Finite Group Databasebeta

← #29835382

Subgroups — #29835382

Order 15616.

622 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
1C2×#16840166C211yesmaximal, characteristic, complemented
2#21456243C211yesmaximal, characteristic
3#21448057C212yesmaximal, complemented
4#21448054C212yesmaximal, complemented
5#21448045C212yesmaximal, complemented
6#21448041C212yesmaximal, complemented
7#21447532C211yesmaximal, characteristic, complemented
8#21447531C211yesmaximal, characteristic, complemented
9#21447529C211yesmaximal, characteristic, complemented
10M32×Dic61C211yesmaximal, characteristic, complemented
11C61×#2139C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519797—21no—
14#16840308—21no—
15C2×#14519776—21no—
16C2×#14519717C2211yescharacteristic
17#16840166—21no—
18#16840166—21no—
19#16840166C2212yes—
20#16840166C2212yes—
21#16840147C2211yescharacteristic
22#16840145C2211yescharacteristic
23#16840143—21no—
24#16840140C2211yescharacteristic
25#16840137C2211yescharacteristic
26#16840133—21no—
27C2×#14519711C2211yescharacteristic
28C2×#14519710—21no—
29C2×C8×Dic61C2211yescharacteristic
30#16839567C2212yes—
31#16839566C2212yes—
32#16839565C2212yes—
33#16839564C2212yes—
34#16839563C2212yes—
35#16839562C2212yes—
36#16839537C2212yes—
37C16×Dic61C2212yes—
38C2×#135×C61C2211yescharacteristic
39C2×M32×C61C2211yescharacteristic
40C2×#107×C61C2211yescharacteristic
41C61×#358C2212yes—
42C61×#357C2212yes—
43C2×#11758222—21no—
44C22×Dic122—41no—
45#14519799—22no—
46#14519799—22no—
47#14519797—44no—
48C2×Dic244C2311yescharacteristic
49C2×#11758202—21no—
50C2×#11758190—21no—
51C2×#11758200—21no—
52C2×#11758201—21no—
53C2×D488C2311yescharacteristic
54C2×#11758199—21no—
55C22×#92418D411yescharacteristic
56#14519776—22no—
57#14519776—22no—
58#14519776—41no—
59#14519776—41no—
60#14519775—21no—
61#14519773—21no—
62#14519766—21no—
63#14519764—21no—
64#14519749—41no—
65#14519749—41no—
66C4×C22⋊D61—41no—
67C4×C22⋊D61—41no—
68C2×C4⋊Dic61—21no—
69C2×#11758196—41no—
70C2×C4×Dic61D411yescharacteristic
71#14519717—22no—
72C2×#11758185C2312yes—
73#14519711—21no—
74#14519711C2312yes—
75#14519710—22no—
76#14519710—22no—
77C8×Dic61D411yescharacteristic
78C8×Dic61D411yescharacteristic
79C8×Dic61C2311yescharacteristic
80C8×Dic61C2311yescharacteristic
81C2×#41×C61—21no—
82#135×C61—22no—
83#135×C61—22no—
84C2×Q16×C61C2311yescharacteristic
85C2×SD16×C61—21no—
86C2×D8×C61C2311yescharacteristic
87C22×C8×C61C2311yescharacteristic
88C2×C4⋊C4×C61—21no—
89M32×C61—22no—
90C2×C16×C61C2312yes—
91#107×C61—21no—
92#107×C61C2312yes—
93#41×C61—42no—
94#41×C61—42no—
95C2×Q8×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C2×C22⋊D61—41no—
102C2×C22⋊D61—41no—
103C2×C22⋊D61—41no—
104C22×Dic61—41no—
105C22×Dic61D811yescharacteristic
106C22×Dic61D811yescharacteristic
107#11758222—42no—
108#11758222—42no—
109C2×D244—21no—
110C2×C4×D61—41no—
111C2×Dic122—21no—
112C2×Dic122—42no—
113C2×Dic122—42no—
114C2×Dic122—84no—
115Q16×C61—21no—
116Q16×C61—21no—
117SD16×C61—42no—
118D8×C61—21no—
119D8×C61—21no—
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61#4112yes—
122C2×C8×C61#4112yes—
123C2×C8×C61C2411yescharacteristic
124C4⋊C4×C61—22no—
125C4⋊C4×C61—41no—
126C22⋊Dic61—41no—
127C22⋊Dic61—41no—
128#11758202—42no—
129#11758201—42no—
130#11758200—42no—
131#11758199—42no—
132#11758198—41no—
133#11758198—41no—
134C4⋊Dic61—22no—
135C4⋊Dic61—41no—
136#11758196—42no—
137#11758196—42no—
138#11758196—41no—
139#11758196—41no—
140C4×Dic61D811yescharacteristic
141C4×Dic61D811yescharacteristic
142C4×Dic61C2×D411yescharacteristic
143C4×Dic61C2×D411yescharacteristic
144C2×#92418SD1614yes—
145C2×#92418C2×D411yescharacteristic
146C2×#92418C2×D411yescharacteristic
147Dic244—21no—
148Dic244—21no—
149D488—21no—
150D488—21no—
151#11758190—42no—
152C16×C61—22no—
153#11758185—22no—
154C23×C61—41no—
155C23×C61C2×D811yescharacteristic
156C22×D61—41no—
157Q8×C61—41no—
158Q8×C61—41no—
159D4×C61—41no—
160D4×C61—41no—
161D4×C61—82no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—82no—
165C2×C4×C61—82no—
166C2×C4×C61#12412yes—
167C2×C4×C61C2×D811yescharacteristic
168C2×C4×C61C2×SD1612yes—
169C2×C4×C61C22×D411yescharacteristic
170C22⋊D61—82no—
171C22⋊D61—82no—
172C22⋊D61—82no—
173C2×Dic61—22no—
174C2×Dic61—22no—
175C2×Dic61—42no—
176C2×Dic61—41no—
177C2×Dic61—82no—
178C2×Dic61C2×D811yescharacteristic
179C2×Dic61C2×D811yescharacteristic
180C2×Dic61C2×D811yescharacteristic
181C2×Dic61C2×D811yescharacteristic
182D244—41no—
183D244—41no—
184C4×D61—82no—
185Dic122—41no—
186Dic122—41no—
187Dic122—84no—
188Dic122—84no—
189C8×C61#12412yes—
190C8×C61C2×#4111yescharacteristic
191C8×C61C2×#4111yescharacteristic
192#92418C2×SD1612yes—
193#92418C2×SD1612yes—
194#33707—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—81no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61#48612yes—
201C22×C61C22×D811yescharacteristic
202C2×D61—81no—
203C2×D61—81no—
204C2×D61—81no—
205C4×C61—81no—
206C4×C61—82no—
207C4×C61#48612yes—
208C4×C61C2×#12411yescharacteristic
209C4×C61C22×SD1611yescharacteristic
210Dic61—21no—
211Dic61—21no—
212Dic61—81no—
213Dic61—82no—
214Dic61#50912yes—
215Dic61#51012yes—
216C2×#500—611no—
217#3147—611no—
218#2243—612no—
219#2243—612no—
220#2241—612no—
221#2240—612no—
222#2139—611no—
223#2139—611no—
224#2139—611no—
225#2139D6111yescharacteristic, complemented
226C4×M32—611no—
227C2×C61—22no—
228C2×C61—161no—
229C2×C61C2×#48611yescharacteristic
230C2×C61C2×#50911yescharacteristic
231C2×C61C2×#51011yescharacteristic
232D61—161no—
233C2×#135—611no—
234C2×#135—1221no—
235C2×#135C2×D6111yescharacteristic
236#532—1221no—
237C2×C4⋊Q8—1221no—
238C2×M32—611no—
239C2×M32C2×D6111yescharacteristic
240#500—612no—
241#500—612no—
242#500—1221no—
243#500—1221no—
244#495—611no—
245#495—611no—
246#495—1221no—
247#494—611no—
248#493—611no—
249#492—1221no—
250C2×#107—611no—
251C2×#107—1221no—
252C2×#107C2×D6111yescharacteristic
253C2×C4×C8—611no—
254#358—612no—
255#358—612no—
256#358—612no—
257#358C2×D6112yes—
258#357—612no—
259#357—612no—
260#357—612no—
261#357C2×D6112yes—
262#346—612no—
263C4×C16—612no—
264C61#3370711yescharacteristic, complemented, Sylow
265C2×#41—21no—
266C2×#41—1221no—
267C22×Q8—2441no—
268#135—22no—
269#135—22no—
270#135—1222no—
271#135—1222no—
272#135—2444no—
273C2×Q16—611no—
274C2×Q16—1221no—
275C2×Q16C22×D6111yescharacteristic
276C2×SD16—21no—
277C2×SD16—1221no—
278C2×SD16—1221no—
279C2×SD16—1221no—
280C2×D8—611no—
281C2×D8—1221no—
282C2×D8C22×D6111yescharacteristic
283C22×C8—611no—
284C22×C8C22×D6111yescharacteristic
285C4⋊Q8—1222no—
286C4⋊Q8—1222no—
287C4⋊Q8—1221no—
288C4⋊Q8—2441no—
289C4⋊Q8—2441no—
290#127—1221no—
291#124—1221no—
292#124—1221no—
293#121—2441no—
294#121—2441no—
295C4×D4—2441no—
296C4×D4—2441no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—1221no—
299C2×C4⋊C4—2441no—
300C2×C42—611no—
301M32—22no—
302M32—1222no—
303C2×C16—612no—
304C2×C16C22×D6112yes—
305#107—21no—
306#107—612no—
307#107—1222no—
308#107—1222no—
309#107—1221no—
310#107C22×D6112yes—
311C4×C8—611no—
312C4×C8—611no—
313C4×C8—611no—
314C4×C8—611no—
315#41—42no—
316#41—42no—
317#41—2442no—
318#41—2442no—
319C2×Q8—21no—
320C2×Q8—1221no—
321C2×Q8—4884no—
322C2×Q8—2442no—
323C2×Q8—2442no—
324C2×D4—21no—
325C2×D4—41no—
326C2×D4—1221no—
327C2×D4—2441no—
328C2×D4—2441no—
329C2×D4—2441no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—611no—
333C22×C4—611no—
334C22×C4—2441no—
335C22×C4—2441no—
336C22×C4D4×D6111yescharacteristic
337Q16—21no—
338Q16—21no—
339Q16—1221no—
340Q16—1221no—
341Q16—2442no—
342SD16—42no—
343SD16—2442no—
344SD16—2442no—
345SD16—2442no—
346D8—21no—
347D8—21no—
348D8—1221no—
349D8—1221no—
350D8—2442no—
351C2×C8—611no—
352C2×C8—611no—
353C2×C8—614no—
354C2×C8#1175822212yes—
355C2×C8D4×D6111yescharacteristic
356C2×C8#1175822412yes—
357C2×C8C23×D6111yescharacteristic
358C4⋊C4—22no—
359C4⋊C4—41no—
360C4⋊C4—1222no—
361C4⋊C4—2442no—
362C4⋊C4—2442no—
363C4⋊C4—2441no—
364C4⋊C4—2441no—
365C4⋊C4—2441no—
366C22⋊C4—2441no—
367C22⋊C4—2441no—
368C22⋊C4—2441no—
369C22⋊C4—2441no—
370C42—611no—
371C42—611no—
372C42—611no—
373C42—611no—
374C16—22no—
375C16—1222no—
376C23—41no—
377C23—2441no—
378C23D8×D6111yescharacteristic
379Q8—41no—
380Q8—41no—
381Q8—4884no—
382Q8—4884no—
383Q8—2441no—
384Q8—2441no—
385D4—41no—
386D4—41no—
387D4—82no—
388D4—4882no—
389D4—4882no—
390D4—4882no—
391D4—2441no—
392D4—2441no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—611no—
398C2×C4—611no—
399C2×C4—611no—
400C2×C4—611no—
401C2×C4—1222no—
402C2×C4—1222no—
403C2×C4—4882no—
404C2×C4—4882no—
405C2×C4—2441no—
406C2×C4—2442no—
407C2×C4#1451976312yes—
408C2×C4D8×D6111yescharacteristic
409C2×C4SD16×D6112yes—
410C2×C4C2×D4×D6111yescharacteristic
411C8—612no—
412C8—612no—
413C8#1451976312yes—
414C8C2×#1175822411yescharacteristic
415C8C2×#1175822211yescharacteristic
416C22—21no—
417C22—21no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—4881no—
422C22—4881no—
423C22—4881no—
424C22#1684010712yes—
425C22C2×D8×D6111yescharacteristic
426C4—81no—
427C4—82no—
428C4—612no—
429C4—612no—
430C4—1221no—
431C4—1221no—
432C4—4881no—
433C4—4882no—
434C4#1684010712yes—
435C4C2×#1451976311yescharacteristic
436C4C2×SD16×D6111yescharacteristic
437C2—22no—
438C2—161no—
439C2—9761no—
440C2C2×#1684010711yescharacteristic
441C2#509×D6111yescharacteristic
442C2#510×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2139C2
socleC22×C61C22×D8
prime radical#21456243C2
hypercenter#2139D61
hypocenterC61#33707
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
1. C2×#16840166C2semidirect
3. #21448057C2semidirect
4. #21448054C2semidirect
5. #21448045C2semidirect
6. #21448041C2semidirect
7. #21447532C2semidirect
8. #21447531C2semidirect
9. #21447529C2semidirect
10. M32×Dic61C2semidirect
11. C61×#2139C2semidirect
225. #2139D61semidirect
264. C61#33707semidirect

Maximal subgroups

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

subgroupindex
C2×#168401662
#214562432
#214480572
#214480542
#214480452
#214480412
#214475322
#214475312
#214475292
M32×Dic612
C61×#21392
#3370761