Finite Group Databasebeta

← #29725101

Subgroups — #29725101

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#21455539C211yesmaximal, characteristic
2C4×#14519799C211yesmaximal, characteristic, complemented
3#21447275C212yesmaximal, complemented
4#21447123C212yesmaximal, complemented
5#21446717C211yesmaximal, characteristic, complemented
6#21446567C211yesmaximal, characteristic, complemented
7#21446323C212yesmaximal, complemented
8#21445943C211yesmaximal, characteristic, complemented
9#21445922C212yesmaximal, complemented
10#21445865C211yesmaximal, characteristic, complemented
11C61×#1764C211yesmaximal, characteristic, complemented
12#16840560C2212yes—
13C2×#14519799C2211yescharacteristic
14C4×#11758222C2212yes—
15#16840037—21no—
16#16840032—21no—
17#16840025—22no—
18C4×Dic244—21no—
19C4×#11758190—22no—
20C4×D488—21no—
21C2×#14519713C2211yescharacteristic
22#16839800C2211yescharacteristic
23#16839800C2211yescharacteristic
24#16839788C2212yes—
25#16839778C2212yes—
26#16839759C2212yes—
27#16839744C2212yes—
28#16839736C2212yes—
29#16839735C2212yes—
30#16839533C2212yes—
31#16839522C2212yes—
32#16839519C2212yes—
33#16839518C2212yes—
34C2×#131×C61C2211yescharacteristic
35C2×#104×C61C2211yescharacteristic
36C61×#416C2211yescharacteristic
37C61×#405C2211yescharacteristic
38C2×C4×C8×C61C2211yescharacteristic
39C61×#341C2211yescharacteristic
40C61×#339C2211yescharacteristic
41C2×#11758222C2312yes—
42#14519808—22no—
43#14519807—22no—
44#14519803—22no—
45#14519802—22no—
46#14519799—21no—
47#14519799—21no—
48#14519799—21no—
49#14519799—21no—
50#14519797—42no—
51C2×Dic244D411yescharacteristic
52C2×#11758190D412yes—
53C2×D488D411yescharacteristic
54C2×#11758194C2312yes—
55C4×Dic122—22no—
56C4×D244—22no—
57C4×C22⋊D61—42no—
58#14519736C2312yes—
59#14519736C2312yes—
60#14519734—22no—
61C42×D61—22no—
62#14519713—21no—
63#14519713—21no—
64#14519711—21no—
65#14519711—21no—
66#14519709—21no—
67#14519709—21no—
68#14519706—42no—
69#14519703—42no—
70#14519702—22no—
71#14519699—22no—
72#14519695—22no—
73#14519693—22no—
74#14519692—42no—
75#14519691—42no—
76C2×#41×C61C2311yescharacteristic
77#131×C61—21no—
78#131×C61—21no—
79#131×C61—21no—
80#131×C61—21no—
81C2×M16×C61—21no—
82C2×M16×C61C2311yescharacteristic
83C2×M16×C61C2311yescharacteristic
84C22×C8×C61—21no—
85C22×C8×C61C2311yescharacteristic
86C22×C8×C61C2311yescharacteristic
87#117×C61C2311yescharacteristic
88C2×C42×C61C2311yescharacteristic
89#104×C61—21no—
90#104×C61—21no—
91#103×C61—21no—
92#102×C61—21no—
93#97×C61—21no—
94C4×C8×C61—21no—
95C4×C8×C61—21no—
96C4×C8×C61—21no—
97#41×C61—21no—
98#41×C61—21no—
99#41×C61—21no—
100#41×C61—21no—
101C2×Q8×C61#4111yescharacteristic
102C2×D4×C61—21no—
103C2×D4×C61#4111yescharacteristic
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61C2411yescharacteristic
107C2×C22⋊D61—22no—
108#11758222—22no—
109#11758222—22no—
110#11758222—22no—
111#11758222—22no—
112C2×D244C2×D412yes—
113C2×C4×D61—22no—
114C2×C4×D61—42no—
115C2×Dic122C2×D412yes—
116M16×C61—21no—
117M16×C61—21no—
118M16×C61—21no—
119M16×C61—21no—
120M16×C61—41no—
121M16×C61—41no—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—21no—
126C2×C8×C61—21no—
127C2×C8×C61—41no—
128C2×C8×C61—41no—
129C2×C8×C61C2×D411yescharacteristic
130C2×C8×C61C2×D411yescharacteristic
131C2×C8×C61#4111yescharacteristic
132C2×C8×C61#4111yescharacteristic
133C2×C8×C61#4111yescharacteristic
134C2×C8×C61#4111yescharacteristic
135C4⋊C4×C61—21no—
136C22⋊C4×C61—21no—
137C42×C61—21no—
138C42×C61—21no—
139C42×C61#4111yescharacteristic
140C42×C61#4111yescharacteristic
141C22⋊Dic61—22no—
142C22⋊Dic61—22no—
143#11758202—42no—
144#11758201—42no—
145#11758200—42no—
146#11758199—42no—
147#11758198—42no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150#11758196—42no—
151C4×Dic61—22no—
152C4×Dic61—22no—
153C4×Dic61—42no—
154#11758194—22no—
155#11758194—22no—
156C2×#92418—22no—
157Dic244—21no—
158Dic244—21no—
159D488—21no—
160D488—21no—
161#11758190—22no—
162#11758190—22no—
163C23×C61—21no—
164C23×C61H(5,𝔽2)11yescharacteristic
165C22×D61—22no—
166Q8×C61—21no—
167Q8×C61—21no—
168D4×C61—21no—
169D4×C61—21no—
170D4×C61—41no—
171D4×C61—41no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181C2×C4×C61C2×#4111yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C2×C4×C61H(5,𝔽2)11yescharacteristic
185C22⋊D61—42no—
186C22⋊D61—42no—
187C2×Dic61—22no—
188C2×Dic61—42no—
189C2×Dic61—42no—
190D244—22no—
191D244—22no—
192C4×D61—42no—
193C4×D61—42no—
194C4×D61—84no—
195Dic122—22no—
196Dic122—22no—
197C8×C61—41no—
198C8×C61—41no—
199C8×C61—41no—
200C8×C61#12311yescharacteristic
201C8×C61#12311yescharacteristic
202C8×C61#12311yescharacteristic
203C8×C61#12311yescharacteristic
204#92418—42no—
205#24985—611nomaximal, Sylow
206C22×C61—41no—
207C22×C61—41no—
208C22×C61—81no—
209C22×C61#54011yescharacteristic
210C22×C61#54611yescharacteristic
211C22×C61#55311yescharacteristic
212C2×D61—42no—
213C2×D61—82no—
214C4×C61—22no—
215C4×C61—41no—
216C4×C61—41no—
217C4×C61—81no—
218C4×C61C2×#12311yescharacteristic
219C4×C61#52911yescharacteristic
220C4×C61#54611yescharacteristic
221C4×C61#55511yescharacteristic
222Dic61—42no—
223Dic61—82no—
224Dic61—82no—
225#3067—611no—
226C4×#135—611no—
227#2080—612no—
228#2053—612no—
229#1976—611no—
230#1945—611no—
231#1906—612no—
232#1833—611no—
233#1827—612no—
234#1812—611no—
235#1764D6111yescharacteristic, complemented
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—21no—
239C2×C61—81no—
240C2×C61#242011yescharacteristic
241D61—82no—
242#575—612no—
243C2×#135—611no—
244C2×#131C2×D6111yescharacteristic
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#427—611no—
255#424—612no—
256#421—612no—
257C2×#104C2×D6111yescharacteristic
258#416—612no—
259#416C2×D6111yescharacteristic
260#413—612no—
261#410—612no—
262#410—612no—
263#405C2×D6111yescharacteristic
264C2×C4×C8C2×D6111yescharacteristic
265#344—612no—
266#341C2×D6111yescharacteristic
267#339—612no—
268#339C2×D6111yescharacteristic
269H(3,ℤ/4)—612no—
270H(3,ℤ/4)—612no—
271C61#2498511yescharacteristic, complemented, Sylow
272C2×#41—612no—
273C2×#41C22×D6111yescharacteristic
274#137—1222no—
275#137—1222no—
276AGL(1,ℤ/8)—1222no—
277AGL(1,ℤ/8)—1222no—
278#135—1221no—
279#135—1221no—
280#135—1221no—
281#135—1221no—
282#135—2442no—
283C2×Q16—611no—
284C2×SD16—612no—
285C2×D8—611no—
286#131—21no—
287#131—21no—
288#131—21no—
289#131—21no—
290C2×M16—21no—
291C2×M16—612no—
292C2×M16C22×D6111yescharacteristic
293C2×M16C22×D6111yescharacteristic
294C22×C8—21no—
295C22×C8C22×D6111yescharacteristic
296C22×C8C22×D6111yescharacteristic
297C4×Q8—1222no—
298C4×D4—1222no—
299C4×D4—2442no—
300#117—612no—
301#117—612no—
302#117—1222no—
303#117C22×D6111yescharacteristic
304C2×C42—1222no—
305C2×C42C22×D6111yescharacteristic
306#108—1221no—
307#108—1221no—
308#107—1221no—
309#107—1221no—
310#106—1221no—
311#106—1221no—
312C4⋊C8—2442no—
313#104—21no—
314#104—21no—
315#104—2442no—
316#103—21no—
317#103—1222no—
318#102—21no—
319#102—1222no—
320#101—1222no—
321#100—1222no—
322#99—2442no—
323#99—2442no—
324#97—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331#41—21no—
332#41—1222no—
333#41—1222no—
334#41—1222no—
335#41—1222no—
336C2×Q8—612no—
337C2×Q8#1175822211yescharacteristic
338C2×D4—21no—
339C2×D4—612no—
340C2×D4—1222no—
341C2×D4#1175822211yescharacteristic
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—1222no—
345C22×C4—2442no—
346C22×C4C23×D6111yescharacteristic
347Q16—1221no—
348Q16—1221no—
349Q16—2442no—
350SD16—1222no—
351SD16—1222no—
352SD16—2442no—
353SD16—2442no—
354D8—1221no—
355D8—1221no—
356D8—2442no—
357M16—21no—
358M16—21no—
359M16—21no—
360M16—21no—
361M16—41no—
362M16—41no—
363M16—1222no—
364M16—1222no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—1222no—
373C2×C8#1175822211yescharacteristic
374C2×C8#1175822211yescharacteristic
375C2×C8D4×D6111yescharacteristic
376C2×C8D4×D6111yescharacteristic
377C2×C8#1175822411yescharacteristic
378C2×C8#1175822411yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—1222no—
381C4⋊C4—1222no—
382C4⋊C4—2442no—
383C22⋊C4—21no—
384C22⋊C4—1222no—
385C22⋊C4—1222no—
386C22⋊C4—2442no—
387C42—21no—
388C42—21no—
389C42—1222no—
390C42—1222no—
391C42—2442no—
392C42#1175822211yescharacteristic
393C42#1175822211yescharacteristic
394C23—21no—
395C23—1222no—
396C23#1451982311yescharacteristic
397Q8—21no—
398Q8—21no—
399Q8—1222no—
400Q8—1222no—
401D4—21no—
402D4—21no—
403D4—41no—
404D4—41no—
405D4—1222no—
406D4—1222no—
407D4—2442no—
408D4—2442no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—1222no—
417C2×C4—4884no—
418C2×C4—2442no—
419C2×C4—2442no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4C2×D4×D6111yescharacteristic
423C2×C4C2×#1175822411yescharacteristic
424C2×C4C2×#1175822211yescharacteristic
425C2×C4C2×#1175822211yescharacteristic
426C2×C4C2×#1175822211yescharacteristic
427C2×C4#1451982311yescharacteristic
428C8—41no—
429C8—41no—
430C8—41no—
431C8—2442no—
432C8#1451975911yescharacteristic
433C8#1451975911yescharacteristic
434C8#1451975911yescharacteristic
435C8#1451975911yescharacteristic
436C22—41no—
437C22—41no—
438C22—81no—
439C22—4882no—
440C22—2442no—
441C22#1684036711yescharacteristic
442C22#1684040811yescharacteristic
443C22#1684045811yescharacteristic
444C4—22no—
445C4—41no—
446C4—41no—
447C4—81no—
448C4—4882no—
449C4—4882no—
450C4—2442no—
451C4C2×#1451975911yescharacteristic
452C4#1684029611yescharacteristic
453C4#1684040811yescharacteristic
454C4#1684047711yescharacteristic
455C2—21no—
456C2—21no—
457C2—21no—
458C2—81no—
459C2—4882no—
460C2#2144931111yescharacteristic
subgroupG/H
centerC4#16840477
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1764C2
socleC2×C61#2420
prime radical#21455539C2
hypercenter#1764D61
hypocenterC61#24985
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1764

Complements

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

NH ≅ G/Ntype
2. C4×#14519799C2semidirect
3. #21447275C2semidirect
4. #21447123C2semidirect
5. #21446717C2semidirect
6. #21446567C2semidirect
7. #21446323C2semidirect
8. #21445943C2semidirect
9. #21445922C2semidirect
10. #21445865C2semidirect
11. C61×#1764C2semidirect
235. #1764D61semidirect
271. C61#24985semidirect

Maximal subgroups

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

subgroupindex
#214555392
C4×#145197992
#214472752
#214471232
#214467172
#214465672
#214463232
#214459432
#214459222
#214458652
C61×#17642
#2498561