Finite Group Databasebeta

← #29725112

Subgroups — #29725112

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455539C211yesmaximal, characteristic
2#21454621C211yesmaximal, characteristic, complemented
3#21447275C212yesmaximal, complemented
4#21447123C212yesmaximal, complemented
5#21446717C211yesmaximal, characteristic, complemented
6#21446572C211yesmaximal, characteristic, complemented
7#21446323C212yesmaximal, complemented
8#21445943C211yesmaximal, characteristic, complemented
9#21445922C212yesmaximal, complemented
10#21445865C211yesmaximal, characteristic, complemented
11C61×#1765C211yesmaximal, characteristic, complemented
12#16840560C2212yes—
13C2×#14519799C2211yescharacteristic
14C4×#11758222C2212yes—
15#16840037—21no—
16#16840032—21no—
17#16840025—22no—
18#16839875—21no—
19#16839870—21no—
20#16839864—22no—
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×#97×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—22no—
90#103×C61—21no—
91#102×C61—21no—
92#97×C61—21no—
93#97×C61—21no—
94#97×C61—21no—
95C4×C8×C61—21no—
96#41×C61—21no—
97#41×C61—21no—
98#41×C61—21no—
99#41×C61—21no—
100C2×Q8×C61#4111yescharacteristic
101C2×D4×C61—21no—
102C2×D4×C61#4111yescharacteristic
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61C2411yescharacteristic
106C2×C22⋊D61—22no—
107#11758222—22no—
108#11758222—22no—
109#11758222—22no—
110#11758222—22no—
111C2×D244C2×D412yes—
112C2×C4×D61—22no—
113C2×C4×D61—42no—
114C2×Dic122C2×D412yes—
115M16×C61—21no—
116M16×C61—21no—
117M16×C61—21no—
118M16×C61—21no—
119M16×C61—41no—
120M16×C61—41no—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—21no—
126C2×C8×C61—41no—
127C2×C8×C61—41no—
128C2×C8×C61C2×D411yescharacteristic
129C2×C8×C61C2×D411yescharacteristic
130C2×C8×C61#4111yescharacteristic
131C2×C8×C61#4111yescharacteristic
132C2×C8×C61#4111yescharacteristic
133C2×C8×C61#4111yescharacteristic
134C4⋊C4×C61—21no—
135C22⋊C4×C61—21no—
136C42×C61—21no—
137C42×C61—21no—
138C42×C61#4112yes—
139C22⋊Dic61—22no—
140C22⋊Dic61—22no—
141#11758202—42no—
142#11758201—42no—
143#11758200—42no—
144#11758199—42no—
145#11758198—42no—
146C4⋊Dic61—22no—
147C4⋊Dic61—22no—
148#11758196—42no—
149C4×Dic61—22no—
150C4×Dic61—22no—
151C4×Dic61—42no—
152#11758194—22no—
153#11758194—22no—
154C2×#92418—22no—
155Dic244—21no—
156Dic244—21no—
157D488—21no—
158D488—21no—
159#11758190—22no—
160#11758190—22no—
161C23×C61—21no—
162C23×C61H(5,𝔽2)11yescharacteristic
163C22×D61—22no—
164Q8×C61—21no—
165Q8×C61—21no—
166D4×C61—21no—
167D4×C61—21no—
168D4×C61—41no—
169D4×C61—41no—
170C2×C4×C61—22no—
171C2×C4×C61—21no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—81no—
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C2×#4111yescharacteristic
178C2×C4×C61C2×#4111yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181C2×C4×C61H(5,𝔽2)11yescharacteristic
182C22⋊D61—42no—
183C22⋊D61—42no—
184C2×Dic61—22no—
185C2×Dic61—42no—
186C2×Dic61—42no—
187D244—22no—
188D244—22no—
189C4×D61—42no—
190C4×D61—42no—
191C4×D61—84no—
192Dic122—22no—
193Dic122—22no—
194C8×C61—41no—
195C8×C61—41no—
196C8×C61—41no—
197C8×C61#12311yescharacteristic
198C8×C61#12311yescharacteristic
199C8×C61#12311yescharacteristic
200C8×C61#12311yescharacteristic
201#92418—42no—
202#24986—611nomaximal, Sylow
203C22×C61—41no—
204C22×C61—41no—
205C22×C61—81no—
206C22×C61#54011yescharacteristic
207C22×C61#54611yescharacteristic
208C22×C61#55311yescharacteristic
209C2×D61—42no—
210C2×D61—82no—
211C4×C61—42no—
212C4×C61—41no—
213C4×C61—81no—
214C4×C61C2×#12311yescharacteristic
215C4×C61#52911yescharacteristic
216C4×C61#54611yescharacteristic
217C4×C61#55511yescharacteristic
218Dic61—42no—
219Dic61—82no—
220Dic61—82no—
221#3067—611no—
222#2943—611no—
223#2080—612no—
224#2053—612no—
225#1976—611no—
226#1946—611no—
227#1906—612no—
228#1833—611no—
229#1827—612no—
230#1812—611no—
231#1765D6111yescharacteristic, complemented
232C2×C61—21no—
233C2×C61—21no—
234C2×C61—21no—
235C2×C61—81no—
236C2×C61#242011yescharacteristic
237D61—82no—
238#575—612no—
239C2×#135—611no—
240C2×#131C2×D6111yescharacteristic
241C4×#41—612no—
242#473—1221no—
243#472—1221no—
244#471—1222no—
245#442—1221no—
246#441—1221no—
247#440—1222no—
248C2×#108—611no—
249#427—611no—
250#427—611no—
251#424—612no—
252#421—612no—
253C2×#104C2×D6111yescharacteristic
254#416—612no—
255#416C2×D6111yescharacteristic
256#413—612no—
257#410—612no—
258#410—612no—
259#405C2×D6111yescharacteristic
260C2×#97C2×D6111yescharacteristic
261#344—612no—
262#341C2×D6111yescharacteristic
263#339—612no—
264#339C2×D6111yescharacteristic
265H(3,ℤ/4)—612no—
266H(3,ℤ/4)—612no—
267C61#2498611yescharacteristic, complemented, Sylow
268C2×#41—612no—
269C2×#41C22×D6111yescharacteristic
270#137—1222no—
271#137—1222no—
272AGL(1,ℤ/8)—1222no—
273AGL(1,ℤ/8)—1222no—
274#135—1221no—
275#135—1221no—
276#135—1221no—
277#135—1221no—
278#135—2442no—
279C2×Q16—611no—
280C2×SD16—612no—
281C2×D8—611no—
282#131—21no—
283#131—21no—
284#131—21no—
285#131—21no—
286C2×M16—21no—
287C2×M16—612no—
288C2×M16C22×D6111yescharacteristic
289C2×M16C22×D6111yescharacteristic
290C22×C8—21no—
291C22×C8C22×D6111yescharacteristic
292C22×C8C22×D6111yescharacteristic
293C4×Q8—1222no—
294C4×D4—1222no—
295C4×D4—2442no—
296#117—612no—
297#117—612no—
298#117—1222no—
299#117C22×D6111yescharacteristic
300C2×C42—1222no—
301C2×C42C22×D6111yescharacteristic
302#108—1221no—
303#108—1221no—
304#107—1221no—
305#107—1221no—
306#106—1221no—
307#106—1221no—
308C4⋊C8—2442no—
309#104—22no—
310#104—2442no—
311#103—21no—
312#103—1222no—
313#102—21no—
314#102—1222no—
315#101—1222no—
316#100—1222no—
317#99—2442no—
318#99—2442no—
319#97—21no—
320#97—21no—
321#97—21no—
322C4×C8—21no—
323#41—21no—
324#41—21no—
325#41—21no—
326#41—21no—
327#41—1222no—
328#41—1222no—
329#41—1222no—
330#41—1222no—
331C2×Q8—612no—
332C2×Q8#1175822211yescharacteristic
333C2×D4—21no—
334C2×D4—612no—
335C2×D4—1222no—
336C2×D4#1175822211yescharacteristic
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—1222no—
340C22×C4—2442no—
341C22×C4C23×D6111yescharacteristic
342Q16—1221no—
343Q16—1221no—
344Q16—2442no—
345SD16—1222no—
346SD16—1222no—
347SD16—2442no—
348SD16—2442no—
349D8—1221no—
350D8—1221no—
351D8—2442no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—21no—
356M16—41no—
357M16—41no—
358M16—1222no—
359M16—1222no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—1222no—
368C2×C8#1175822211yescharacteristic
369C2×C8#1175822211yescharacteristic
370C2×C8D4×D6111yescharacteristic
371C2×C8D4×D6111yescharacteristic
372C2×C8#1175822411yescharacteristic
373C2×C8#1175822411yescharacteristic
374C4⋊C4—21no—
375C4⋊C4—1222no—
376C4⋊C4—1222no—
377C4⋊C4—2442no—
378C22⋊C4—21no—
379C22⋊C4—1222no—
380C22⋊C4—1222no—
381C22⋊C4—2442no—
382C42—21no—
383C42—21no—
384C42—1222no—
385C42—1222no—
386C42—2442no—
387C42#1175822212yes—
388C23—21no—
389C23—1222no—
390C23#1451982311yescharacteristic
391Q8—21no—
392Q8—21no—
393Q8—1222no—
394Q8—1222no—
395D4—21no—
396D4—21no—
397D4—41no—
398D4—41no—
399D4—1222no—
400D4—1222no—
401D4—2442no—
402D4—2442no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—81no—
409C2×C4—1222no—
410C2×C4—4884no—
411C2×C4—2442no—
412C2×C4—2442no—
413C2×C4—2442no—
414C2×C4—2442no—
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4C2×#1175822411yescharacteristic
417C2×C4C2×#1175822211yescharacteristic
418C2×C4C2×#1175822211yescharacteristic
419C2×C4C2×#1175822211yescharacteristic
420C2×C4#1451982311yescharacteristic
421C8—41no—
422C8—41no—
423C8—41no—
424C8—2442no—
425C8#1451975911yescharacteristic
426C8#1451975911yescharacteristic
427C8#1451975911yescharacteristic
428C8#1451975911yescharacteristic
429C22—41no—
430C22—41no—
431C22—81no—
432C22—4882no—
433C22—2442no—
434C22#1684036711yescharacteristic
435C22#1684040811yescharacteristic
436C22#1684045811yescharacteristic
437C4—42no—
438C4—41no—
439C4—81no—
440C4—4882no—
441C4—4882no—
442C4—2442no—
443C4C2×#1451975911yescharacteristic
444C4#1684029611yescharacteristic
445C4#1684040811yescharacteristic
446C4#1684047711yescharacteristic
447C2—21no—
448C2—21no—
449C2—21no—
450C2—81no—
451C2—4882no—
452C2#2144931111yescharacteristic
subgroupG/H
centerC4#16840477
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1765C2
socleC2×C61#2420
prime radical#21455539C2
hypercenter#1765D61
hypocenterC61#24986
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1765

Complements

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

NH ≅ G/Ntype
2. #21454621C2semidirect
3. #21447275C2semidirect
4. #21447123C2semidirect
5. #21446717C2semidirect
6. #21446572C2semidirect
7. #21446323C2semidirect
8. #21445943C2semidirect
9. #21445922C2semidirect
10. #21445865C2semidirect
11. C61×#1765C2semidirect
231. #1765D61semidirect
267. C61#24986semidirect

Maximal subgroups

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

subgroupindex
#214555392
#214546212
#214472752
#214471232
#214467172
#214465722
#214463232
#214459432
#214459222
#214458652
C61×#17652
#2498661