Finite Group Databasebeta

← #29837703

Subgroups — #29837703

Order 15616.

472 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458804C211yesmaximal, characteristic
2#21458710C211yesmaximal, characteristic, complemented
3#21448050C211yesmaximal, characteristic
4#21448049C211yesmaximal, characteristic
5#21448045C211yesmaximal, characteristic, complemented
6#21447984C211yesmaximal, characteristic, complemented
7#21447981C211yesmaximal, characteristic, complemented
8#21447954C211yesmaximal, characteristic
9#21447879C211yesmaximal, characteristic
10#21447867C211yesmaximal, characteristic, complemented
11#21447721C211yesmaximal, characteristic, complemented
12Q32×Dic61C211yesmaximal, characteristic, complemented
13C4×#14519723C211yesmaximal, characteristic, complemented
14SD32×Dic61C211yesmaximal, characteristic
15C61×#2211C211yesmaximal, characteristic
16#16840488—21no—
17#16840481—21no—
18C2×#14519723C2211yescharacteristic
19#16840166C2211yescharacteristic
20#16840164C2211yescharacteristic
21#16840163C2211yescharacteristic
22#16840145C2211yescharacteristic
23#16840140C2211yescharacteristic
24#16840069—21no—
25#16840067—21no—
26#16840045—21no—
27#16839982—21no—
28#16839981—21no—
29#16839965—21no—
30C8×Dic122C2211yescharacteristic
31C4×Dic244C2211yescharacteristic
32Q16×Dic61C2211yescharacteristic
33Q16×Dic61C2211yescharacteristic
34Q16×Dic61C2211yescharacteristic
35D8×Dic61C2211yescharacteristic
36#16839590C2211yescharacteristic
37#16839588C2211yescharacteristic
38#16839587C2211yescharacteristic
39#16839580C2211yescharacteristic
40#16839580C2211yescharacteristic
41#16839567C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839566C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839565C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic
49C4×#11758185C2211yescharacteristic
50C16×Dic61C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×Q32×C61C2211yescharacteristic
53C2×SD32×C61C2211yescharacteristic
54C61×#495C2211yescharacteristic
55C4×Q16×C61C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×Dic244C2311yescharacteristic
60C2×#11758202—21no—
61C2×#11758200—21no—
62#14519776—21no—
63#14519776—41no—
64#14519775—21no—
65#14519770—41no—
66#14519770—41no—
67#14519769—21no—
68#14519769—41no—
69#14519768—21no—
70#14519768—41no—
71#14519764—21no—
72#14519754—41no—
73Q8×Dic61—21no—
74Q8×Dic61—21no—
75Q8×Dic61—21no—
76C4×Dic122—21no—
77C4×Dic122—41no—
78C4×Dic122D411yescharacteristic
79D4×Dic61—21no—
80#14519733—41no—
81#14519723—22no—
82C2×#11758185C2311yescharacteristic
83C2×#11758185C2311yescharacteristic
84#14519711C2311yescharacteristic
85#14519711C2311yescharacteristic
86#14519711C2311yescharacteristic
87#14519710—21no—
88#14519709—21no—
89#14519708—21no—
90C4⋊#92418D411yescharacteristic
91#14519706D411yescharacteristic
92#14519706D411yescharacteristic
93#14519702—21no—
94#14519701—21no—
95#14519701—21no—
96#14519701—21no—
97#14519698—21no—
98C8×Dic61C2311yescharacteristic
99C8×Dic61C2311yescharacteristic
100C2×Q16×C61C2311yescharacteristic
101C2×Q16×C61C2311yescharacteristic
102C2×SD16×C61—21no—
103C2×D8×C61C2311yescharacteristic
104#124×C61—21no—
105C4×Q8×C61—21no—
106Q32×C61—22no—
107SD32×C61—22no—
108C2×C16×C61C2311yescharacteristic
109C2×C16×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#103×C61—21no—
112C4×C8×C61C2311yescharacteristic
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×D4×C61—21no—
116C22×Dic61—41no—
117C2×Dic122—21no—
118C2×Dic122—41no—
119C2×Dic122D811yescharacteristic
120Q16×C61—21no—
121Q16×C61—21no—
122Q16×C61—21no—
123Q16×C61#4112yes—
124SD16×C61—81no—
125D8×C61—21no—
126D8×C61—21no—
127C2×C8×C61C2×D411yescharacteristic
128C2×C8×C61C2411yescharacteristic
129C4⋊C4×C61—21no—
130C4⋊C4×C61—41no—
131C22⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133C42×C61—41no—
134C42×C61C2×D411yescharacteristic
135C22⋊Dic61—41no—
136C22⋊Dic61—41no—
137C22⋊Dic61—41no—
138#11758202—81no—
139#11758200—81no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—41no—
144C4⋊Dic61—41no—
145C4⋊Dic61—41no—
146C4⋊Dic61—41no—
147C4⋊Dic61D811yescharacteristic
148#11758196—41no—
149#11758196—41no—
150#11758196—41no—
151#11758196—41no—
152#11758196—41no—
153#11758196—41no—
154#11758196—41no—
155#11758196D811yescharacteristic
156#11758196D811yescharacteristic
157C4×Dic61—41no—
158C4×Dic61—41no—
159C4×Dic61—41no—
160C4×Dic61—41no—
161C4×Dic61C2×D411yescharacteristic
162C4×Dic61C2×D411yescharacteristic
163C2×#92418C2×D411yescharacteristic
164C2×#92418C2×D411yescharacteristic
165Dic244—21no—
166Dic244—21no—
167C16×C61—21no—
168C16×C61—21no—
169#11758185—21no—
170#11758185—21no—
171C23×C61—41no—
172Q8×C61—22no—
173Q8×C61—41no—
174Q8×C61—41no—
175Q8×C61—41no—
176D4×C61—41no—
177D4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61C2×D811yescharacteristic
182C2×C4×C61C2×D811yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—82no—
189C2×Dic61C2×D811yescharacteristic
190C2×Dic61C2×D811yescharacteristic
191C2×Dic61C2×D811yescharacteristic
192C2×Dic61C2×D811yescharacteristic
193Dic122—22no—
194Dic122—41no—
195Dic122—41no—
196Dic122—82no—
197C8×C61—21no—
198C8×C61C2×#4111yescharacteristic
199C8×C612−1+411yescharacteristic
200#92418—21no—
201#92418—21no—
202#33931—611nomaximal, Sylow
203C22×C61—81no—
204C22×C61—81no—
205C22×C61—81no—
206C22×C61C22×D811yescharacteristic
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—42no—
210C4×C61—81no—
211C4×C61—81no—
212C4×C61#54811yescharacteristic
213C4×C61#57711yescharacteristic
214Dic61—21no—
215Dic61—21no—
216Dic61—21no—
217Dic61—81no—
218Dic61—81no—
219Dic61—81no—
220Dic61—81no—
221Dic61#50812yes—
222#3393—611no—
223#3383—611no—
224#2242—611no—
225#2242—611no—
226#2241—611no—
227#2229—611no—
228#2229—611no—
229#2224—611no—
230#2211—611no—
231#2211D6111yescharacteristic
232#2209—611no—
233Q8×C16—611no—
234C4×Q32—611no—
235C4×Q32—611no—
236C4×SD32—611no—
237C2×C61—161no—
238C2×C61#335811yescharacteristic
239C2×C61C2×#50811yescharacteristic
240C2×C61#341711yescharacteristic
241#557—1221no—
242#556—1221no—
243C2×Q32—611no—
244C2×Q32C2×D6111yescharacteristic
245C2×SD32C2×D6111yescharacteristic
246#500—611no—
247#499—611no—
248#499—611no—
249#495—611no—
250#495C2×D6111yescharacteristic
251#494—611no—
252#479—1221no—
253#479—1221no—
254#475—1221no—
255#464—1221no—
256#464—1221no—
257#461—1221no—
258C8×Q8—611no—
259C4×Q16—611no—
260C4×Q16—611no—
261C4×Q16—611no—
262C4×Q16—611no—
263C4×Q16C2×D6111yescharacteristic
264C4×D8—611no—
265#367—611no—
266#366—611no—
267#366—611no—
268C4⋊C16—611no—
269C4⋊C16—611no—
270C4⋊C16C2×D6111yescharacteristic
271#358—611no—
272#358—611no—
273#358—611no—
274#358—611no—
275#358—611no—
276#358—611no—
277#358C2×D6111yescharacteristic
278#357—611no—
279#357—611no—
280#357C2×D6111yescharacteristic
281C4×C16—611no—
282C4×C16—611no—
283C4×C16—611no—
284C61#3393111yescharacteristic, complemented, Sylow
285C2×Q16—611no—
286C2×Q16—1221no—
287C2×Q16C22×D6111yescharacteristic
288C2×Q16C22×D6111yescharacteristic
289C2×SD16—21no—
290C2×SD16—1221no—
291C2×D8C22×D6111yescharacteristic
292C4⋊Q8—1221no—
293C4⋊Q8—1221no—
294C4⋊Q8—2441no—
295#126—2441no—
296#126—2441no—
297#125—1221no—
298#125—1221no—
299#125—2441no—
300#125—2441no—
301#124—21no—
302#124—1221no—
303C22⋊Q8—2441no—
304C4×Q8—21no—
305C4×Q8—611no—
306C4×Q8—1221no—
307C4×Q8—1221no—
308C4×Q8—1221no—
309C4×Q8—1221no—
310C4×Q8—2441no—
311C4×D4—1221no—
312#117—2441no—
313Q32—22no—
314Q32—1222no—
315SD32—22no—
316C2×C16—611no—
317C2×C16—611no—
318C2×C16C22×D6111yescharacteristic
319C2×C16C22×D6111yescharacteristic
320#107—611no—
321#107—611no—
322#107—611no—
323#107—1221no—
324#107C22×D6111yescharacteristic
325#106—1221no—
326#106—1221no—
327C4⋊C8—611no—
328C4⋊C8—611no—
329C4⋊C8—611no—
330#103—21no—
331#103—1221no—
332#103—1221no—
333#103—1221no—
334#103—1221no—
335#102—1221no—
336C4×C8—611no—
337C4×C8—611no—
338C4×C8C22×D6111yescharacteristic
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—611no—
342C2×Q8—1221no—
343C2×Q8—2441no—
344C2×D4—21no—
345C22×C4—2441no—
346Q16—21no—
347Q16—21no—
348Q16—21no—
349Q16—1221no—
350Q16—1221no—
351Q16—4881no—
352Q16#1175822412yes—
353SD16—81no—
354SD16—4881no—
355D8—21no—
356D8—21no—
357C2×C8—611no—
358C2×C8—611no—
359C2×C8D4×D6111yescharacteristic
360C2×C8C23×D6111yescharacteristic
361C4⋊C4—21no—
362C4⋊C4—41no—
363C4⋊C4—611no—
364C4⋊C4—611no—
365C4⋊C4—611no—
366C4⋊C4—1221no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C4⋊C4—2441no—
370C4⋊C4—2441no—
371C4⋊C4—2441no—
372C4⋊C4—2441no—
373C4⋊C4—2441no—
374C4⋊C4—2441no—
375C4⋊C4—2441no—
376C4⋊C4—2441no—
377C4⋊C4—2441no—
378C4⋊C4—2441no—
379C4⋊C4—2441no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—2441no—
383C22⋊C4—2441no—
384C22⋊C4—2441no—
385C42—41no—
386C42—611no—
387C42—611no—
388C42—2441no—
389C42—2441no—
390C42—2441no—
391C42—2441no—
392C42D4×D6111yescharacteristic
393C16—21no—
394C16—21no—
395C16—1221no—
396C16—1221no—
397C23—41no—
398Q8—22no—
399Q8—41no—
400Q8—41no—
401Q8—41no—
402Q8—1222no—
403Q8—4882no—
404Q8—2441no—
405Q8—2441no—
406D4—41no—
407D4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—611no—
412C2×C4—611no—
413C2×C4—611no—
414C2×C4—611no—
415C2×C4—4882no—
416C2×C4—2441no—
417C2×C4—2441no—
418C2×C4—2441no—
419C2×C4—2441no—
420C2×C4D8×D6111yescharacteristic
421C2×C4D8×D6111yescharacteristic
422C2×C4C2×D4×D6111yescharacteristic
423C8—21no—
424C8—1221no—
425C8—1221no—
426C8C2×#1175822411yescharacteristic
427C8#1451982411yescharacteristic
428C22—81no—
429C22—81no—
430C22—81no—
431C22C2×D8×D6111yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—81no—
436C4—81no—
437C4—612no—
438C4—1221no—
439C4—1221no—
440C4—1221no—
441C4—4881no—
442C4—4881no—
443C4—4881no—
444C4—4881no—
445C4#1684041811yescharacteristic
446C4#1684057211yescharacteristic
447C2—161no—
448C2#2145844911yescharacteristic
449C2#508×D6111yescharacteristic
450C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2211C2
socleC22×C61C22×D8
prime radicalC2×D8×C61C23
hypercenter#2211D61
hypocenterC61#33931
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2211

Complements

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

NH ≅ G/Ntype
2. #21458710C2semidirect
5. #21448045C2semidirect
6. #21447984C2semidirect
7. #21447981C2semidirect
10. #21447867C2semidirect
11. #21447721C2semidirect
12. Q32×Dic61C2semidirect
13. C4×#14519723C2semidirect
284. C61#33931semidirect

Maximal subgroups

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

subgroupindex
#214588042
#214587102
#214480502
#214480492
#214480452
#214479842
#214479812
#214479542
#214478792
#214478672
#214477212
Q32×Dic612
C4×#145197232
SD32×Dic612
C61×#22112
#3393161