Finite Group Databasebeta

← #29663146

Subgroups — #29663146

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457137C212yesmaximal
2#21456330C211yesmaximal, characteristic, complemented
3#21454526C211yesmaximal, characteristic
4#21445321C212yesmaximal, complemented
5#21445320C212yesmaximal, complemented
6#21445215C211yesmaximal, characteristic, complemented
7#21445209C211yesmaximal, characteristic, complemented
8#21445208C211yesmaximal, characteristic, complemented
9#335×D61C211yesmaximal, characteristic, complemented
10#21444506C212yesmaximal, complemented
11C61×#1675C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13#16840317C2212yes—
14#16840309C2211yescharacteristic
15#16840167C2211yescharacteristic
16#16840163—21no—
17#16840163—21no—
18#16840158C2212yes—
19#16840049C2212yes—
20#16840046—22no—
21#16839992—22no—
22#16839990—22no—
23#16839968C2212yes—
24#16839967C2212yes—
25#16839961—22no—
26#16839840C2211yescharacteristic
27#16839839C2211yescharacteristic
28#16839837—21no—
29C8×D244C2211yescharacteristic
30C8×C22⋊D61—21no—
31#16839799—21no—
32#16839799—21no—
33#16839784C2211yescharacteristic
34C4⋊C8×D61C2212yes—
35#16839727—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839514C2212yes—
39#16839511C2212yes—
40#16839509C2212yes—
41#16839507C2212yes—
42C61×#500C2211yescharacteristic
43C61×#476C2212yes—
44C8×D4×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#333C2212yes—
47C2×#11758222—21no—
48C2×#11758190—22no—
49C2×#11758200—22no—
50#14519783—44no—
51C2×#11758194—21no—
52C2×#11758189—21no—
53C2×#11758189—21no—
54C2×C8×D61—21no—
55C2×C8×D61—22no—
56C2×C8×D61D411yescharacteristic
57C2×C8×D61D411yescharacteristic
58#14519777C2312yes—
59#14519776D411yescharacteristic
60#14519773D411yescharacteristic
61#14519768—22no—
62#14519763—21no—
63#14519757—22no—
64#14519753—22no—
65#14519752D411yescharacteristic
66#14519750—21no—
67#14519748D411yescharacteristic
68Q8×Dic61—22no—
69#14519742—22no—
70C4×D244C2311yescharacteristic
71C4×C22⋊D61—21no—
72#14519735—22no—
73C42×D61C2311yescharacteristic
74#14519711—21no—
75#14519711—21no—
76#14519708—22no—
77#14519708—22no—
78#14519708—22no—
79C4⋊#92418C2312yes—
80C4⋊#92418C2311yescharacteristic
81#14519706—22no—
82#14519706—21no—
83#14519702—22no—
84#14519700—22no—
85#14519697—22no—
86C22⋊#92418—21no—
87#14519689—21no—
88#14519689—21no—
89C4×#92418C2311yescharacteristic
90C8×Dic61D411yescharacteristic
91C8×Dic61D411yescharacteristic
92C22×C8×C61—21no—
93C4⋊Q8×C61C2312yes—
94C4×D4×C61C2311yescharacteristic
95#107×C61—21no—
96#107×C61—21no—
97#106×C61—22no—
98C4⋊C8×C61C2312yes—
99C4⋊C8×C61C2311yescharacteristic
100#102×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—
119C2×C8×C61—21no—
120C2×C8×C61—22no—
121C2×C8×C61—42no—
122C2×C8×C61C2×D411yescharacteristic
123C2×C8×C61C2×D411yescharacteristic
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61C2×D411yescharacteristic
127C22⋊C4×C61—21no—
128C42×C61C2411yescharacteristic
129C22⋊Dic61—21no—
130#11758200—82no—
131#11758198—21no—
132#11758198—21no—
133#11758198—42no—
134C4⋊Dic61—22no—
135C4⋊Dic61—22no—
136C4⋊Dic61—42no—
137C4⋊Dic61C2×D411yescharacteristic
138#11758196—21no—
139#11758196—42no—
140C4×Dic61—42no—
141C4×Dic61C2×D411yescharacteristic
142C4×Dic61C2×D411yescharacteristic
143C4×Dic61#4111yescharacteristic
144#11758194—42no—
145C2×#92418—21no—
146C2×#92418—22no—
147C2×#92418C2×D411yescharacteristic
148C2×#92418C2×D411yescharacteristic
149#11758190—82no—
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—82no—
160D4×C61—21no—
161D4×C61—21no—
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—21no—
178D244—21no—
179D244—42no—
180C4×D61—21no—
181C4×D61—21no—
182C4×D61—41no—
183C4×D61—41no—
184C4×D61—41no—
185C4×D61C2×SD1612yes—
186C4×D61#13712yes—
187Dic122—42no—
188Dic122—42no—
189Dic122—42no—
190C8×C61—21no—
191C8×C61—41no—
192C8×C61—42no—
193C8×C61C2×SD1612yes—
194#92418—21no—
195#92418—41no—
196#92418—42no—
197#92418C2×SD1612yes—
198#19195—611nomaximal, Sylow
199C22×C61—41no—
200C22×C61—41no—
201C22×C61—41no—
202C22×C61C2×#12111yescharacteristic
203C2×D61—21no—
204C2×D61—21no—
205C2×D61—41no—
206C2×D61—41no—
207C2×D61—41no—
208C2×D61#46112yes—
209C4×C61—21no—
210C4×C61—41no—
211C4×C61—82no—
212C4×C61C22×SD1611yescharacteristic
213C4×C61C2×#13711yescharacteristic
214C4×C61#57611yescharacteristic
215C4×C61#57811yescharacteristic
216Dic61—21no—
217Dic61—21no—
218Dic61—41no—
219Dic61—82no—
220Dic61#46112yes—
221#3246—612no—
222#3160—611no—
223#2931—611no—
224#1691—612no—
225#1691—612no—
226#1675—611no—
227#1675D6111yescharacteristic, complemented
228#1673—611no—
229#1673—611no—
230C2×#335—611no—
231#1550—612no—
232C2×C61—81no—
233C2×C61C2×#46111yescharacteristic
234C2×C61#306111yescharacteristic
235C2×C61#306711yescharacteristic
236D61—22no—
237D61—81no—
238C2×#131—1221no—
239#533—612no—
240#532—611no—
241#500—611no—
242#500C2×D6111yescharacteristic
243#499—1221no—
244#499—1221no—
245#498—612no—
246#476—612no—
247#476C2×D6112yes—
248#475—1222no—
249#465—1222no—
250#465—1222no—
251#461—612no—
252#461—612no—
253#460—1222no—
254#436—611no—
255#436—611no—
256#436—1221no—
257C8×D4—611no—
258C8×D4—1221no—
259C8×D4C2×D6111yescharacteristic
260#427—1221no—
261#427—1221no—
262#423—611no—
263C2×C4⋊C8—612no—
264#408—1221no—
265C2×C4×C8—611no—
266#335—612no—
267#335—611no—
268#335C2×D6111yescharacteristic
269#333—612no—
270#333C2×D6112yes—
271#332—612no—
272#332—612no—
273C61#1919511yescharacteristic, complemented, Sylow
274C2×#41—1221no—
275C2×SD16—1222no—
276C2×SD16—1222no—
277#131—2444no—
278C2×M16—1221no—
279C2×M16—1221no—
280C2×M16—1221no—
281C22×C8—21no—
282C22×C8—611no—
283C22×C8—611no—
284C22×C8—1222no—
285C22×C8—1221no—
286C4⋊Q8—611no—
287C4⋊Q8—612no—
288C4⋊Q8C22×D6112yes—
289#127—611no—
290#125—1222no—
291#124—1221no—
292C22⋊Q8—1222no—
293C22⋊Q8—1222no—
294#121—611no—
295#121—611no—
296#121—1221no—
297C4×Q8—1222no—
298C4×Q8—1222no—
299C4×D4—611no—
300C4×D4—1221no—
301C4×D4C22×D6111yescharacteristic
302#117—1222no—
303C2×C42—611no—
304#107—21no—
305#107—21no—
306#107—1221no—
307#107—1221no—
308#106—22no—
309#106—1222no—
310#106—1222no—
311#106—1222no—
312C4⋊C8—611no—
313C4⋊C8—612no—
314C4⋊C8—1222no—
315C4⋊C8—1221no—
316C4⋊C8C22×D6111yescharacteristic
317C4⋊C8C22×D6112yes—
318#103—1222no—
319#103—1222no—
320#102—22no—
321#102—1222no—
322C22⋊C8—21no—
323C22⋊C8—1221no—
324C22⋊C8—1221no—
325C22⋊C8—1221no—
326C4×C8—611no—
327C4×C8—611no—
328C4×C8—611no—
329C4×C8C22×D6111yescharacteristic
330#41—2442no—
331#41—2442no—
332C2×Q8—22no—
333C2×Q8—1221no—
334C2×Q8—1222no—
335C2×D4—611no—
336C2×D4—1221no—
337C2×D4—1221no—
338C2×D4—1221no—
339C2×D4—1221no—
340C2×D4D4×D6111yescharacteristic
341C22×C4—21no—
342C22×C4—611no—
343C22×C4—611no—
344C22×C4—611no—
345C22×C4—1221no—
346SD16—4882no—
347SD16—4882no—
348M16—2442no—
349M16—2442no—
350M16—2442no—
351C2×C8—21no—
352C2×C8—22no—
353C2×C8—42no—
354C2×C8—611no—
355C2×C8—614no—
356C2×C8—611no—
357C2×C8—1222no—
358C2×C8—1221no—
359C2×C8—1222no—
360C2×C8—4882no—
361C2×C8—2442no—
362C2×C8D4×D6111yescharacteristic
363C2×C8D4×D6111yescharacteristic
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—611no—
367C4⋊C4—1221no—
368C4⋊C4—1222no—
369C4⋊C4—1222no—
370C4⋊C4—2442no—
371C4⋊C4—2442no—
372C4⋊C4D4×D6111yescharacteristic
373C22⋊C4—21no—
374C22⋊C4—1221no—
375C22⋊C4—1221no—
376C22⋊C4—1221no—
377C22⋊C4—2442no—
378C42—611no—
379C42—611no—
380C42—611no—
381C42—2442no—
382C42C23×D6111yescharacteristic
383C23—21no—
384C23—611no—
385C23—1221no—
386Q8—82no—
387Q8—2442no—
388Q8—2442no—
389Q8—2442no—
390D4—21no—
391D4—21no—
392D4—1221no—
393D4—1221no—
394D4—2442no—
395D4—2442no—
396D4—2442no—
397D4—2442no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—611no—
403C2×C4—612no—
404C2×C4—612no—
405C2×C4—611no—
406C2×C4—611no—
407C2×C4—1221no—
408C2×C4—1221no—
409C2×C4—1221no—
410C2×C4—2441no—
411C2×C4—2442no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4C2×D4×D6111yescharacteristic
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4#41×D6111yescharacteristic
417C8—21no—
418C8—41no—
419C8—42no—
420C8—612no—
421C8—1221no—
422C8—2441no—
423C8—2442no—
424C8SD16×D6112yes—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—612no—
429C22—1221no—
430C22—1221no—
431C22—2441no—
432C22—2441no—
433C22—2441no—
434C22#121×D6111yescharacteristic
435C4—21no—
436C4—41no—
437C4—82no—
438C4—612no—
439C4—1221no—
440C4—1221no—
441C4—4882no—
442C4—2441no—
443C4C2×SD16×D6111yescharacteristic
444C4#137×D6111yescharacteristic
445C4#1684056411yescharacteristic
446C4#1684057511yescharacteristic
447C2—81no—
448C2—1222no—
449C2—4881no—
450C2#461×D6111yescharacteristic
451C2#2145549011yescharacteristic
452C2#2145553711yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1675C2
socleC22×C61C2×#121
prime radical#16840309C22
hypercenter#1675D61
hypocenterC61#19195
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1675

Complements

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

NH ≅ G/Ntype
2. #21456330C2semidirect
4. #21445321C2semidirect
5. #21445320C2semidirect
6. #21445215C2semidirect
7. #21445209C2semidirect
8. #21445208C2semidirect
9. #335×D61C2semidirect
10. #21444506C2semidirect
11. C61×#1675C2semidirect
155. C8×D61SD16semidirect
227. #1675D61semidirect
273. C61#19195semidirect

Maximal subgroups

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

subgroupindex
#214571372
#214563302
#214545262
#214453212
#214453202
#214452152
#214452092
#214452082
#335×D612
#214445062
C61×#16752
#1919561