Finite Group Databasebeta

← #29835165

Subgroups — #29835165

Order 15616.

532 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#21456455C211yesmaximal, characteristic
2#21454682C211yesmaximal, characteristic, complemented
3#21448026C211yesmaximal, characteristic, complemented
4#21448008C211yesmaximal, characteristic
5#21447972C211yesmaximal, characteristic, complemented
6#21447961C211yesmaximal, characteristic, complemented
7#21447878C211yesmaximal, characteristic, complemented
8#21447751C211yesmaximal, characteristic, complemented
9#21447734C211yesmaximal, characteristic
10#21447687C211yesmaximal, characteristic, complemented
11#21447684C211yesmaximal, characteristic, complemented
12Q32×Dic61C211yesmaximal, characteristic, complemented
13#21447674C211yesmaximal, characteristic, complemented
14#21447625C211yesmaximal, characteristic, complemented
15C61×#2211C211yesmaximal, characteristic, complemented
16C2×Q16×D61C2211yescharacteristic
17#16840381—21no—
18#16840238—21no—
19C2×#14519723C2211yescharacteristic
20C2×#14519721C2211yescharacteristic
21#16840167C2211yescharacteristic
22#16840121—21no—
23#16840005—21no—
24#16840000C2211yescharacteristic
25#16839995C2211yescharacteristic
26#16839867—21no—
27#16839866—21no—
28#16839863—21no—
29C4×Dic244C2211yescharacteristic
30Q16×Dic61C2211yescharacteristic
31Q16×Dic61C2211yescharacteristic
32#16839856C2211yescharacteristic
33D8×Dic61C2211yescharacteristic
34#16839799C2211yescharacteristic
35#16839769—21no—
36#16839716C2211yescharacteristic
37#16839589C2211yescharacteristic
38#16839589C2211yescharacteristic
39#16839588C2211yescharacteristic
40#16839588C2211yescharacteristic
41C4⋊#11758185C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839566C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839541C2211yescharacteristic
49#16839541C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×Q32×C61C2211yescharacteristic
53C2×SD32×C61C2211yescharacteristic
54C61×#495C2211yescharacteristic
55C4×Q16×C61C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×Q8×D61—21no—
60C2×Dic244C2311yescharacteristic
61C2×#11758202—21no—
62Q16×D61—22no—
63Q16×D61—22no—
64C2×#11758189D411yescharacteristic
65C2×C8×D61C2311yescharacteristic
66#14519777—21no—
67#14519770—41no—
68#14519763—41no—
69#14519761—41no—
70#14519754—41no—
71#14519753—21no—
72#14519748—21no—
73Q8×Dic61—21no—
74Q8×Dic61—21no—
75C4×Dic122—21no—
76C4×Dic122—41no—
77#14519742—21no—
78#14519742—41no—
79D4×Dic61—21no—
80#14519735—21no—
81#14519735—41no—
82#14519734—41no—
83#14519734D411yescharacteristic
84#14519723—22no—
85#14519721—22no—
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88#14519711—21no—
89#14519711C2311yescharacteristic
90#14519711C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519708—21no—
93#14519702—21no—
94#14519701—21no—
95#14519701—21no—
96#14519701—21no—
97#14519700—21no—
98#14519698—21no—
99#14519698—21no—
100#14519688D411yescharacteristic
101#14519687D411yescharacteristic
102C8×Dic61C2311yescharacteristic
103C2×Q16×C61C2311yescharacteristic
104C2×Q16×C61C2311yescharacteristic
105C2×SD16×C61—21no—
106C2×D8×C61C2311yescharacteristic
107#124×C61—21no—
108C4×Q8×C61—21no—
109Q32×C61—22no—
110SD32×C61—22no—
111C2×C16×C61C2311yescharacteristic
112C2×C16×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#103×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×D4×C61—21no—
119C2×C22⋊D61—41no—
120C22×Dic61—41no—
121Q8×D61—24no—
122Q8×D61—42no—
123C2×C4×D61—41no—
124C2×C4×D61C2×D411yescharacteristic
125C2×Dic122—21no—
126C2×Dic122—41no—
127C2×Dic122—41no—
128Q16×C61—21no—
129Q16×C61—21no—
130Q16×C61—21no—
131Q16×C61#4112yes—
132SD16×C61—42no—
133D8×C61—21no—
134D8×C61—21no—
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61C2411yescharacteristic
137C4⋊C4×C61—21no—
138C4⋊C4×C61—41no—
139C22⋊C4×C61—41no—
140C22⋊C4×C61—41no—
141C42×C61—41no—
142C42×C61C2×D411yescharacteristic
143C22⋊Dic61—41no—
144C22⋊Dic61—41no—
145#11758202—24no—
146#11758198—41no—
147#11758198—41no—
148#11758198D811yescharacteristic
149#11758198D811yescharacteristic
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—41no—
154C4⋊Dic61—41no—
155C4⋊Dic61—41no—
156#11758196—41no—
157#11758196—41no—
158#11758196—41no—
159#11758196D811yescharacteristic
160#11758196D811yescharacteristic
161C4×Dic61—41no—
162C4×Dic61—41no—
163C4×Dic61—41no—
164C4×Dic61—41no—
165C4×Dic61—41no—
166C4×Dic61C2×D411yescharacteristic
167C2×#92418C2×D411yescharacteristic
168C2×#92418C2×D411yescharacteristic
169Dic244—21no—
170Dic244#4112yes—
171#11758189—41no—
172C8×D61—21no—
173C8×D61—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C22×D61C2×D811yescharacteristic
180Q8×C61—22no—
181Q8×C61—41no—
182Q8×C61—41no—
183Q8×C61—41no—
184D4×C61—41no—
185D4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61C2×D811yescharacteristic
190C2×C4×C61C2×D811yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C22⋊D61—161no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—161no—
198C2×Dic61C2×D811yescharacteristic
199C2×Dic61C2×D811yescharacteristic
200C2×Dic61C2×D811yescharacteristic
201C4×D61—21no—
202C4×D61—21no—
203C4×D61—44no—
204Dic122—22no—
205Dic122—44no—
206Dic122—41no—
207Dic122—161no—
208C8×C61—21no—
209C8×C61C2×#4111yescharacteristic
210C8×C61C2×#4111yescharacteristic
211#92418—21no—
212#92418—21no—
213#33681—611nomaximal, Sylow
214C22×C61—81no—
215C22×C61—81no—
216C22×C61—81no—
217C22×C61C22×D811yescharacteristic
218C2×D61—21no—
219C2×D61—21no—
220C2×D61—21no—
221C4×C61—21no—
222C4×C61—21no—
223C4×C61—42no—
224C4×C61—81no—
225C4×C61—81no—
226C4×C61#52511yescharacteristic
227C4×C61#57511yescharacteristic
228Dic61—21no—
229Dic61—21no—
230Dic61—21no—
231Dic61—42no—
232Dic61—81no—
233Dic61—81no—
234Dic61—81no—
235#3175—611no—
236#2950—611no—
237#2237—611no—
238#2234—611no—
239#2227—611no—
240#2225—611no—
241#2211—611no—
242#2211D6111yescharacteristic, complemented
243#2188—611no—
244#2185—611no—
245#2176—611no—
246#2175—611no—
247C4×Q32—611no—
248C4×SD32—611no—
249#2164—611no—
250C2×C61—161no—
251C2×C61#310011yescharacteristic
252C2×C61#341611yescharacteristic
253C2×C61#341711yescharacteristic
254D61—41no—
255C22×Q16—611no—
256#542—1221no—
257#519—1221no—
258C2×Q32—611no—
259C2×Q32C2×D6111yescharacteristic
260C2×SD32—611no—
261C2×SD32C2×D6111yescharacteristic
262#500—611no—
263#495C2×D6111yescharacteristic
264#489—1221no—
265#468—1221no—
266#467—611no—
267#466—611no—
268#441—1221no—
269#441—1221no—
270#440—1221no—
271C4×Q16—611no—
272C4×Q16—611no—
273C4×Q16—611no—
274C4×Q16—611no—
275C4×Q16C2×D6111yescharacteristic
276C4×D8—611no—
277#427—611no—
278#419—1221no—
279#405—611no—
280#367—611no—
281#367—611no—
282#366—611no—
283#366—611no—
284C4⋊C16—611no—
285C4⋊C16C2×D6111yescharacteristic
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⋊C16—611no—
296#346—611no—
297C4×C16—611no—
298C61#3368111yescharacteristic, complemented, Sylow
299C22×Q8—1221no—
300C2×Q16—611no—
301C2×Q16—1221no—
302C2×Q16—1222no—
303C2×Q16—1222no—
304C2×Q16C22×D6111yescharacteristic
305C2×Q16C22×D6111yescharacteristic
306C2×SD16—21no—
307C2×D8C22×D6111yescharacteristic
308C2×M16—611no—
309C22×C8—611no—
310C4⋊Q8—1221no—
311#126—2441no—
312#124—21no—
313#124—2441no—
314#123—2441no—
315C22⋊Q8—1221no—
316C22⋊Q8—2441no—
317#121—1221no—
318C4×Q8—21no—
319C4×Q8—1221no—
320C4×Q8—1221no—
321C4×Q8—1221no—
322C4×Q8—1221no—
323C4×Q8—2441no—
324C4×Q8—2441no—
325C4×D4—1221no—
326#117—611no—
327#117—1221no—
328#117—2441no—
329#117—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×C8C22×D6111yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—1224no—
359C2×Q8—1221no—
360C2×Q8—2441no—
361C2×Q8—2442no—
362C2×Q8—2441no—
363C2×D4—21no—
364C2×D4—2441no—
365C22×C4—611no—
366C22×C4—2441no—
367C22×C4—2441no—
368Q16—21no—
369Q16—21no—
370Q16—21no—
371Q16—612no—
372Q16—1224no—
373Q16—1221no—
374Q16#1175822412yes—
375SD16—42no—
376D8—21no—
377D8—21no—
378M16—2441no—
379C2×C8—611no—
380C2×C8—611no—
381C2×C8—1221no—
382C2×C8—1221no—
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—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—611no—
401C22⋊C4—611no—
402C22⋊C4—2441no—
403C22⋊C4—2441no—
404C22⋊C4—2441no—
405C22⋊C4—2441no—
406C42—41no—
407C42—611no—
408C42—2441no—
409C42—2441no—
410C42—2441no—
411C42—2441no—
412C42—2441no—
413C42D4×D6111yescharacteristic
414C16—21no—
415C16—21no—
416C16—1221no—
417C16—1221no—
418C23—41no—
419C23—611no—
420Q8—22no—
421Q8—41no—
422Q8—41no—
423Q8—41no—
424Q8—1222no—
425Q8—9761no—
426Q8—2444no—
427Q8—2441no—
428D4—41no—
429D4—41no—
430D4—9761no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—611no—
435C2×C4—611no—
436C2×C4—611no—
437C2×C4—1221no—
438C2×C4—1221no—
439C2×C4—9761no—
440C2×C4—2444no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4D8×D6111yescharacteristic
446C2×C4D8×D6111yescharacteristic
447C2×C4C2×D4×D6111yescharacteristic
448C8—21no—
449C8—1221no—
450C8—1221no—
451C8#41×D6111yescharacteristic
452C8C2×#1175822411yescharacteristic
453C22—81no—
454C22—81no—
455C22—81no—
456C22—1221no—
457C22—1221no—
458C22—1221no—
459C22C2×D8×D6111yescharacteristic
460C4—21no—
461C4—21no—
462C4—42no—
463C4—81no—
464C4—81no—
465C4—1221no—
466C4—1221no—
467C4—1221no—
468C4—4881no—
469C4—4881no—
470C4—4881no—
471C4—2442no—
472C4#1684027211yescharacteristic
473C4#1684055811yescharacteristic
474C2—161no—
475C2—2441no—
476C2#2145581811yescharacteristic
477C2#2145896411yescharacteristic
478C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2211C2
socleC22×C61C22×D8
prime radical#16839995C22
hypercenter#2211D61
hypocenterC61#33681
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. #21454682C2semidirect
3. #21448026C2semidirect
5. #21447972C2semidirect
6. #21447961C2semidirect
7. #21447878C2semidirect
8. #21447751C2semidirect
10. #21447687C2semidirect
11. #21447684C2semidirect
12. Q32×Dic61C2semidirect
13. #21447674C2semidirect
14. #21447625C2semidirect
15. C61×#2211C2semidirect
242. #2211D61semidirect
298. C61#33681semidirect

Maximal subgroups

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

subgroupindex
#214564552
#214546822
#214480262
#214480082
#214479722
#214479612
#214478782
#214477512
#214477342
#214476872
#214476842
Q32×Dic612
#214476742
#214476252
C61×#22112
#3368161