Finite Group Databasebeta

← #29663135

Subgroups — #29663135

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457132C212yesmaximal
2#21456330C211yesmaximal, characteristic, complemented
3#21454528C211yesmaximal, characteristic
4#21445343C211yesmaximal, characteristic, complemented
5#21445342C211yesmaximal, characteristic, complemented
6#21445329C211yesmaximal, characteristic, complemented
7#21445193C212yesmaximal, complemented
8#21445192C212yesmaximal, complemented
9#334×D61C211yesmaximal, characteristic, complemented
10#21444508C212yesmaximal, complemented
11C61×#1693C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840317C2212yes—
14#16840309C2211yescharacteristic
15#16840167C2211yescharacteristic
16#16840163—21no—
17#16840163—21no—
18#16840158C2212yes—
19#16840053C2212yes—
20#16840052C2212yes—
21#16840045—22no—
22#16839994—22no—
23#16839988—22no—
24#16839964C2212yes—
25#16839960—22no—
26#16839841C2211yescharacteristic
27#16839838—21no—
28D4×#92418C2211yescharacteristic
29#16839827C2211yescharacteristic
30#16839825—21no—
31#16839799—21no—
32#16839799—21no—
33#16839784C2211yescharacteristic
34C4⋊C8×D61C2212yes—
35#16839727—21no—
36C4×C8×D61C2211yescharacteristic
37#16839513C2211yescharacteristic
38#16839512C2212yes—
39#16839510C2212yes—
40#16839510C2212yes—
41#16839508C2212yes—
42C61×#500C2211yescharacteristic
43C61×#461C2212yes—
44C61×#436C2211yescharacteristic
45C61×#334C2211yescharacteristic
46C61×#332C2212yes—
47C2×#11758222—21no—
48C2×#11758201—22no—
49#14519785—44no—
50C2×#11758189—21no—
51C2×#11758189—21no—
52C22×#92418—21no—
53C2×C8×D61—21no—
54C2×C8×D61—22no—
55C2×C8×D61D411yescharacteristic
56C2×C8×D61D411yescharacteristic
57#14519777C2312yes—
58#14519776D411yescharacteristic
59#14519773D411yescharacteristic
60#14519768—22no—
61#14519763—21no—
62#14519757—22no—
63#14519753—22no—
64#14519752D411yescharacteristic
65#14519750—21no—
66#14519748D411yescharacteristic
67Q8×Dic61—22no—
68#14519742—22no—
69C4×D244C2311yescharacteristic
70C4×C22⋊D61—21no—
71#14519735—22no—
72C42×D61C2311yescharacteristic
73#14519711—21no—
74#14519711—21no—
75#14519709—22no—
76#14519708—22no—
77#14519708—22no—
78#14519708—22no—
79C4⋊#92418C2312yes—
80C4⋊#92418C2311yescharacteristic
81#14519706—22no—
82#14519706—21no—
83#14519701—22no—
84#14519699—22no—
85#14519698—22no—
86C22⋊#92418—21no—
87#14519689—21no—
88#14519689—21no—
89C4×#92418C2311yescharacteristic
90C8×Dic61D411yescharacteristic
91C8×Dic61D411yescharacteristic
92C2×SD16×C61—22no—
93C2×M16×C61—21no—
94C4⋊Q8×C61C2312yes—
95C4×D4×C61C2311yescharacteristic
96#107×C61—21no—
97#107×C61—21no—
98C4⋊C8×C61C2311yescharacteristic
99C4⋊C8×C61C2312yes—
100#103×C61—22no—
101C22⋊C8×C61—21no—
102C4×C8×C61C2311yescharacteristic
103C2×Q8×C61—22no—
104C2×D4×C61C2×D411yescharacteristic
105C22×C4×C61—21no—
106C2×C22⋊D61—21no—
107C2×C22⋊D61—21no—
108C2×C22⋊D61—21no—
109#11758222—42no—
110#11758222—42no—
111C2×D244—21no—
112C2×D244C2×D411yescharacteristic
113C2×C4×D61—21no—
114C2×C4×D61C2×D411yescharacteristic
115C2×C4×D61C2×D411yescharacteristic
116C2×C4×D61#4111yescharacteristic
117C2×Dic122—21no—
118C2×Dic122—22no—
119SD16×C61—82no—
120M16×C61—42no—
121C2×C8×C61—21no—
122C2×C8×C61—22no—
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2×D411yescharacteristic
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61C2×D411yescharacteristic
128C22⋊C4×C61—21no—
129C42×C61C2411yescharacteristic
130C22⋊Dic61—21no—
131#11758201—82no—
132#11758198—21no—
133#11758198—21no—
134#11758198—42no—
135C4⋊Dic61—22no—
136C4⋊Dic61—22no—
137C4⋊Dic61—42no—
138C4⋊Dic61C2×D411yescharacteristic
139#11758196—21no—
140#11758196—42no—
141C4×Dic61—42no—
142C4×Dic61C2×D411yescharacteristic
143C4×Dic61C2×D411yescharacteristic
144C4×Dic61#4111yescharacteristic
145C2×#92418—21no—
146C2×#92418—22no—
147C2×#92418—42no—
148C2×#92418C2×D411yescharacteristic
149C2×#92418C2×D411yescharacteristic
150#11758189—42no—
151#11758189—42no—
152C8×D61—22no—
153C8×D61—42no—
154C8×D61—82no—
155C8×D61SD1614yescomplemented
156C23×C61—21no—
157C22×D61—21no—
158C22×D61#12111yescharacteristic
159Q8×C61—42no—
160Q8×C61—42no—
161D4×C61—22no—
162C2×C4×C61—21no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—42no—
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C2×#4111yescharacteristic
169C22⋊D61—42no—
170C22⋊D61—42no—
171C22⋊D61—42no—
172C2×Dic61—21no—
173C2×Dic61—42no—
174C2×Dic61#12111yescharacteristic
175C2×Dic61#12111yescharacteristic
176C2×Dic61#12111yescharacteristic
177D244—22no—
178D244—42no—
179C4×D61—22no—
180C4×D61—41no—
181C4×D61—41no—
182C4×D61—41no—
183C4×D61C2×SD1612yes—
184C4×D61#13712yes—
185Dic122—42no—
186Dic122—82no—
187C8×C61—21no—
188C8×C61—41no—
189C8×C61—42no—
190C8×C61C2×SD1612yes—
191#92418—21no—
192#92418—41no—
193#92418—42no—
194#92418C2×SD1612yes—
195#19194—611nomaximal, Sylow
196C22×C61—41no—
197C22×C61—41no—
198C22×C61—41no—
199C22×C61C2×#12111yescharacteristic
200C2×D61—21no—
201C2×D61—21no—
202C2×D61—41no—
203C2×D61—41no—
204C2×D61—41no—
205C2×D61#46112yes—
206C4×C61—21no—
207C4×C61—41no—
208C4×C61—82no—
209C4×C61C22×SD1611yescharacteristic
210C4×C61C2×#13711yescharacteristic
211C4×C61#57712yes—
212Dic61—21no—
213Dic61—21no—
214Dic61—41no—
215Dic61—82no—
216Dic61#46112yes—
217#3246—612no—
218#3160—611no—
219#2931—611no—
220#1695—611no—
221#1695—611no—
222#1693—611no—
223#1693D6111yescharacteristic, complemented
224#1671—612no—
225#1671—612no—
226C2×#334—611no—
227#1550—612no—
228C2×C61—81no—
229C2×C61C2×#46111yescharacteristic
230C2×C61#306011yescharacteristic
231C2×C61#306711yescharacteristic
232D61—22no—
233D61—81no—
234C2×#131—1221no—
235#533—612no—
236#532—611no—
237#500—611no—
238#500C2×D6111yescharacteristic
239#499—1221no—
240#499—1221no—
241#498—612no—
242#476—612no—
243#476—612no—
244#475—1222no—
245#465—1222no—
246#465—1222no—
247#461—612no—
248#461C2×D6112yes—
249#460—1222no—
250#436—611no—
251#436—1221no—
252#436C2×D6111yescharacteristic
253C8×D4—611no—
254C8×D4—611no—
255C8×D4—1221no—
256#427—1221no—
257#427—1221no—
258#423—611no—
259C2×C4⋊C8—612no—
260#408—1221no—
261C2×C4×C8—611no—
262#334—612no—
263#334—611no—
264#334C2×D6111yescharacteristic
265#333—612no—
266#333—612no—
267#332—612no—
268#332C2×D6112yes—
269C61#1919411yescharacteristic, complemented, Sylow
270C2×#41—1221no—
271C2×SD16—22no—
272C2×SD16—1222no—
273#131—2444no—
274C2×M16—21no—
275C2×M16—1221no—
276C2×M16—1221no—
277C22×C8—611no—
278C22×C8—611no—
279C22×C8—1221no—
280C22×C8—1222no—
281C22×C8—1221no—
282C4⋊Q8—611no—
283C4⋊Q8—612no—
284C4⋊Q8C22×D6112yes—
285#127—611no—
286#125—1222no—
287#124—1221no—
288C22⋊Q8—1222no—
289C22⋊Q8—1222no—
290#121—611no—
291#121—611no—
292#121—1221no—
293C4×Q8—1222no—
294C4×Q8—1222no—
295C4×D4—611no—
296C4×D4—1221no—
297C4×D4C22×D6111yescharacteristic
298#117—1222no—
299C2×C42—611no—
300#107—21no—
301#107—21no—
302#107—1221no—
303#107—1221no—
304#106—1222no—
305#106—1222no—
306#106—1222no—
307#106—1222no—
308C4⋊C8—611no—
309C4⋊C8—612no—
310C4⋊C8—1222no—
311C4⋊C8—1221no—
312C4⋊C8C22×D6111yescharacteristic
313C4⋊C8C22×D6112yes—
314#103—22no—
315#103—1222no—
316#102—1222no—
317#102—1222no—
318C22⋊C8—21no—
319C22⋊C8—1221no—
320C22⋊C8—1221no—
321C22⋊C8—1221no—
322C4×C8—611no—
323C4×C8—611no—
324C4×C8—611no—
325C4×C8C22×D6111yescharacteristic
326#41—2442no—
327#41—2442no—
328C2×Q8—22no—
329C2×Q8—1221no—
330C2×Q8—1222no—
331C2×D4—611no—
332C2×D4—1221no—
333C2×D4—1221no—
334C2×D4—1221no—
335C2×D4—1221no—
336C2×D4D4×D6111yescharacteristic
337C22×C4—21no—
338C22×C4—611no—
339C22×C4—611no—
340C22×C4—611no—
341C22×C4—1221no—
342SD16—82no—
343SD16—4882no—
344M16—42no—
345M16—2442no—
346M16—2442no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—611no—
350C2×C8—614no—
351C2×C8—611no—
352C2×C8—1222no—
353C2×C8—1221no—
354C2×C8—1222no—
355C2×C8—4882no—
356C2×C8—2442no—
357C2×C8—2442no—
358C2×C8D4×D6111yescharacteristic
359C2×C8D4×D6111yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—611no—
363C4⋊C4—1221no—
364C4⋊C4—1222no—
365C4⋊C4—1222no—
366C4⋊C4—2442no—
367C4⋊C4—2442no—
368C4⋊C4D4×D6111yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—1221no—
371C22⋊C4—1221no—
372C22⋊C4—1221no—
373C22⋊C4—2442no—
374C42—611no—
375C42—611no—
376C42—611no—
377C42—2442no—
378C42C23×D6111yescharacteristic
379C23—21no—
380C23—611no—
381C23—1221no—
382Q8—42no—
383Q8—42no—
384Q8—4882no—
385Q8—2442no—
386D4—22no—
387D4—1222no—
388D4—2442no—
389D4—2442no—
390D4—2442no—
391D4—2442no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—611no—
397C2×C4—612no—
398C2×C4—612no—
399C2×C4—611no—
400C2×C4—611no—
401C2×C4—1222no—
402C2×C4—1221no—
403C2×C4—2441no—
404C2×C4—2442no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4C2×D4×D6111yescharacteristic
408C2×C4C2×D4×D6111yescharacteristic
409C2×C4#41×D6111yescharacteristic
410C8—21no—
411C8—41no—
412C8—42no—
413C8—612no—
414C8—1221no—
415C8—2441no—
416C8—2442no—
417C8SD16×D6112yes—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—612no—
422C22—1221no—
423C22—1221no—
424C22—2441no—
425C22—2441no—
426C22—2441no—
427C22#121×D6111yescharacteristic
428C4—21no—
429C4—41no—
430C4—82no—
431C4—612no—
432C4—1221no—
433C4—1221no—
434C4—4882no—
435C4—2441no—
436C4C2×SD16×D6111yescharacteristic
437C4#137×D6111yescharacteristic
438C4#1684057212yes—
439C2—81no—
440C2—1222no—
441C2—4881no—
442C2#461×D6111yescharacteristic
443C2#2145547911yescharacteristic
444C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1693C2
socleC22×C61C2×#121
prime radical#16840309C22
hypercenter#1693D61
hypocenterC61#19194
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1693

Complements

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

NH ≅ G/Ntype
2. #21456330C2semidirect
4. #21445343C2semidirect
5. #21445342C2semidirect
6. #21445329C2semidirect
7. #21445193C2semidirect
8. #21445192C2semidirect
9. #334×D61C2semidirect
10. #21444508C2semidirect
11. C61×#1693C2semidirect
155. C8×D61SD16semidirect
223. #1693D61semidirect
269. C61#19194semidirect

Maximal subgroups

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

subgroupindex
#214571322
#214563302
#214545282
#214453432
#214453422
#214453292
#214451932
#214451922
#334×D612
#214445082
C61×#16932
#1919461