Finite Group Databasebeta

← #29835152

Subgroups — #29835152

Order 15616.

534 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456471C211yesmaximal, characteristic, complemented
2#21454695C211yesmaximal, characteristic
3#21448022C211yesmaximal, characteristic, complemented
4#21448014C211yesmaximal, characteristic, complemented
5#21447985C211yesmaximal, characteristic, complemented
6#21447974C211yesmaximal, characteristic, complemented
7#21447867C211yesmaximal, characteristic, complemented
8#21447862C211yesmaximal, characteristic, complemented
9#21447759C211yesmaximal, characteristic
10#21447687C211yesmaximal, characteristic, complemented
11#21447684C211yesmaximal, characteristic, complemented
12Q32×Dic61C211yesmaximal, characteristic, complemented
13#21447674C211yesmaximal, characteristic, complemented
14#21447623C211yesmaximal, characteristic, complemented
15C61×#2189C211yesmaximal, characteristic
16#16840372—21no—
17#16840245—21no—
18C2×#14519723C2211yescharacteristic
19C2×#14519721C2211yescharacteristic
20#16840163C2211yescharacteristic
21#16840140C2211yescharacteristic
22#16840129—21no—
23#16840018—21no—
24#16840003C2211yescharacteristic
25#16839866—21no—
26#16839863—21no—
27#16839863—21no—
28Q16×Dic61C2211yescharacteristic
29Q16×Dic61C2211yescharacteristic
30Q16×Dic61C2211yescharacteristic
31#16839856C2211yescharacteristic
32D8×Dic61C2211yescharacteristic
33D8×Dic61C2211yescharacteristic
34C2×#14519711C2211yescharacteristic
35#16839763—21no—
36#16839717C2211yescharacteristic
37#16839589C2211yescharacteristic
38#16839589C2211yescharacteristic
39#16839588C2211yescharacteristic
40#16839588C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839566C2211yescharacteristic
47#16839566C2211yescharacteristic
48#16839563C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×#135×C61C2211yescharacteristic
53C2×Q32×C61C2211yescharacteristic
54C2×SD32×C61C2211yescharacteristic
55C61×#467C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic
59C2×Dic244C2311yescharacteristic
60C2×#11758202—21no—
61C2×#11758194D411yescharacteristic
62#14519775—21no—
63#14519770—41no—
64#14519770—41no—
65#14519768—21no—
66#14519768—41no—
67#14519767—41no—
68#14519756—21no—
69Q8×Dic61—21no—
70Q8×Dic61—21no—
71Q8×Dic61—21no—
72#14519742—21no—
73D4×Dic61—21no—
74D4×Dic61—21no—
75D4×Dic61—41no—
76D4×Dic61—41no—
77#14519736—41no—
78#14519736D411yescharacteristic
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—41no—
81#14519723—22no—
82#14519721—22no—
83C2×#11758185C2311yescharacteristic
84C2×#11758185C2311yescharacteristic
85#14519711—21no—
86#14519711C2311yescharacteristic
87#14519711C2311yescharacteristic
88#14519711C2311yescharacteristic
89#14519708—21no—
90#14519702—21no—
91#14519701—21no—
92#14519701—21no—
93#14519701—21no—
94#14519700—21no—
95#14519698—21no—
96#14519698—21no—
97#14519687D411yescharacteristic
98#14519687D411yescharacteristic
99C8×Dic61C2311yescharacteristic
100C8×Dic61C2311yescharacteristic
101C2×#41×C61—21no—
102#135×C61—22no—
103#135×C61—22no—
104C2×Q16×C61C2311yescharacteristic
105C2×Q16×C61C2311yescharacteristic
106C2×SD16×C61—21no—
107C2×D8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22⋊Q8×C61—21no—
110Q32×C61—22no—
111SD32×C61—22no—
112C2×C16×C61C2311yescharacteristic
113C2×C16×C61C2311yescharacteristic
114#107×C61C2311yescharacteristic
115#103×C61—21no—
116#41×C61—24no—
117#41×C61—42no—
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61C2×D411yescharacteristic
124C22×Dic61—41no—
125C22×Dic61—41no—
126C22×Dic61—41no—
127C2×Dic122—21no—
128Q16×C61—21no—
129Q16×C61—21no—
130Q16×C61—21no—
131Q16×C61#4112yes—
132SD16×C61—24no—
133D8×C61—21no—
134D8×C61#4112yes—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2411yescharacteristic
139C4⋊C4×C61—21no—
140C4⋊C4×C61—41no—
141C22⋊C4×C61—41no—
142C22⋊Dic61—41no—
143C22⋊Dic61—41no—
144C22⋊Dic61—41no—
145C22⋊Dic61D811yescharacteristic
146C22⋊Dic61D811yescharacteristic
147#11758202—42no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—41no—
152C4⋊Dic61—41no—
153C4⋊Dic61—41no—
154C4⋊Dic61—41no—
155C4⋊Dic61—41no—
156C4⋊Dic61D811yescharacteristic
157C4⋊Dic61D811yescharacteristic
158#11758196—41no—
159#11758196—41no—
160#11758196—41no—
161#11758196—41no—
162#11758196—41no—
163C4×Dic61—41no—
164C4×Dic61—41no—
165C4×Dic61—41no—
166C4×Dic61—41no—
167C4×Dic61C2×D411yescharacteristic
168C4×Dic61C2×D411yescharacteristic
169#11758194—41no—
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172Dic244—21no—
173Dic244—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C23×C61C2×D811yescharacteristic
180Q8×C61—22no—
181Q8×C61—41no—
182Q8×C61—41no—
183Q8×C61—41no—
184D4×C61—22no—
185D4×C61—44no—
186D4×C61—41no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—44no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—161no—
200C2×Dic61—161no—
201C2×Dic61—161no—
202C2×Dic61C2×D811yescharacteristic
203C2×Dic61C2×D811yescharacteristic
204C2×Dic61C2×D811yescharacteristic
205C2×Dic61C2×D811yescharacteristic
206Dic122—41no—
207Dic122—41no—
208C8×C61—21no—
209C8×C61C2×#4111yescharacteristic
210C8×C61C2×#4111yescharacteristic
211#92418—21no—
212#92418—21no—
213#33680—611nomaximal, Sylow
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—42no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61C22×D811yescharacteristic
221C4×C61—21no—
222C4×C61—42no—
223C4×C61—81no—
224C4×C61—81no—
225C4×C61#52511yescharacteristic
226C4×C61#57511yescharacteristic
227Dic61—21no—
228Dic61—21no—
229Dic61—21no—
230Dic61—21no—
231Dic61—81no—
232Dic61—81no—
233Dic61—81no—
234Dic61—81no—
235#3177—611no—
236#2951—611no—
237#2236—611no—
238#2235—611no—
239#2229—611no—
240#2228—611no—
241#2209—611no—
242#2208—611no—
243#2189—611no—
244#2189D6111yescharacteristic
245#2176—611no—
246#2175—611no—
247C4×Q32—611no—
248C4×SD32—611no—
249#2164—611no—
250C2×C61—41no—
251C2×C61—82no—
252C2×C61#310011yescharacteristic
253C2×C61#341611yescharacteristic
254C2×C61#341711yescharacteristic
255C2×#135C2×D6111yescharacteristic
256#541—1221no—
257#520—1221no—
258C2×Q32—611no—
259C2×Q32C2×D6111yescharacteristic
260C2×SD32—611no—
261C2×SD32C2×D6111yescharacteristic
262#499—611no—
263#494—611no—
264#491—1221no—
265#470—1221no—
266#467—611no—
267#467C2×D6111yescharacteristic
268#441—1221no—
269#440—1221no—
270#440—1221no—
271C4×Q16—611no—
272C4×Q16—611no—
273C4×Q16—611no—
274C4×Q16—611no—
275C4×D8—611no—
276C4×D8—611no—
277C2×#107—611no—
278#417—1221no—
279#405—611no—
280#367—611no—
281#367—611no—
282#366—611no—
283#366—611no—
284C4⋊C16—611no—
285C4⋊C16—611no—
286#358—611no—
287#358—611no—
288#358—611no—
289#358—611no—
290#358—611no—
291#358C2×D6111yescharacteristic
292#357—611no—
293#357C2×D6111yescharacteristic
294C22⋊C16—611no—
295C22⋊C16C2×D6111yescharacteristic
296#346—611no—
297C4×C16—611no—
298C61#3368011yescharacteristic, complemented, Sylow
299C2×#41—21no—
300#135—22no—
301#135—22no—
302C2×Q16—611no—
303C2×Q16—1221no—
304C2×Q16C22×D6111yescharacteristic
305C2×Q16C22×D6111yescharacteristic
306C2×SD16—21no—
307C2×D8C22×D6111yescharacteristic
308C2×M16—611no—
309C22×C8C22×D6111yescharacteristic
310C4⋊Q8—1221no—
311#126—2441no—
312#126—2441no—
313#125—1221no—
314#125—2441no—
315#125—2441no—
316C22⋊Q8—21no—
317C22⋊Q8—1221no—
318C4×Q8—1221no—
319C4×Q8—1221no—
320C4×Q8—1221no—
321C4×Q8—1221no—
322C4×D4—1221no—
323C4×D4—1221no—
324C4×D4—2441no—
325C4×D4—2441no—
326#117—611no—
327#117—2441no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—2441no—
330Q32—22no—
331Q32—1222no—
332SD32—22no—
333SD32—1222no—
334C2×C16—611no—
335C2×C16—611no—
336C2×C16C22×D6111yescharacteristic
337C2×C16C22×D6111yescharacteristic
338#107—611no—
339#107—611no—
340#107—611no—
341#107—1221no—
342#107C22×D6111yescharacteristic
343#106—1221no—
344#103—21no—
345#103—1221no—
346#103—1221no—
347#103—1221no—
348#103—1221no—
349#103—1221no—
350#102—1221no—
351#102—1221no—
352#97—611no—
353#97—611no—
354C4×C8—611no—
355C4×C8—611no—
356#41—24no—
357#41—42no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—1221no—
361C2×D4—21no—
362C2×D4—41no—
363C22×C4—41no—
364C22×C4—2441no—
365C22×C4—2441no—
366C22×C4—2441no—
367C22×C4D4×D6111yescharacteristic
368Q16—21no—
369Q16—21no—
370Q16—21no—
371Q16—1221no—
372Q16—1221no—
373Q16—2442no—
374Q16#1175822412yes—
375SD16—24no—
376D8—21no—
377D8#1175822412yes—
378M16—2441no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—611no—
382C2×C8—611no—
383C2×C8D4×D6111yescharacteristic
384C2×C8C23×D6111yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—41no—
387C4⋊C4—611no—
388C4⋊C4—611no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—2441no—
393C4⋊C4—2441no—
394C4⋊C4—2441no—
395C4⋊C4—2441no—
396C4⋊C4—2441no—
397C4⋊C4—2441no—
398C4⋊C4—2441no—
399C4⋊C4—2441no—
400C4⋊C4—2441no—
401C4⋊C4—2441no—
402C22⋊C4—41no—
403C22⋊C4—611no—
404C22⋊C4—611no—
405C22⋊C4—2441no—
406C22⋊C4—2441no—
407C22⋊C4—2441no—
408C42—611no—
409C42—611no—
410C42—2441no—
411C42—2441no—
412C42—2441no—
413C42—2441no—
414C16—21no—
415C16—21no—
416C16—1221no—
417C16—1221no—
418C23—41no—
419C23D8×D6111yescharacteristic
420Q8—22no—
421Q8—41no—
422Q8—41no—
423Q8—41no—
424Q8—2441no—
425Q8—2441no—
426D4—22no—
427D4—44no—
428D4—41no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—611no—
436C2×C4—611no—
437C2×C4—611no—
438C2×C4—611no—
439C2×C4—9761no—
440C2×C4—9761no—
441C2×C4—9761no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4D8×D6111yescharacteristic
447C2×C4C2×D4×D6111yescharacteristic
448C8—21no—
449C8—1221no—
450C8—1221no—
451C8C2×#1175822411yescharacteristic
452C8C2×#1175822411yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—42no—
457C22—81no—
458C22—81no—
459C22C2×D8×D6111yescharacteristic
460C4—21no—
461C4—42no—
462C4—81no—
463C4—81no—
464C4—1221no—
465C4—1221no—
466C4—1221no—
467C4—1221no—
468C4—4881no—
469C4—4881no—
470C4—4881no—
471C4—4881no—
472C4#1684026811yescharacteristic
473C4#1684055811yescharacteristic
474C2—41no—
475C2—82no—
476C2#2145581411yescharacteristic
477C2#2145896411yescharacteristic
478C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2189C2
socleC22×C61C22×D8
prime radicalC2×#135×C61C22
hypercenter#2189D61
hypocenterC61#33680
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2189

Complements

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

NH ≅ G/Ntype
1. #21456471C2semidirect
3. #21448022C2semidirect
4. #21448014C2semidirect
5. #21447985C2semidirect
6. #21447974C2semidirect
7. #21447867C2semidirect
8. #21447862C2semidirect
10. #21447687C2semidirect
11. #21447684C2semidirect
12. Q32×Dic61C2semidirect
13. #21447674C2semidirect
14. #21447623C2semidirect
298. C61#33680semidirect

Maximal subgroups

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

subgroupindex
#214564712
#214546952
#214480222
#214480142
#214479852
#214479742
#214478672
#214478622
#214477592
#214476872
#214476842
Q32×Dic612
#214476742
#214476232
C61×#21892
#3368061