Finite Group Databasebeta

← #29836933

Subgroups — #29836933

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458246C211yesmaximal, characteristic, complemented
2#21457535C211yesmaximal, characteristic
3#21448045C211yesmaximal, characteristic, complemented
4#21448028C211yesmaximal, characteristic, complemented
5#21448025C211yesmaximal, characteristic, complemented
6#21447954C211yesmaximal, characteristic, complemented
7#21447908C211yesmaximal, characteristic, complemented
8#21447907C211yesmaximal, characteristic, complemented
9#21447867C211yesmaximal, characteristic, complemented
10#21447756C211yesmaximal, characteristic, complemented
11SD32×Dic61C211yesmaximal, characteristic, complemented
12#21447636C211yesmaximal, characteristic, complemented
13#21447526C211yesmaximal, characteristic, complemented
14#21447523C211yesmaximal, characteristic, complemented
15C61×#2189C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17#16840483—21no—
18#16840411—21no—
19C2×#14519723C2211yescharacteristic
20C22×#11758185C2211yescharacteristic
21C2×C16×D61C2211yescharacteristic
22#16840166C2211yescharacteristic
23#16840140C2211yescharacteristic
24#16840101—21no—
25#16840100—21no—
26#16840045—21no—
27#16840003C2211yescharacteristic
28#16840000C2211yescharacteristic
29#16840000C2211yescharacteristic
30#16839965—21no—
31#16839924—21no—
32#16839921—21no—
33Q16×Dic61C2211yescharacteristic
34D8×Dic61C2211yescharacteristic
35C8×C22⋊D61C2211yescharacteristic
36#16839800C2211yescharacteristic
37#16839799C2211yescharacteristic
38#16839590C2211yescharacteristic
39#16839588C2211yescharacteristic
40#16839587C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839566C2211yescharacteristic
47#16839565C2211yescharacteristic
48#16839563C2211yescharacteristic
49#16839563C2211yescharacteristic
50#16839541C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×#135×C61C2211yescharacteristic
53C2×Q32×C61C2211yescharacteristic
54C2×SD32×C61C2211yescharacteristic
55C61×#467C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic
59C2×#11758224—21no—
60#14519796—22no—
61#14519796—41no—
62C2×Dic244C2311yescharacteristic
63C2×#11758202—21no—
64C2×#11758200—21no—
65C2×C8×D61C2311yescharacteristic
66#14519776—21no—
67#14519775—21no—
68#14519772—41no—
69#14519770—41no—
70#14519763—41no—
71#14519760—41no—
72#14519758—41no—
73#14519756—21no—
74#14519753—21no—
75#14519753—21no—
76#14519749—41no—
77Q8×Dic61—21no—
78#14519742—41no—
79D4×Dic61—21no—
80C4×C22⋊D61D411yescharacteristic
81#14519736—21no—
82#14519735—21no—
83C2×#11758196—41no—
84#14519723—41no—
85C2×#11758185—41no—
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88C16×D61—41no—
89#14519711C2311yescharacteristic
90#14519711C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519710—21no—
93#14519709—21no—
94#14519708—21no—
95#14519706D411yescharacteristic
96#14519702—21no—
97#14519701—21no—
98#14519701—21no—
99#14519701—21no—
100#14519698—21no—
101C22⋊#92418D411yescharacteristic
102#14519689D411yescharacteristic
103C8×Dic61C2311yescharacteristic
104C2×#41×C61—21no—
105#135×C61—22no—
106#135×C61—41no—
107C2×Q16×C61C2311yescharacteristic
108C2×Q16×C61C2311yescharacteristic
109C2×SD16×C61—21no—
110C2×D8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22⋊Q8×C61—21no—
113Q32×C61—41no—
114SD32×C61—22no—
115C2×C16×C61C2311yescharacteristic
116C2×C16×C61C2311yescharacteristic
117#107×C61C2311yescharacteristic
118#103×C61—21no—
119#41×C61—42no—
120#41×C61—81no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—41no—
125C22×C4×C61—41no—
126C22×C4×C61C2×D411yescharacteristic
127C2×C22⋊D61—41no—
128C2×C22⋊D61D811yescharacteristic
129C22×Dic61—41no—
130C22×Dic61—41no—
131#11758224—42no—
132#11758224—81no—
133C2×C4×D61C2×D411yescharacteristic
134C2×Dic122—21no—
135C2×Dic122—41no—
136Q16×C61—21no—
137Q16×C61—21no—
138Q16×C61—21no—
139Q16×C61—21no—
140SD16×C61—42no—
141D8×C61—21no—
142D8×C61#4112yes—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2411yescharacteristic
147C4⋊C4×C61—21no—
148C4⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C22⋊Dic61—41no—
151C22⋊Dic61—41no—
152C22⋊Dic61—41no—
153C22⋊Dic61D811yescharacteristic
154#11758202—81no—
155#11758200—42no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159#11758198D811yescharacteristic
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—41no—
164#11758196—41no—
165#11758196—41no—
166#11758196—41no—
167#11758196—41no—
168#11758196—41no—
169#11758196—41no—
170#11758196D811yescharacteristic
171C4×Dic61—41no—
172C4×Dic61—41no—
173C4×Dic61—41no—
174C4×Dic61C2×D411yescharacteristic
175C2×#92418C2×D411yescharacteristic
176C2×#92418C2×D411yescharacteristic
177Dic244—21no—
178Dic244—21no—
179C8×D61—21no—
180C8×D61—21no—
181C16×C61—21no—
182C16×C61—21no—
183#11758185—21no—
184#11758185—21no—
185C23×C61—41no—
186C23×C61C2×D811yescharacteristic
187C22×D61C2×D811yescharacteristic
188Q8×C61—41no—
189Q8×C61—41no—
190Q8×C61—41no—
191Q8×C61—41no—
192D4×C61—22no—
193D4×C61—41no—
194D4×C61—82no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—82no—
201C2×C4×C61C2×D811yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C22⋊D61—22no—
204C22⋊D61—82no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—82no—
210C2×Dic61—82no—
211C2×Dic61C2×D811yescharacteristic
212C2×Dic61C2×D811yescharacteristic
213C2×Dic61C2×D811yescharacteristic
214C4×D61—21no—
215C4×D61—21no—
216Dic122—41no—
217Dic122—41no—
218Dic122—82no—
219C8×C61—21no—
220C8×C61C2×#4111yescharacteristic
221C8×C61H(5,𝔽2)11yescharacteristic
222#92418—21no—
223#92418—21no—
224#33869—611nomaximal, Sylow
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—42no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61C22×D811yescharacteristic
232C2×D61—21no—
233C2×D61—21no—
234C2×D61—21no—
235C4×C61—21no—
236C4×C61—81no—
237C4×C61—81no—
238C4×C61—81no—
239C4×C61#54611yescharacteristic
240C4×C61#57711yescharacteristic
241Dic61—21no—
242Dic61—21no—
243Dic61—81no—
244Dic61—81no—
245Dic61—81no—
246Dic61—81no—
247Dic61#50812yes—
248#3342—611no—
249#3283—611no—
250#2241—611no—
251#2237—611no—
252#2237—611no—
253#2224—611no—
254#2216—611no—
255#2216—611no—
256#2209—611no—
257#2189—611no—
258#2189D6111yescharacteristic, complemented
259C4×SD32—611no—
260D4×C16—611no—
261#2138—611no—
262#2138—611no—
263C2×C61—41no—
264C2×C61—82no—
265C2×C61#329411yescharacteristic
266C2×C61C2×#50811yescharacteristic
267C2×C61#341711yescharacteristic
268D61—41no—
269C2×#135—611no—
270C2×#135C2×D6111yescharacteristic
271#556—1221no—
272#547—1221no—
273C2×Q32—611no—
274C2×Q32C2×D6111yescharacteristic
275C2×SD32C2×D6111yescharacteristic
276C22×C16—611no—
277C22×C16—611no—
278#500—611no—
279#494—611no—
280#485—1221no—
281#485—1221no—
282#475—1221no—
283#467—611no—
284#467—611no—
285#467—611no—
286#467C2×D6111yescharacteristic
287#461—1221no—
288#452—1221no—
289#452—1221no—
290C4×Q16—611no—
291C4×D8—611no—
292C8×D4—611no—
293#427—611no—
294#427—611no—
295#367—611no—
296#366—611no—
297#366—611no—
298C4⋊C16—611no—
299#358—611no—
300#358—611no—
301#358—611no—
302#358—611no—
303#358—611no—
304#358—611no—
305#358C2×D6111yescharacteristic
306#357—611no—
307#357—611no—
308#357C2×D6111yescharacteristic
309C22⋊C16—611no—
310C22⋊C16C2×D6111yescharacteristic
311C4×C16—611no—
312C61#3386911yescharacteristic, complemented, Sylow
313C2×#41—21no—
314C2×#41—1221no—
315#135—22no—
316#135—41no—
317#135—1222no—
318#135—2441no—
319C2×Q16—611no—
320C2×Q16—1221no—
321C2×Q16C22×D6111yescharacteristic
322C2×Q16C22×D6111yescharacteristic
323C2×SD16—21no—
324C2×SD16—1221no—
325C2×D8C22×D6111yescharacteristic
326C22×C8—611no—
327C22×C8C22×D6111yescharacteristic
328C4⋊Q8—1221no—
329C4⋊Q8—1221no—
330#126—2441no—
331#126—2441no—
332#124—2441no—
333#123—2441no—
334#123—2441no—
335C22⋊Q8—21no—
336C22⋊Q8—1221no—
337C22⋊Q8—1221no—
338C22⋊Q8—1221no—
339#121—2441no—
340C4×Q8—1221no—
341C4×Q8—2441no—
342C4×D4—611no—
343C4×D4—1221no—
344#117—1221no—
345#117—1221no—
346C2×C4⋊C4—2441no—
347Q32—41no—
348Q32—2441no—
349SD32—22no—
350C2×C16—611no—
351C2×C16—611no—
352C2×C16—2441no—
353C2×C16—2441no—
354C2×C16C22×D6111yescharacteristic
355C2×C16C22×D6111yescharacteristic
356#107—611no—
357#107—611no—
358#107—611no—
359#107—1221no—
360#107C22×D6111yescharacteristic
361#106—1221no—
362#106—1221no—
363C4⋊C8—611no—
364#103—21no—
365#103—1221no—
366#103—1221no—
367#103—1221no—
368#103—1221no—
369#102—1221no—
370C22⋊C8—611no—
371C22⋊C8—611no—
372C4×C8—611no—
373#41—42no—
374#41—81no—
375#41—4881no—
376#41—2442no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—1221no—
380C2×Q8—2441no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—611no—
384C2×D4—2441no—
385C22×C4—41no—
386C22×C4—611no—
387C22×C4—2441no—
388C22×C4—2441no—
389C22×C4D4×D6111yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—21no—
393Q16—21no—
394Q16—1221no—
395Q16—1221no—
396Q16—4881no—
397SD16—42no—
398SD16—2442no—
399D8—21no—
400D8#1175822412yes—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—611no—
404C2×C8—611no—
405C2×C8—1221no—
406C2×C8—1221no—
407C2×C8D4×D6111yescharacteristic
408C2×C8C23×D6111yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—611no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C4⋊C4—1221no—
415C4⋊C4—2441no—
416C4⋊C4—2441no—
417C4⋊C4—2441no—
418C4⋊C4—2441no—
419C4⋊C4—2441no—
420C4⋊C4—2441no—
421C4⋊C4—2441no—
422C22⋊C4—41no—
423C22⋊C4—611no—
424C22⋊C4—611no—
425C22⋊C4—2441no—
426C22⋊C4—2441no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C22⋊C4—2441no—
430C22⋊C4—2441no—
431C42—611no—
432C42—2441no—
433C42—2441no—
434C42—2441no—
435C16—21no—
436C16—21no—
437C16—1221no—
438C16—1221no—
439C23—41no—
440C23—611no—
441C23D8×D6111yescharacteristic
442Q8—41no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—4882no—
447Q8—2441no—
448Q8—2441no—
449D4—22no—
450D4—41no—
451D4—82no—
452D4—1222no—
453D4—4882no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—82no—
460C2×C4—611no—
461C2×C4—611no—
462C2×C4—611no—
463C2×C4—1221no—
464C2×C4—1221no—
465C2×C4—4882no—
466C2×C4—4882no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4D8×D6111yescharacteristic
472C2×C4C2×D4×D6111yescharacteristic
473C8—21no—
474C8—1221no—
475C8—1221no—
476C8C2×#1175822411yescharacteristic
477C8#1451982211yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—42no—
482C22—81no—
483C22—81no—
484C22—1221no—
485C22—1221no—
486C22—1221no—
487C22C2×D8×D6111yescharacteristic
488C4—21no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—612no—
493C4—1221no—
494C4—1221no—
495C4—4881no—
496C4—4881no—
497C4—4881no—
498C4—4881no—
499C4#1684039911yescharacteristic
500C4#1684057211yescharacteristic
501C2—41no—
502C2—82no—
503C2—2441no—
504C2#2145762511yescharacteristic
505C2#508×D6111yescharacteristic
506C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2189C2
socleC22×C61C22×D8
prime radical#21457535C2
hypercenter#2189D61
hypocenterC61#33869
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2189

Complements

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

NH ≅ G/Ntype
1. #21458246C2semidirect
3. #21448045C2semidirect
4. #21448028C2semidirect
5. #21448025C2semidirect
6. #21447954C2semidirect
7. #21447908C2semidirect
8. #21447907C2semidirect
9. #21447867C2semidirect
10. #21447756C2semidirect
11. SD32×Dic61C2semidirect
12. #21447636C2semidirect
13. #21447526C2semidirect
14. #21447523C2semidirect
15. C61×#2189C2semidirect
258. #2189D61semidirect
312. C61#33869semidirect

Maximal subgroups

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

subgroupindex
#214582462
#214575352
#214480452
#214480282
#214480252
#214479542
#214479082
#214479072
#214478672
#214477562
SD32×Dic612
#214476362
#214475262
#214475232
C61×#21892
#3386961