758196—21no—
148C4×Dic61—21no—
149C2×#92418—21no—
150C2×#92418—22no—
151C2×#92418—22no—
152C2×#92418—21no—
153#11758189—42no—
154C8×D61—42no—
155C23×C61—42no—
156C23×C61#12111yescharacteristic
157C22×D61—21no—
158Q8×C61—42no—
159Q8×C61—42no—
160D4×C61—42no—
161D4×C61—42no—
162D4×C61—42no—
163D4×C61—42no—
164D4×C61—82no—
165D4×C61—84no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—41no—
170C2×C4×C61—42no—
171C2×C4×C61—84no—
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61C22×D411yescharacteristic
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C2×#4111yescharacteristic
178C22⋊D61—42no—
179C2×Dic61—21no—
180C2×Dic61—21no—
181C2×Dic61—21no—
182D244—22no—
183C4×D61—41no—
184C4×D61—41no—
185Dic122—22no—
186C8×C61—21no—
187C8×C61—21no—
188C8×C61—21no—
189C8×C61#13512yes—
190#92418—41no—
191#92418—41no—
192#92418—42no—
193#92418—42no—
194#19225—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—82no—
199C22×C61—82no—
200C22×C61—82no—
201C22×C61C2×#12111yescharacteristic
202C2×D61—41no—
203C2×D61—41no—
204C2×D61—41no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—82no—
210C4×C61C2×#13511yescharacteristic
211C4×C61#57511yescharacteristic
212C4×C61#57612yes—
213Dic61—41no—
214Dic61—41no—
215Dic61—41no—
216#3212—612no—
217#3169D6111yescharacteristic, complemented
218#2966—611no—
219#1694—611no—
220#1693—611no—
221#1688—611no—
222#1687—611no—
223#1682—612no—
224#1681—612no—
225#1564—611no—
226#1553—612no—
227C2×C61—41no—
228C2×C61—162no—
229C2×C61#303511yescharacteristic
230C2×C61#305811yescharacteristic
231C2×C61#306711yescharacteristic
232D61—81no—
233#532C2×D6112yes—
234#529—611no—
235#500C2×D6111yescharacteristic
236#495—1222no—
237#493C2×D6111yescharacteristic
238#488—1222no—
239#486C2×D6112yes—
240#466—21no—
241#466—21no—
242#462—612no—
243#461—612no—
244#460—612no—
245#459—612no—
246#452—1222no—
247#449—1222no—
248#446—611no—
249C8×Q8—611no—
250C4×D8—21no—
251C4×D8—21no—
252#436—611no—
253#435—1221no—
254C8×D4—611no—
255C8×D4—1221no—
256#433—1221no—
257#432—611no—
258#431—612no—
259#405—1221no—
260C2×C4×C8C2×D6111yescharacteristic
261#335—612no—
262#334—611no—
263#334—611no—
264#332—612no—
265#332—612no—
266#331—612no—
267#331—612no—
268C61#1922511yescharacteristic, complemented, Sylow
269C2×#41—22no—
270C2×Q16—1222no—
271C2×SD16—1222no—
272C2×SD16—1222no—
273C2×D8—21no—
274C2×D8—21no—
275C2×D8—1222no—
276C2×M16—1221no—
277C22×C8—1221no—
278C22×C8C22⋊D6111yescharacteristic
279C22×C8C22⋊D6111yescharacteristic
280C4⋊Q8C22×D6112yes—
281#127C22×D6112yes—
282#126—1221no—
283#125—611no—
284#124—22no—
285#124—611no—
286#123—1221no—
287C22⋊Q8—611no—
288#121—22no—
289#121—22no—
290#121—611no—
291C4×Q8—611no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—611no—
295C4×D4—1221no—
296#117—1221no—
297C2×C42C22×D6111yescharacteristic
298#107—21no—
299#107—21no—
300C4⋊C8—611no—
301C4⋊C8—611no—
302C4⋊C8—612no—
303C4⋊C8—612no—
304C4⋊C8—1221no—
305C4⋊C8—1221no—
306#103—1222no—
307#103—1222no—
308#102—22no—
309#102—22no—
310#102—1222no—
311#102—1222no—
312C22⋊C8—1222no—
313C22⋊C8—1221no—
314C22⋊C8—1221no—
315C22⋊C8—1221no—
316#97—1221no—
317#97—1221no—
318C4×C8—1221no—
319C4×C8—1222no—
320C4×C8C22⋊D6111yescharacteristic
321C4×C8C22⋊D6111yescharacteristic
322C4×C8C22×D6111yescharacteristic
323C4×C8C22×D6111yescharacteristic
324#41—82no—
325#41—82no—
326C2×Q8—22no—
327C2×Q8—611no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—42no—
332C2×D4—611no—
333C2×D4—1221no—
334C22×C4—42no—
335C22×C4—1221no—
336C22×C4D4×D6111yescharacteristic
337C22×C4#1175822411yescharacteristic
338C22×C4C2×C22⋊D6111yescharacteristic
339Q16—4882no—
340SD16—4882no—
341SD16—4882no—
342D8—42no—
343D8—42no—
344D8—4882no—
345M16—2442no—
346C2×C8—22no—
347C2×C8—41no—
348C2×C8—1222no—
349C2×C8—1222no—
350C2×C8—1221no—
351C2×C8—1221no—
352C2×C8—2442no—
353C2×C8C2×C22⋊D6111yescharacteristic
354C2×C8C2×C22⋊D6111yescharacteristic
355C2×C8C2×C22⋊D6111yescharacteristic
356C2×C8C2×C22⋊D6111yescharacteristic
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—611no—
360C4⋊C4—611no—
361C4⋊C4—1221no—
362C4⋊C4—1221no—
363C4⋊C4—1221no—
364C4⋊C4—1221no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—1221no—
368C22⋊C4—1221no—
369C22⋊C4—1221no—
370C22⋊C4—1221no—
371C22⋊C4—1221no—
372C42—1221no—
373C42D4×D6111yescharacteristic
374C42#1175822411yescharacteristic
375C42C2×C22⋊D6111yescharacteristic
376C42C23×D6111yescharacteristic
377C23—42no—
378C23—1221no—
379C23#1451974811yescharacteristic
380Q8—42no—
381Q8—42no—
382Q8—1222no—
383D4—42no—
384D4—42no—
385D4—42no—
386D4—42no—
387D4—82no—
388D4—84no—
389D4—1222no—
390D4—2442no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—84no—
397C2×C4—1221no—
398C2×C4—1221no—
399C2×C4—1221no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4#1451974811yescharacteristic
403C2×C4#1451974811yescharacteristic
404C2×C4#1451974811yescharacteristic
405C2×C4C2×D4×D6111yescharacteristic
406C2×C4C22×C22⋊D6111yescharacteristic
407C2×C4C2×#1175822411yescharacteristic
408C8—21no—
409C8—21no—
410C8—21no—
411C8—2441no—
412C8—2441no—
413C8—2442no—
414C8—2442no—
415C8#1451979712yes—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—2441no—
423C22—2441no—
424C22—2441no—
425C22C2×#1451974811yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—82no—
431C4—2441no—
432C4—2441no—
433C4—2441no—
434C4C2×#1451979711yescharacteristic
435C4#1684056011yescharacteristic
436C4#1684056312yes—
437C2—41no—
438C2—162no—
439C2—4881no—
440C2#2145526811yescharacteristic
441C2#2145546311yescharacteristic
442C2#2145553511yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3169C2
socleC22×C61C2×#121
hypercenter#3169D61
hypocenterC61#19225
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. #21456795C2semidirect
2. #21454794C2semidirect
3. #21445334C2semidirect
4. #21445329C2semidirect
5. #21445300C2semidirect
6. #21445295C2semidirect
7. #21445260C2semidirect
8. #21445253C2semidirect
9. #21444567C2semidirect
10. #21444524C2semidirect
11. C61×#3169C2semidirect
217. #3169D61semidirect
268. C61#19225semidirect

Maximal subgroups

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

subgroupindex
#214567952
#214547942
#214453342
#214453292
#214453002
#214452952
#214452602
#214452532
#214445672
#214445242
C61×#31692
#1922561