Finite Group Databasebeta

← #29725108

Subgroups — #29725108

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#21454620C211yesmaximal, characteristic, complemented
2#21447272C212yesmaximal, complemented
3#21447122C212yesmaximal, complemented
4#21446718C211yesmaximal, characteristic, complemented
5#21446570C211yesmaximal, characteristic, complemented
6#21446322C212yesmaximal, complemented
7#21445944C211yesmaximal, characteristic, complemented
8#21445924C212yesmaximal, complemented
9#21445865C211yesmaximal, characteristic, complemented
10#21445661C211yesmaximal, characteristic, complemented
11C61×#3067C211yesmaximal, characteristic
12#41×Dic61C2212yes—
13#16839874—21no—
14#16839869—21no—
15#16839863—22no—
16#16839800C2211yescharacteristic
17#16839800C2211yescharacteristic
18#16839790C2212yes—
19#16839779C2212yes—
20C2×#14519704C2211yescharacteristic
21#16839758C2212yes—
22#16839758C2211yescharacteristic
23#16839736C2212yes—
24#16839736C2212yes—
25#16839717C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27#16839534C2212yes—
28#16839526C2211yescharacteristic
29#16839523C2211yescharacteristic
30#16839523C2212yes—
31#16839519C2212yes—
32#16839519C2212yes—
33C61×#575C2212yes—
34C2×#135×C61C2211yescharacteristic
35C2×#131×C61C2211yescharacteristic
36C61×#473—21no—
37C61×#472—21no—
38C61×#471—22no—
39C2×#108×C61C2211yescharacteristic
40C61×#413C2212yes—
41C2×#11758194C2311yescharacteristic
42C22×#92418C2311yescharacteristic
43Q8×Dic61—22no—
44D4×Dic61—22no—
45D4×Dic61—42no—
46#14519736—22no—
47#14519736C2311yescharacteristic
48#14519736C2312yes—
49#14519736C2312yes—
50C2×C4×Dic61—22no—
51C2×C4×Dic61C2311yescharacteristic
52#14519711—21no—
53#14519711—21no—
54#14519709—21no—
55#14519709—21no—
56#14519706—42no—
57#14519704—22no—
58#14519704—42no—
59#14519701—22no—
60#14519701—21no—
61#14519698—22no—
62#14519698—21no—
63#14519692—42no—
64#14519692—42no—
65#14519687—21no—
66#14519687—21no—
67#14519687—21no—
68C8×Dic61—21no—
69C2×#41×C61C2312yes—
70C2×#41×C61C2311yescharacteristic
71#137×C61—22no—
72#137×C61—22no—
73AGL(1,ℤ/8)×C61—22no—
74AGL(1,ℤ/8)×C61—22no—
75#135×C61—21no—
76#135×C61—21no—
77#135×C61—21no—
78#135×C61—21no—
79#135×C61—42no—
80C2×Q16×C61D411yescharacteristic
81C2×SD16×C61D412yes—
82C2×D8×C61D411yescharacteristic
83#131×C61—21no—
84#131×C61—21no—
85#131×C61—21no—
86#131×C61—21no—
87C2×M16×C61—21no—
88C2×M16×C61C2311yescharacteristic
89C2×M16×C61C2312yes—
90C22×C8×C61—21no—
91C22×C8×C61C2311yescharacteristic
92#108×C61—21no—
93#108×C61—21no—
94#101×C61—22no—
95#100×C61—22no—
96#41×C61—21no—
97#41×C61—21no—
98#41×C61—22no—
99#41×C61—22no—
100#41×C61—22no—
101#41×C61—21no—
102#41×C61—21no—
103#41×C61—22no—
104C2×Q8×C61C2×D412yes—
105C2×Q8×C61#4111yescharacteristic
106C2×D4×C61—21no—
107C2×D4×C61—22no—
108C2×D4×C61C2×D412yes—
109C2×D4×C61#4111yescharacteristic
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61C2411yescharacteristic
113C22×Dic61—21no—
114C22×Dic61—42no—
115Q16×C61—21no—
116Q16×C61—21no—
117Q16×C61—42no—
118SD16×C61—22no—
119SD16×C61—22no—
120SD16×C61—42no—
121SD16×C61—42no—
122D8×C61—21no—
123D8×C61—21no—
124D8×C61—42no—
125M16×C61—21no—
126M16×C61—22no—
127M16×C61—21no—
128M16×C61—22no—
129M16×C61—41no—
130M16×C61—41no—
131C2×C8×C61—21no—
132C2×C8×C61—22no—
133C2×C8×C61—41no—
134C2×C8×C61—41no—
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61C2×D411yescharacteristic
137C2×C8×C61#4111yescharacteristic
138C2×C8×C61#4111yescharacteristic
139C2×C8×C61#4111yescharacteristic
140C2×C8×C61#4111yescharacteristic
141C22⋊Dic61—22no—
142C22⋊Dic61—21no—
143C22⋊Dic61—22no—
144C22⋊Dic61—42no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—22no—
148C4⋊Dic61—42no—
149C4×Dic61—21no—
150C4×Dic61—21no—
151C4×Dic61—22no—
152C4×Dic61—22no—
153C4×Dic61—42no—
154C4×Dic61#4112yes—
155#11758194—21no—
156#11758194—21no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—21no—
161C23×C61—22no—
162C23×C61—21no—
163C23×C61H(5,𝔽2)11yescharacteristic
164Q8×C61—22no—
165Q8×C61—21no—
166Q8×C61—21no—
167Q8×C61—22no—
168D4×C61—22no—
169D4×C61—21no—
170D4×C61—21no—
171D4×C61—22no—
172D4×C61—41no—
173D4×C61—42no—
174D4×C61—41no—
175D4×C61—42no—
176C2×C4×C61—22no—
177C2×C4×C61—21no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C2×C4×C61C2×#4111yescharacteristic
185C2×C4×C61C2×#4111yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C2×C4×C61H(5,𝔽2)11yescharacteristic
188C2×Dic61—22no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192C2×Dic61—81no—
193C2×Dic61—84no—
194C8×C61—41no—
195C8×C61—42no—
196C8×C61#12311yescharacteristic
197C8×C61#12311yescharacteristic
198C8×C61#12311yescharacteristic
199C8×C61#12311yescharacteristic
200#92418—41no—
201#92418—41no—
202#24986—611nomaximal, Sylow
203C22×C61—41no—
204C22×C61—41no—
205C22×C61—42no—
206C22×C61—82no—
207C22×C61—81no—
208C22×C61#54011yescharacteristic
209C22×C61#54611yescharacteristic
210C22×C61#55311yescharacteristic
211C4×C61—42no—
212C4×C61—41no—
213C4×C61C2×#12311yescharacteristic
214C4×C61#52911yescharacteristic
215C4×C61#54611yescharacteristic
216C4×C61#55511yescharacteristic
217Dic61—42no—
218Dic61—82no—
219Dic61—81no—
220Dic61—82no—
221#3067D6111yescharacteristic
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#1765—611no—
232C2×C61—21no—
233C2×C61—21no—
234C2×C61—21no—
235C2×C61—82no—
236C2×C61—81no—
237C2×C61#242011yescharacteristic
238#575C2×D6112yes—
239C2×#135C2×D6111yescharacteristic
240C2×#131C2×D6111yescharacteristic
241C4×#41—612no—
242#473—21no—
243#472—21no—
244#471—22no—
245#442—1221no—
246#441—1221no—
247#440—1222no—
248C2×#108C2×D6111yescharacteristic
249#427—611no—
250#427—611no—
251#424—612no—
252#421—612no—
253C2×#104—611no—
254#416—611no—
255#416—612no—
256#413C2×D6112yes—
257#410—612no—
258#410—612no—
259#405—611no—
260C2×#97—611no—
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×#41C22×D6111yescharacteristic
269C2×#41C22×D6112yes—
270#137—22no—
271#137—22no—
272AGL(1,ℤ/8)—22no—
273AGL(1,ℤ/8)—22no—
274#135—21no—
275#135—21no—
276#135—21no—
277#135—21no—
278#135—42no—
279C2×Q16C22⋊D6111yescharacteristic
280C2×SD16C22⋊D6112yes—
281C2×D8C22⋊D6111yescharacteristic
282#131—21no—
283#131—21no—
284#131—21no—
285#131—21no—
286C2×M16—21no—
287C2×M16—611no—
288C2×M16C22×D6111yescharacteristic
289C2×M16C22×D6112yes—
290C22×C8—21no—
291C22×C8—611no—
292C22×C8C22×D6111yescharacteristic
293C4×Q8—1222no—
294C4×D4—1222no—
295C4×D4—2442no—
296#117—612no—
297#117—611no—
298#117—612no—
299#117—1222no—
300C2×C42—611no—
301C2×C42—1222no—
302#108—21no—
303#108—21no—
304#107—1221no—
305#107—1221no—
306#106—1221no—
307#106—1221no—
308C4⋊C8—2442no—
309#104—1222no—
310#104—2442no—
311#103—1221no—
312#103—1222no—
313#102—1221no—
314#102—1222no—
315#101—22no—
316#100—22no—
317#99—2442no—
318#99—2442no—
319#97—1221no—
320#97—1221no—
321#97—1221no—
322C4×C8—1221no—
323#41—21no—
324#41—21no—
325#41—21no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—22no—
330#41—21no—
331C2×Q8#1175822411yescharacteristic
332C2×Q8C2×C22⋊D6112yes—
333C2×D4—21no—
334C2×D4—22no—
335C2×D4#1175822411yescharacteristic
336C2×D4C2×C22⋊D6112yes—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—1221no—
340C22×C4—2442no—
341C22×C4C23×D6111yescharacteristic
342Q16—21no—
343Q16—21no—
344Q16—42no—
345SD16—22no—
346SD16—22no—
347SD16—42no—
348SD16—42no—
349D8—21no—
350D8—21no—
351D8—42no—
352M16—21no—
353M16—22no—
354M16—21no—
355M16—22no—
356M16—41no—
357M16—41no—
358M16—1221no—
359M16—1221no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—1221no—
365C2×C8—1221no—
366C2×C8—1221no—
367C2×C8—1221no—
368C2×C8#1175822411yescharacteristic
369C2×C8#1175822411yescharacteristic
370C2×C8#1175822411yescharacteristic
371C2×C8#1175822411yescharacteristic
372C2×C8C2×C22⋊D6111yescharacteristic
373C2×C8C2×C22⋊D6111yescharacteristic
374C4⋊C4—1222no—
375C4⋊C4—1221no—
376C4⋊C4—1222no—
377C4⋊C4—2442no—
378C22⋊C4—1221no—
379C22⋊C4—1222no—
380C22⋊C4—1222no—
381C22⋊C4—2442no—
382C42—612no—
383C42—1221no—
384C42—1222no—
385C42—1221no—
386C42—1222no—
387C42—2442no—
388C23—21no—
389C23—22no—
390C23#1451982311yescharacteristic
391Q8—22no—
392Q8—21no—
393Q8—21no—
394Q8—22no—
395D4—22no—
396D4—21no—
397D4—21no—
398D4—22no—
399D4—42no—
400D4—41no—
401D4—42no—
402D4—41no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—1222no—
410C2×C4—4884no—
411C2×C4—4881no—
412C2×C4—2441no—
413C2×C4—2442no—
414C2×C4—2442no—
415C2×C4C22×C22⋊D6111yescharacteristic
416C2×C4#41×D6111yescharacteristic
417C2×C4C2×#1175822411yescharacteristic
418C2×C4C2×#1175822411yescharacteristic
419C2×C4C2×#1175822411yescharacteristic
420C2×C4#1451982211yescharacteristic
421C8—41no—
422C8—42no—
423C8—2441no—
424C8—2441no—
425C8#1451975811yescharacteristic
426C8#1451975811yescharacteristic
427C8#1451975811yescharacteristic
428C8#1451975811yescharacteristic
429C22—41no—
430C22—41no—
431C22—42no—
432C22—82no—
433C22—81no—
434C22#1684036511yescharacteristic
435C22#1684040011yescharacteristic
436C22#1684046411yescharacteristic
437C4—41no—
438C4—42no—
439C4—4882no—
440C4—4882no—
441C4—4881no—
442C4—2442no—
443C4C2×#1451975811yescharacteristic
444C4#1684029311yescharacteristic
445C4#1684040611yescharacteristic
446C4#1684047611yescharacteristic
447C2—21no—
448C2—21no—
449C2—21no—
450C2—82no—
451C2—81no—
452C2#2144930711yescharacteristic
subgroupG/H
centerC4#16840476
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3067C2
socleC2×C61#2420
hypercenter#3067D61
hypocenterC61#24986
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #21454620C2semidirect
2. #21447272C2semidirect
3. #21447122C2semidirect
4. #21446718C2semidirect
5. #21446570C2semidirect
6. #21446322C2semidirect
7. #21445944C2semidirect
8. #21445924C2semidirect
9. #21445865C2semidirect
10. #21445661C2semidirect
267. C61#24986semidirect

Maximal subgroups

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

subgroupindex
#214546202
#214472722
#214471222
#214467182
#214465702
#214463222
#214459442
#214459242
#214458652
#214456612
C61×#30672
#2498661