Finite Group Databasebeta

← #29725106

Subgroups — #29725106

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#21455535C211yesmaximal, characteristic
2#21447273C212yesmaximal, complemented
3#21447123C212yesmaximal, complemented
4#21446718C211yesmaximal, characteristic, complemented
5#21446572C211yesmaximal, characteristic, complemented
6#21446320C212yesmaximal, complemented
7#21445942C211yesmaximal, characteristic, complemented
8#21445923C212yesmaximal, complemented
9#21445862C211yesmaximal, characteristic, complemented
10#21445663C211yesmaximal, characteristic, complemented
11C61×#2943C211yesmaximal, characteristic, complemented
12#16840560C2212yes—
13C2×#14519783C2211yescharacteristic
14#16840034—21no—
15#16840029—21no—
16#16840022—22no—
17C2×#14519713C2211yescharacteristic
18#16839800C2211yescharacteristic
19#16839788C2212yes—
20#16839779C2212yes—
21C2×#14519705C2211yescharacteristic
22#16839759C2211yescharacteristic
23#16839746C2212yes—
24#16839736C2212yes—
25#16839735C2212yes—
26#16839717C2211yescharacteristic
27#16839532C2212yes—
28#16839527C2211yescharacteristic
29#16839522C2211yescharacteristic
30#16839522C2212yes—
31#16839518C2212yes—
32#16839518C2212yes—
33C2×#135×C61C2211yescharacteristic
34C4×#41×C61C2212yes—
35C61×#442—21no—
36C61×#441—21no—
37C61×#440—22no—
38C61×#427C2211yescharacteristic
39C61×#416C2212yes—
40C2×#97×C61C2211yescharacteristic
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×#11758194C2311yescharacteristic
53C2×#11758194C2312yes—
54C2×#11758189—21no—
55C2×C8×D61—21no—
56#14519736C2311yescharacteristic
57#14519736C2312yes—
58#14519713—21no—
59#14519713—21no—
60#14519711—21no—
61#14519709—21no—
62#14519706—42no—
63#14519705—22no—
64#14519704—42no—
65#14519702—21no—
66#14519699—21no—
67#14519696—22no—
68#14519694—22no—
69#14519692—42no—
70#14519691—42no—
71#14519687—21no—
72C8×Dic61—21no—
73C2×#41×C61C2312yes—
74#135×C61—21no—
75#135×C61—21no—
76#135×C61—21no—
77#135×C61—21no—
78C2×Q16×C61D411yescharacteristic
79C2×SD16×C61D412yes—
80C2×D8×C61D411yescharacteristic
81C22×C8×C61C2311yescharacteristic
82C22×C8×C61C2311yescharacteristic
83C4×Q8×C61—22no—
84C4×D4×C61—22no—
85C4×D4×C61—42no—
86#117×C61—22no—
87#117×C61C2312yes—
88C2×C42×C61—22no—
89C2×C42×C61C2311yescharacteristic
90#107×C61—21no—
91#106×C61—21no—
92#103×C61—22no—
93#102×C61—22no—
94#97×C61—21no—
95#97×C61—21no—
96#41×C61—22no—
97#41×C61—22no—
98#41×C61—22no—
99#41×C61—22no—
100C2×Q8×C61C2×D412yes—
101C2×D4×C61—22no—
102C2×D4×C61C2×D412yes—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—42no—
106C22×C4×C61C2411yescharacteristic
107C2×C22⋊D61—21no—
108#11758222—21no—
109#11758222—21no—
110#11758222—21no—
111#11758222—21no—
112C2×D244#4111yescharacteristic
113C2×C4×D61—21no—
114C2×Dic122#4111yescharacteristic
115Q16×C61—21no—
116Q16×C61—21no—
117SD16×C61—22no—
118SD16×C61—22no—
119D8×C61—21no—
120D8×C61—21no—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61#4111yescharacteristic
127C2×C8×C61#4111yescharacteristic
128C2×C8×C61#4111yescharacteristic
129C2×C8×C61#4111yescharacteristic
130C4⋊C4×C61—22no—
131C4⋊C4×C61—42no—
132C22⋊C4×C61—22no—
133C22⋊C4×C61—42no—
134C42×C61—21no—
135C42×C61—22no—
136C42×C61—42no—
137C42×C61#4112yes—
138C22⋊Dic61—22no—
139C22⋊Dic61—21no—
140#11758202—42no—
141#11758201—42no—
142#11758200—42no—
143#11758199—42no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4×Dic61—21no—
147C4×Dic61—22no—
148#11758194—22no—
149#11758194—21no—
150#11758194—21no—
151#11758194—21no—
152#11758194—22no—
153#11758194—21no—
154C2×#92418—21no—
155C2×#92418—21no—
156C2×#92418—22no—
157#11758189—41no—
158#11758189—41no—
159C8×D61—41no—
160C8×D61—41no—
161C23×C61—22no—
162C23×C61H(5,𝔽2)11yescharacteristic
163C22×D61—21no—
164Q8×C61—22no—
165Q8×C61—22no—
166D4×C61—22no—
167D4×C61—22no—
168D4×C61—42no—
169D4×C61—42no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61—81no—
176C2×C4×C61—84no—
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—42no—
187C2×Dic61—41no—
188D244—21no—
189D244—21no—
190C4×D61—41no—
191C4×D61—41no—
192Dic122—21no—
193Dic122—21no—
194C8×C61—41no—
195C8×C61#12311yescharacteristic
196C8×C61#12311yescharacteristic
197C8×C61#12311yescharacteristic
198C8×C61#12311yescharacteristic
199#92418—41no—
200#92418—42no—
201#92418—41no—
202#24986—611nomaximal, Sylow
203C22×C61—41no—
204C22×C61—42no—
205C22×C61—82no—
206C22×C61#54011yescharacteristic
207C22×C61#54611yescharacteristic
208C22×C61#55311yescharacteristic
209C2×D61—41no—
210C2×D61—81no—
211C4×C61—42no—
212C4×C61—42no—
213C4×C61—82no—
214C4×C61C2×#12311yescharacteristic
215C4×C61#52911yescharacteristic
216C4×C61#54611yescharacteristic
217C4×C61#55511yescharacteristic
218Dic61—41no—
219Dic61—82no—
220Dic61—81no—
221#3067—611no—
222#2943D6111yescharacteristic, complemented
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#1765—611no—
232C2×C61—21no—
233C2×C61—21no—
234C2×C61—21no—
235C2×C61—82no—
236C2×C61#242011yescharacteristic
237D61—81no—
238#575—612no—
239C2×#135C2×D6111yescharacteristic
240C2×#131—611no—
241C4×#41C2×D6112yes—
242#473—1221no—
243#472—1221no—
244#471—1222no—
245#442—21no—
246#441—21no—
247#440—22no—
248C2×#108—611no—
249#427—611no—
250#427C2×D6111yescharacteristic
251#424—612no—
252#421—612no—
253C2×#104—611no—
254#416—611no—
255#416C2×D6112yes—
256#413—612no—
257#410—612no—
258#410—612no—
259#405—611no—
260C2×#97C2×D6111yescharacteristic
261#344—612no—
262#341—611no—
263#339—611no—
264#339—612no—
265H(3,ℤ/4)—612no—
266H(3,ℤ/4)—612no—
267C61#2498611yescharacteristic, complemented, Sylow
268C2×#41—611no—
269C2×#41C22×D6112yes—
270#137—1222no—
271#137—1222no—
272AGL(1,ℤ/8)—1222no—
273AGL(1,ℤ/8)—1222no—
274#135—21no—
275#135—21no—
276#135—21no—
277#135—21no—
278#135—2442no—
279C2×Q16C22⋊D6111yescharacteristic
280C2×SD16C22⋊D6112yes—
281C2×D8C22⋊D6111yescharacteristic
282#131—1221no—
283#131—1221no—
284#131—1221no—
285#131—1221no—
286C2×M16—611no—
287C2×M16—612no—
288C2×M16—611no—
289C2×M16—1221no—
290C22×C8—1221no—
291C22×C8C22×D6111yescharacteristic
292C22×C8C22×D6111yescharacteristic
293C4×Q8—22no—
294C4×D4—22no—
295C4×D4—42no—
296#117—22no—
297#117—611no—
298#117—612no—
299#117C22×D6112yes—
300C2×C42—22no—
301C2×C42C22×D6111yescharacteristic
302#108—1221no—
303#108—1221no—
304#107—21no—
305#107—1221no—
306#106—21no—
307#106—1221no—
308C4⋊C8—2442no—
309#104—1222no—
310#104—2442no—
311#103—22no—
312#103—1221no—
313#102—22no—
314#102—1221no—
315#101—1222no—
316#100—1222no—
317#99—2442no—
318#99—2442no—
319#97—21no—
320#97—21no—
321#97—1221no—
322C4×C8—1221no—
323#41—22no—
324#41—22no—
325#41—22no—
326#41—22no—
327#41—1221no—
328#41—1221no—
329#41—1221no—
330#41—1221no—
331C2×Q8—611no—
332C2×Q8C2×C22⋊D6112yes—
333C2×D4—22no—
334C2×D4—611no—
335C2×D4—1221no—
336C2×D4C2×C22⋊D6112yes—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—42no—
340C22×C4—1221no—
341C22×C4C23×D6111yescharacteristic
342Q16—21no—
343Q16—21no—
344Q16—2442no—
345SD16—22no—
346SD16—22no—
347SD16—2442no—
348SD16—2442no—
349D8—21no—
350D8—21no—
351D8—2442no—
352M16—1221no—
353M16—1222no—
354M16—1222no—
355M16—1221no—
356M16—1221no—
357M16—1221no—
358M16—2441no—
359M16—2441no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—1221no—
364C2×C8—1221no—
365C2×C8—1222no—
366C2×C8—2441no—
367C2×C8—2441no—
368C2×C8#1175822211yescharacteristic
369C2×C8#1175822211yescharacteristic
370C2×C8#1175822211yescharacteristic
371C2×C8#1175822211yescharacteristic
372C2×C8C2×C22⋊D6111yescharacteristic
373C2×C8C2×C22⋊D6111yescharacteristic
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—1221no—
377C4⋊C4—1222no—
378C22⋊C4—22no—
379C22⋊C4—42no—
380C22⋊C4—1221no—
381C22⋊C4—1222no—
382C42—21no—
383C42—22no—
384C42—42no—
385C42—1221no—
386C42—1222no—
387C42#1175822412yes—
388C23—22no—
389C23—1221no—
390C23#1451982211yescharacteristic
391Q8—22no—
392Q8—22no—
393Q8—1221no—
394Q8—1221no—
395D4—22no—
396D4—22no—
397D4—42no—
398D4—42no—
399D4—1221no—
400D4—1221no—
401D4—2441no—
402D4—2441no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—81no—
409C2×C4—84no—
410C2×C4—1221no—
411C2×C4—2441no—
412C2×C4—2442no—
413C2×C4—2441no—
414C2×C4—2441no—
415C2×C4C22×C22⋊D6111yescharacteristic
416C2×C4#41×D6111yescharacteristic
417C2×C4C2×#1175822411yescharacteristic
418C2×C4C2×#1175822211yescharacteristic
419C2×C4C2×#1175822211yescharacteristic
420C2×C4#1451982311yescharacteristic
421C8—41no—
422C8—2441no—
423C8—2441no—
424C8—2442no—
425C8#1451976011yescharacteristic
426C8#1451976011yescharacteristic
427C8#1451976011yescharacteristic
428C8#1451976011yescharacteristic
429C22—41no—
430C22—42no—
431C22—82no—
432C22—4881no—
433C22—2441no—
434C22#1684036311yescharacteristic
435C22#1684040611yescharacteristic
436C22#1684046211yescharacteristic
437C4—42no—
438C4—42no—
439C4—82no—
440C4—4882no—
441C4—4881no—
442C4—2441no—
443C4C2×#1451976011yescharacteristic
444C4#1684029411yescharacteristic
445C4#1684040011yescharacteristic
446C4#1684047711yescharacteristic
447C2—21no—
448C2—21no—
449C2—21no—
450C2—82no—
451C2—4881no—
452C2#2144930511yescharacteristic
subgroupG/H
centerC4#16840477
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2943C2
socleC2×C61#2420
prime radical#21455535C2
hypercenter#2943D61
hypocenterC61#24986
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2943

Complements

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

NH ≅ G/Ntype
2. #21447273C2semidirect
3. #21447123C2semidirect
4. #21446718C2semidirect
5. #21446572C2semidirect
6. #21446320C2semidirect
7. #21445942C2semidirect
8. #21445923C2semidirect
9. #21445862C2semidirect
10. #21445663C2semidirect
11. C61×#2943C2semidirect
222. #2943D61semidirect
267. C61#24986semidirect

Maximal subgroups

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

subgroupindex
#214555352
#214472732
#214471232
#214467182
#214465722
#214463202
#214459422
#214459232
#214458622
#214456632
C61×#29432
#2498661