Finite Group Databasebeta

← #29725099

Subgroups — #29725099

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455535C211yesmaximal, characteristic
2#135×Dic61C211yesmaximal, characteristic, complemented
3#21447269C212yesmaximal, complemented
4#21447124C212yesmaximal, complemented
5#21446568C211yesmaximal, characteristic, complemented
6#21446318C212yesmaximal, complemented
7#21445943C211yesmaximal, characteristic, complemented
8#21445920C212yesmaximal, complemented
9#21445862C211yesmaximal, characteristic, complemented
10#21445658C211yesmaximal, characteristic, complemented
11C61×#1976C211yesmaximal, characteristic, complemented
12#16840560C2212yes—
13C2×#14519783C2211yescharacteristic
14#41×Dic61C2212yes—
15#16840034—21no—
16#16840029—21no—
17#16840022—22no—
18Q16×Dic61—21no—
19SD16×Dic61—22no—
20D8×Dic61—21no—
21C2×#14519713C2211yescharacteristic
22#16839800C2211yescharacteristic
23#16839789C2212yes—
24C2×#14519703C2211yescharacteristic
25#16839758C2212yes—
26#16839756C2211yescharacteristic
27#16839746C2212yes—
28#16839735C2212yes—
29C2×C8×Dic61C2211yescharacteristic
30#16839533C2212yes—
31#16839526C2211yescharacteristic
32#16839521C2211yescharacteristic
33#16839521C2212yes—
34#16839518C2212yes—
35#16839518C2212yes—
36C2×#135×C61C2211yescharacteristic
37C61×#427C2211yescharacteristic
38C61×#421C2212yes—
39C61×#410C2212yes—
40C61×#405C2211yescharacteristic
41C2×#11758222C2311yescharacteristic
42#14519808—22no—
43#14519807—22no—
44#14519803—22no—
45#14519802—22no—
46#14519797—42no—
47#14519783—21no—
48#14519783—21no—
49#14519783—21no—
50#14519783—21no—
51C2×#11758194C2311yescharacteristic
52C2×#11758194C2312yes—
53C2×#11758189—21no—
54C22×#92418C2311yescharacteristic
55C2×C8×D61—21no—
56Q8×Dic61—22no—
57D4×Dic61—22no—
58D4×Dic61—42no—
59#14519736—22no—
60#14519736C2312yes—
61C2×C4×Dic61—22no—
62C2×C4×Dic61C2311yescharacteristic
63#14519713—21no—
64#14519713—21no—
65#14519711—21no—
66#14519709—21no—
67C4⋊#92418—42no—
68#14519703—21no—
69#14519703—21no—
70#14519701—22no—
71#14519700—21no—
72#14519698—22no—
73#14519697—21no—
74#14519696—22no—
75#14519694—22no—
76#14519691—42no—
77C8×Dic61—21no—
78C8×Dic61—21no—
79C2×#41×C61C2312yes—
80#135×C61—21no—
81#135×C61—21no—
82#135×C61—21no—
83#135×C61—21no—
84C2×Q16×C61D411yescharacteristic
85C2×SD16×C61D412yes—
86C2×D8×C61D411yescharacteristic
87C2×M16×C61C2311yescharacteristic
88C22×C8×C61C2311yescharacteristic
89#117×C61C2312yes—
90#117×C61C2311yescharacteristic
91#107×C61—21no—
92#106×C61—21no—
93#104×C61—42no—
94#99×C61—42no—
95#97×C61—21no—
96C4×C8×C61—21no—
97#41×C61—22no—
98#41×C61—22no—
99#41×C61—22no—
100#41×C61—22no—
101C2×Q8×C61C2×D412yes—
102C2×D4×C61—22no—
103C2×D4×C61C2×D412yes—
104C22×C4×C61—22no—
105C22×C4×C61C2411yescharacteristic
106C2×C22⋊D61—21no—
107C22×Dic61—21no—
108C22×Dic61—42no—
109#11758222—21no—
110#11758222—21no—
111#11758222—21no—
112#11758222—21no—
113C2×D244#4111yescharacteristic
114C2×C4×D61—21no—
115C2×Dic122#4111yescharacteristic
116Q16×C61—21no—
117Q16×C61—21no—
118SD16×C61—22no—
119SD16×C61—22no—
120D8×C61—21no—
121D8×C61—21no—
122M16×C61—21no—
123M16×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61#4111yescharacteristic
128C2×C8×C61#4111yescharacteristic
129C2×C8×C61#4111yescharacteristic
130C2×C8×C61#4111yescharacteristic
131C4⋊C4×C61—22no—
132C4⋊C4×C61—21no—
133C22⋊C4×C61—21no—
134C22⋊C4×C61—22no—
135C42×C61—21no—
136C42×C61—22no—
137C22⋊Dic61—22no—
138C22⋊Dic61—42no—
139#11758202—42no—
140#11758201—42no—
141#11758200—42no—
142#11758199—42no—
143C4⋊Dic61—22no—
144C4⋊Dic61—42no—
145C4×Dic61—21no—
146C4×Dic61—22no—
147C4×Dic61—42no—
148C4×Dic61#4111yescharacteristic
149C4×Dic61#4111yescharacteristic
150#11758194—22no—
151#11758194—21no—
152#11758194—21no—
153#11758194—22no—
154C2×#92418—21no—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—22no—
159#11758189—41no—
160#11758189—41no—
161C8×D61—41no—
162C8×D61—41no—
163C23×C61—22no—
164C23×C61H(5,𝔽2)11yescharacteristic
165C22×D61—21no—
166Q8×C61—22no—
167Q8×C61—22no—
168D4×C61—22no—
169D4×C61—22no—
170D4×C61—42no—
171D4×C61—42no—
172C2×C4×C61—22no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C2×#4111yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181C2×C4×C61C2×#4111yescharacteristic
182C2×C4×C61H(5,𝔽2)11yescharacteristic
183C22⋊D61—41no—
184C22⋊D61—41no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—42no—
189C2×Dic61—42no—
190C2×Dic61—84no—
191D244—21no—
192D244—21no—
193C4×D61—41no—
194C4×D61—41no—
195Dic122—21no—
196Dic122—21no—
197C8×C61—41no—
198C8×C61#12311yescharacteristic
199C8×C61#12311yescharacteristic
200C8×C61#12311yescharacteristic
201C8×C61#12311yescharacteristic
202#92418—41no—
203#92418—41no—
204#92418—42no—
205#24985—611nomaximal, Sylow
206C22×C61—41no—
207C22×C61—42no—
208C22×C61—82no—
209C22×C61#54011yescharacteristic
210C22×C61#54611yescharacteristic
211C22×C61#55311yescharacteristic
212C2×D61—41no—
213C2×D61—81no—
214C4×C61—42no—
215C4×C61—82no—
216C4×C61—81no—
217C4×C61C2×#12311yescharacteristic
218C4×C61#52911yescharacteristic
219C4×C61#54611yescharacteristic
220C4×C61#55511yescharacteristic
221Dic61—22no—
222Dic61—41no—
223Dic61—41no—
224Dic61—82no—
225#3067—611no—
226C4×#135—611no—
227#2080—612no—
228#2053—612no—
229#1976D6111yescharacteristic, complemented
230#1945—611no—
231#1906—612no—
232#1833—611no—
233#1827—612no—
234#1812—611no—
235#1764—611no—
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—21no—
239C2×C61—82no—
240C2×C61#242011yescharacteristic
241D61—81no—
242#575—612no—
243C2×#135C2×D6111yescharacteristic
244C2×#131—611no—
245C4×#41—612no—
246#473—1221no—
247#472—1221no—
248#471—1222no—
249C4×Q16—1221no—
250C4×SD16—1222no—
251C4×D8—1221no—
252C2×#108—611no—
253#427—611no—
254#427C2×D6111yescharacteristic
255#424—612no—
256#421C2×D6112yes—
257C2×#104—611no—
258#416—612no—
259#416—611no—
260#413—612no—
261#410—612no—
262#410C2×D6112yes—
263#405C2×D6111yescharacteristic
264C2×C4×C8—611no—
265#344—612no—
266#341—611no—
267#339—611no—
268#339—612no—
269H(3,ℤ/4)—612no—
270H(3,ℤ/4)—612no—
271C61#2498511yescharacteristic, complemented, Sylow
272C2×#41—611no—
273C2×#41C22×D6112yes—
274#137—1222no—
275#137—1222no—
276AGL(1,ℤ/8)—1222no—
277AGL(1,ℤ/8)—1222no—
278#135—21no—
279#135—21no—
280#135—21no—
281#135—21no—
282#135—2442no—
283C2×Q16D24411yescharacteristic
284C2×SD16D24412yes—
285C2×D8D24411yescharacteristic
286#131—1221no—
287#131—1221no—
288#131—1221no—
289#131—1221no—
290C2×M16—611no—
291C2×M16—612no—
292C2×M16—1221no—
293C2×M16C22×D6111yescharacteristic
294C22×C8—611no—
295C22×C8—1221no—
296C22×C8C22×D6111yescharacteristic
297C4×Q8—1222no—
298C4×D4—1222no—
299C4×D4—2442no—
300#117—612no—
301#117—1222no—
302#117C22×D6111yescharacteristic
303#117C22×D6112yes—
304C2×C42—611no—
305C2×C42—1222no—
306#108—1221no—
307#108—1221no—
308#107—21no—
309#107—1221no—
310#106—21no—
311#106—1221no—
312C4⋊C8—2442no—
313#104—42no—
314#104—1221no—
315#104—1221no—
316#103—1221no—
317#103—1222no—
318#102—1221no—
319#102—1222no—
320#101—1222no—
321#100—1222no—
322#99—42no—
323#99—2442no—
324#97—21no—
325C4×C8—21no—
326C4×C8—1221no—
327C4×C8—1221no—
328#41—22no—
329#41—22no—
330#41—22no—
331#41—22no—
332#41—1221no—
333#41—1221no—
334#41—1221no—
335#41—1221no—
336C2×Q8—611no—
337C2×Q8C2×D24412yes—
338C2×D4—22no—
339C2×D4—611no—
340C2×D4—1221no—
341C2×D4C2×D24412yes—
342C22×C4—22no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4—2442no—
346C22×C4C23×D6111yescharacteristic
347Q16—21no—
348Q16—21no—
349Q16—2442no—
350SD16—22no—
351SD16—22no—
352SD16—2442no—
353SD16—2442no—
354D8—21no—
355D8—21no—
356D8—2442no—
357M16—21no—
358M16—21no—
359M16—1221no—
360M16—1222no—
361M16—1222no—
362M16—1221no—
363M16—2441no—
364M16—2441no—
365C2×C8—21no—
366C2×C8—1221no—
367C2×C8—1221no—
368C2×C8—1221no—
369C2×C8—1221no—
370C2×C8—1222no—
371C2×C8—2441no—
372C2×C8—2441no—
373C2×C8C2×D24411yescharacteristic
374C2×C8C2×D24411yescharacteristic
375C2×C8#1175822411yescharacteristic
376C2×C8#1175822411yescharacteristic
377C2×C8#1175822411yescharacteristic
378C2×C8#1175822411yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—1222no—
382C4⋊C4—2442no—
383C22⋊C4—21no—
384C22⋊C4—22no—
385C22⋊C4—1222no—
386C22⋊C4—2442no—
387C42—21no—
388C42—22no—
389C42—611no—
390C42—611no—
391C42—1221no—
392C42—1222no—
393C42—2442no—
394C23—22no—
395C23—1221no—
396C23#1451982211yescharacteristic
397Q8—22no—
398Q8—22no—
399Q8—1221no—
400Q8—1221no—
401D4—22no—
402D4—22no—
403D4—42no—
404D4—42no—
405D4—1221no—
406D4—1221no—
407D4—2441no—
408D4—2441no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—1221no—
415C2×C4—1221no—
416C2×C4—1221no—
417C2×C4—4884no—
418C2×C4—2442no—
419C2×C4—2442no—
420C2×C4—2441no—
421C2×C4—2441no—
422C2×C4C22×D24411yescharacteristic
423C2×C4#41×D6111yescharacteristic
424C2×C4#41×D6111yescharacteristic
425C2×C4C2×#1175822411yescharacteristic
426C2×C4C2×#1175822411yescharacteristic
427C2×C4#1451982211yescharacteristic
428C8—41no—
429C8—2441no—
430C8—2441no—
431C8—2442no—
432C8#1451976111yescharacteristic
433C8#1451976111yescharacteristic
434C8#1451976111yescharacteristic
435C8#1451976111yescharacteristic
436C22—41no—
437C22—42no—
438C22—82no—
439C22—4881no—
440C22—2441no—
441C22#1684036611yescharacteristic
442C22#1684040711yescharacteristic
443C22#1684045911yescharacteristic
444C4—42no—
445C4—82no—
446C4—81no—
447C4—1222no—
448C4—4882no—
449C4—2441no—
450C4—2441no—
451C4C2×#1451976111yescharacteristic
452C4#1684029111yescharacteristic
453C4#1684040711yescharacteristic
454C4#1684047611yescharacteristic
455C2—21no—
456C2—21no—
457C2—21no—
458C2—82no—
459C2—4881no—
460C2#2144930911yescharacteristic
subgroupG/H
centerC4#16840476
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1976C2
socleC2×C61#2420
prime radical#21455535C2
hypercenter#1976D61
hypocenterC61#24985
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
2. #135×Dic61C2semidirect
3. #21447269C2semidirect
4. #21447124C2semidirect
5. #21446568C2semidirect
6. #21446318C2semidirect
7. #21445943C2semidirect
8. #21445920C2semidirect
9. #21445862C2semidirect
10. #21445658C2semidirect
11. C61×#1976C2semidirect
229. #1976D61semidirect
271. C61#24985semidirect

Maximal subgroups

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

subgroupindex
#214555352
#135×Dic612
#214472692
#214471242
#214465682
#214463182
#214459432
#214459202
#214458622
#214456582
C61×#19762
#2498561