Finite Group Databasebeta

← #29834966

Subgroups — #29834966

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456395C211yesmaximal, characteristic
2C4×Q16×D61C211yesmaximal, characteristic, complemented
3#21448006C211yesmaximal, characteristic, complemented
4#21448003C211yesmaximal, characteristic, complemented
5#21447960C211yesmaximal, characteristic, complemented
6#21447959C211yesmaximal, characteristic, complemented
7#21447858C211yesmaximal, characteristic, complemented
8#21447857C211yesmaximal, characteristic, complemented
9#21447737C211yesmaximal, characteristic, complemented
10#21447734C211yesmaximal, characteristic, complemented
11C4×#14519722C211yesmaximal, characteristic, complemented
12C4×#14519720C211yesmaximal, characteristic, complemented
13C4×#14519721C211yesmaximal, characteristic, complemented
14#21447616C211yesmaximal, characteristic, complemented
15C4×SD32×C61C211yesmaximal, characteristic, complemented
16C2×Q16×D61C2211yescharacteristic
17#16840309—21no—
18C4×Q8×D61—21no—
19C2×#14519722C2211yescharacteristic
20C2×#14519721C2211yescharacteristic
21C2×#14519720C2211yescharacteristic
22#16840166C2211yescharacteristic
23#16840137C2211yescharacteristic
24#16840107—21no—
25#16839999C2211yescharacteristic
26#16839997—21no—
27#16839995C2211yescharacteristic
28C4×Dic244C2211yescharacteristic
29Q16×Dic61C2211yescharacteristic
30#16839856C2211yescharacteristic
31C4×#11758202—21no—
32#16839854—21no—
33C4×D488C2211yescharacteristic
34C4×#11758199—21no—
35#107×D61C2211yescharacteristic
36#103×D61—21no—
37C4×C8×D61C2211yescharacteristic
38#16839590C2211yescharacteristic
39#16839589C2211yescharacteristic
40#16839589C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839565C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839562C2211yescharacteristic
49#16839541C2211yescharacteristic
50#16839541C2211yescharacteristic
51C4×#11758185C2211yescharacteristic
52C2×SD32×C61C2211yescharacteristic
53C4×Q16×C61C2211yescharacteristic
54C4×D8×C61C2211yescharacteristic
55C61×#367C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C4×C16×C61C2211yescharacteristic
59C2×#11758222—41no—
60C2×Q8×D61—21no—
61C2×Dic244C2311yescharacteristic
62C2×#11758202—21no—
63Q16×D61—22no—
64Q16×D61—22no—
65C2×D488C2311yescharacteristic
66C2×#11758199—21no—
67C2×C8×D61D411yescharacteristic
68C2×C8×D61C2311yescharacteristic
69#14519776—21no—
70#14519773—21no—
71#14519763—41no—
72#14519752—21no—
73#14519750—41no—
74#14519748—21no—
75Q8×Dic61—21no—
76C4×Dic122—21no—
77C4×Dic122—41no—
78#14519742—21no—
79#14519742—41no—
80C4×D244—21no—
81C4×C22⋊D61—41no—
82C4⋊C4×D61—21no—
83C4⋊C4×D61—41no—
84C42×D61—41no—
85C42×D61D411yescharacteristic
86#14519722—22no—
87#14519721—22no—
88#14519720—22no—
89C2×#11758185C2311yescharacteristic
90C2×#11758185C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519710—21no—
94#14519710—21no—
95#14519702—21no—
96#14519701—21no—
97#14519700—21no—
98#14519699—21no—
99#14519698—21no—
100#14519697—21no—
101C4×#92418D411yescharacteristic
102C8×Dic61D411yescharacteristic
103C8×Dic61C2311yescharacteristic
104C2×Q16×C61C2311yescharacteristic
105C2×D8×C61C2311yescharacteristic
106C4×Q8×C61—21no—
107C4×D4×C61—21no—
108SD32×C61—22no—
109C2×C16×C61C2311yescharacteristic
110C2×C16×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#103×C61—21no—
114#102×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C2×Q8×C61—21no—
117C2×D4×C61—21no—
118C22×C4×C61—41no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122Q8×D61—24no—
123Q8×D61—42no—
124#11758222—84no—
125C2×D244—21no—
126C2×D244—41no—
127C2×C4×D61—41no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×C4×D61D811yescharacteristic
131C2×C4×D61D811yescharacteristic
132C2×C4×D61C2×D411yescharacteristic
133C2×Dic122—21no—
134C2×Dic122—41no—
135C2×Dic122—41no—
136Q16×C61—21no—
137Q16×C61#4112yescomplemented
138D8×C61—21no—
139D8×C61—21no—
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2411yescharacteristic
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—41no—
145C22⋊C4×C61—41no—
146C42×C61—41no—
147C42×C61C2×D411yescharacteristic
148C22⋊Dic61—41no—
149#11758202—24no—
150#11758199—42no—
151#11758198—41no—
152#11758198—41no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—41no—
156#11758196—41no—
157#11758196—41no—
158#11758196—41no—
159#11758196—41no—
160C4×Dic61—41no—
161C4×Dic61—41no—
162C4×Dic61—41no—
163C4×Dic61D811yescharacteristic
164C4×Dic61D811yescharacteristic
165C4×Dic61C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167C2×#92418C2×D411yescharacteristic
168Dic244—21no—
169Dic244#4112yescomplemented
170D488—21no—
171D488—21no—
172C8×D61—21no—
173C8×D61—21no—
174C8×D61—41no—
175C16×C61—21no—
176C16×C61—21no—
177#11758185—21no—
178#11758185—21no—
179C23×C61—41no—
180C22×D61—41no—
181C22×D61C2×D811yescharacteristic
182Q8×C61—22no—
183Q8×C61—41no—
184D4×C61—41no—
185D4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—82no—
190C2×C4×C61C2×D811yescharacteristic
191C2×C4×C61C2×D811yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C22⋊D61—82no—
194C22⋊D61—82no—
195C22⋊D61—82no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61C2×D811yescharacteristic
200C2×Dic61C2×D811yescharacteristic
201C2×Dic61C2×D811yescharacteristic
202D244—41no—
203D244—41no—
204D244—82no—
205C4×D61—21no—
206C4×D61—21no—
207C4×D61—41no—
208C4×D61—44no—
209C4×D61—82no—
210C4×D61—82no—
211C4×D61SD3214yescomplemented
212Dic122—22no—
213Dic122—44no—
214Dic122—41no—
215Dic122—82no—
216C8×C61—21no—
217C8×C61C2×#4111yescharacteristic
218C8×C61C2×#4111yescharacteristic
219#92418—21no—
220#92418—21no—
221#33663—611nomaximal, Sylow
222C22×C61—81no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61C22×D811yescharacteristic
226C2×D61—21no—
227C2×D61—21no—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C2×D61C2×SD3212yes—
232C4×C61—21no—
233C4×C61—42no—
234C4×C61—81no—
235C4×C61—81no—
236C4×C61C2×SD3212yes—
237C4×C61#52511yescharacteristic
238C4×C61#57511yescharacteristic
239Dic61—21no—
240Dic61—21no—
241Dic61—42no—
242Dic61—81no—
243Dic61—81no—
244Dic61C2×SD3212yes—
245#3169—611no—
246C2×C4×Q16—611no—
247#2233—611no—
248#2233—611no—
249#2225—611no—
250#2225—611no—
251#2207—611no—
252#2207—611no—
253#2185—611no—
254#2185—611no—
255C4×SD32—611no—
256C4×SD32—611no—
257C4×SD32—611no—
258C4×SD32D6111yescharacteristic, complemented
259#2162—611no—
260C2×C61—161no—
261C2×C61#310011yescharacteristic
262C2×C61C22×SD3211yescharacteristic
263C2×C61#341411yescharacteristic
264D61—22no—
265D61—161no—
266C22×Q16—611no—
267#532—1221no—
268C2×C4×Q8—1221no—
269C2×SD32—611no—
270C2×SD32—611no—
271C2×SD32—611no—
272C2×SD32C2×D6111yescharacteristic
273#500—611no—
274#493—611no—
275#486—1221no—
276#466—611no—
277#466—611no—
278#466—1221no—
279C4×Q16—611no—
280C4×Q16—611no—
281C4×Q16—611no—
282C4×Q16—1221no—
283C4×Q16—1221no—
284C4×Q16C2×D6111yescharacteristic
285C4×D8—611no—
286C4×D8—1221no—
287C4×D8C2×D6111yescharacteristic
288C2×#107—611no—
289C2×#103—1221no—
290C2×C4×C8—611no—
291#367—611no—
292#367—611no—
293#367—611no—
294#367C2×D6111yescharacteristic
295C4⋊C16—611no—
296C4⋊C16—611no—
297#358—611no—
298#358—611no—
299#358—611no—
300#358C2×D6111yescharacteristic
301#357—611no—
302#357—611no—
303#357—611no—
304#357C2×D6111yescharacteristic
305C22⋊C16—611no—
306C22⋊C16—611no—
307C4×C16—611no—
308C4×C16C2×D6111yescharacteristic
309C61#3366311yescharacteristic, complemented, Sylow
310C2×#41—2441no—
311C22×Q8—1221no—
312C2×Q16—611no—
313C2×Q16—1221no—
314C2×Q16—1222no—
315C2×Q16—1222no—
316C2×Q16C22×D6111yescharacteristic
317C2×D8—611no—
318C2×D8—1221no—
319C2×D8C22×D6111yescharacteristic
320C22×C8—611no—
321C22×C8—611no—
322C4⋊Q8—1221no—
323#127—1221no—
324#124—2441no—
325#121—1221no—
326#121—1221no—
327#121—2441no—
328C4×Q8—21no—
329C4×Q8—1221no—
330C4×Q8—1221no—
331C4×Q8—1221no—
332C4×Q8—2441no—
333C4×Q8—2441no—
334C4×D4—21no—
335C4×D4—1221no—
336C4×D4—2441no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—2441no—
339C2×C42—611no—
340C2×C42—2441no—
341SD32—22no—
342SD32—1222no—
343SD32—1222no—
344SD32—1222no—
345C2×C16—611no—
346C2×C16—611no—
347C2×C16C22×D6111yescharacteristic
348C2×C16C22×D6111yescharacteristic
349#107—611no—
350#107—611no—
351#107—1221no—
352#107—1221no—
353#107C22×D6111yescharacteristic
354#107C22×D6111yescharacteristic
355#103—21no—
356#103—1221no—
357#103—1221no—
358#103—1221no—
359#102—21no—
360#102—1221no—
361#102—1221no—
362#102—1221no—
363C4×C8—611no—
364C4×C8—611no—
365C4×C8—611no—
366C4×C8C22×D6111yescharacteristic
367#41—4884no—
368C2×Q8—21no—
369C2×Q8—1224no—
370C2×Q8—1221no—
371C2×Q8—2441no—
372C2×Q8—2441no—
373C2×Q8—2442no—
374C2×D4—21no—
375C2×D4—1221no—
376C2×D4—2441no—
377C2×D4—2441no—
378C2×D4—2441no—
379C2×D4—2441no—
380C22×C4—41no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—2441no—
385C22×C4—2441no—
386C22×C4—2441no—
387Q16—21no—
388Q16—612no—
389Q16—1224no—
390Q16—1221no—
391Q16#1175822212yescomplemented
392D8—21no—
393D8—21no—
394D8—1221no—
395D8—1221no—
396D8—2442no—
397C2×C8—611no—
398C2×C8—611no—
399C2×C8—1221no—
400C2×C8—1221no—
401C2×C8—2441no—
402C2×C8D4×D6111yescharacteristic
403C2×C8C23×D6111yescharacteristic
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—41no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—2441no—
410C4⋊C4—2441no—
411C4⋊C4—2441no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C22⋊C4—41no—
415C22⋊C4—2441no—
416C22⋊C4—2441no—
417C22⋊C4—2441no—
418C42—41no—
419C42—611no—
420C42—611no—
421C42—611no—
422C42—2441no—
423C42—2441no—
424C42—2441no—
425C42D4×D6111yescharacteristic
426C16—21no—
427C16—21no—
428C16—1221no—
429C16—1221no—
430C23—41no—
431C23—611no—
432C23—2441no—
433Q8—22no—
434Q8—41no—
435Q8—1222no—
436Q8—4882no—
437Q8—2444no—
438Q8—2441no—
439D4—41no—
440D4—41no—
441D4—4882no—
442D4—4882no—
443D4—4882no—
444D4—4882no—
445D4—2441no—
446D4—2441no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—82no—
451C2×C4—611no—
452C2×C4—614no—
453C2×C4—611no—
454C2×C4—611no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—4882no—
458C2×C4—4882no—
459C2×C4—2441no—
460C2×C4—2444no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4D8×D6111yescharacteristic
465C2×C4D8×D6111yescharacteristic
466C2×C4C2×D4×D6111yescharacteristic
467C8—21no—
468C8—1221no—
469C8—1221no—
470C8#41×D6111yescharacteristic
471C8C2×#1175822211yescharacteristic
472C22—81no—
473C22—81no—
474C22—81no—
475C22—612no—
476C22—1221no—
477C22—1221no—
478C22—4881no—
479C22—4881no—
480C22—4881no—
481C22C2×D8×D6111yescharacteristic
482C4—21no—
483C4—42no—
484C4—81no—
485C4—81no—
486C4—612no—
487C4—1221no—
488C4—1221no—
489C4—4881no—
490C4—4881no—
491C4—2442no—
492C4SD32×D6112yes—
493C4#1684027111yescharacteristic
494C4#1684055911yescharacteristic
495C2—161no—
496C2—1222no—
497C2—9761no—
498C2#2145581711yescharacteristic
499C2C2×SD32×D6111yescharacteristic
500C2#2145895211yescharacteristic
subgroupG/H
centerC2×C4D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC4×SD32×C61C2
socleC22×C61C22×D8
prime radical#21456395C2
hypercenterC4×SD32D61
hypocenterC61#33663
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ C4×SD32

Complements

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

NH ≅ G/Ntype
2. C4×Q16×D61C2semidirect
3. #21448006C2semidirect
4. #21448003C2semidirect
5. #21447960C2semidirect
6. #21447959C2semidirect
7. #21447858C2semidirect
8. #21447857C2semidirect
9. #21447737C2semidirect
10. #21447734C2semidirect
11. C4×#14519722C2semidirect
12. C4×#14519720C2semidirect
13. C4×#14519721C2semidirect
14. #21447616C2semidirect
15. C4×SD32×C61C2semidirect
137. Q16×C61#41semidirect
169. Dic244#41semidirect
211. C4×D61SD32semidirect
258. C4×SD32D61semidirect
309. C61#33663semidirect
391. Q16#11758222semidirect

Maximal subgroups

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

subgroupindex
#214563952
C4×Q16×D612
#214480062
#214480032
#214479602
#214479592
#214478582
#214478572
#214477372
#214477342
C4×#145197222
C4×#145197202
C4×#145197212
#214476162
C4×SD32×C612
#3366361