Finite Group Databasebeta

← #29668001

Subgroups — #29668001

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456964C211yesmaximal, characteristic
2#21456959C211yesmaximal, characteristic
3#21456404C211yesmaximal, characteristic
4#500×D61C211yesmaximal, characteristic
5#21456196C211yesmaximal, characteristic
6#21456196C211yesmaximal, characteristic
7#21445468C211yesmaximal, characteristic, complemented
8#21445465C211yesmaximal, characteristic, complemented
9#21445464C211yesmaximal, characteristic, complemented
10#21445427C211yesmaximal, characteristic, complemented
11#21445427C211yesmaximal, characteristic, complemented
12#21445415C211yesmaximal, characteristic, complemented
13#21445414C211yesmaximal, characteristic, complemented
14#21444098C211yesmaximal, characteristic
15C61×#1718C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840317C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20#16840167C2211yescharacteristic
21#16840166—22no—
22#16840165C2211yescharacteristic
23#16840165C2211yescharacteristic
24#16840165C2211yescharacteristic
25#16840160C2211yescharacteristic
26#16840158C2211yescharacteristic
27#16840158C2211yescharacteristic
28#16840141C2211yescharacteristic
29#16840129—22no—
30#16840129—22no—
31#16840114C2211yescharacteristic
32#16840114C2211yescharacteristic
33#16840114C2211yescharacteristic
34#16840113C2211yescharacteristic
35#16840113C2211yescharacteristic
36#16840113C2211yescharacteristic
37#16840113C2211yescharacteristic
38#16840112C2211yescharacteristic
39#16840112C2211yescharacteristic
40#16840112C2211yescharacteristic
41#16840094—22no—
42#16840091—22no—
43#16840046—22no—
44#16840045—22no—
45#16840002—22no—
46#16839854—22no—
47#16839813C2211yescharacteristic
48#107×D61—22no—
49#16839769—22no—
50#16839769—22no—
51C4×#11758189C2211yescharacteristic
52C4×C8×D61C2211yescharacteristic
53#16839484C2211yescharacteristic
54C8×#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#500C2211yescharacteristic
57C61×#498C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#487C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#487C2211yescharacteristic
62C61×#322C2211yescharacteristic
63C2×Q8×D61—21no—
64C2×Q8×D61—21no—
65C2×#11758202—22no—
66C2×#11758189D412yes—
67C2×C8×D61D412yes—
68#14519777C2311yescharacteristic
69#14519777C2311yescharacteristic
70#14519777C2311yescharacteristic
71#14519777C2311yescharacteristic
72#14519776—21no—
73#14519776—21no—
74#14519776—21no—
75#14519776—21no—
76#14519775—21no—
77#14519775—21no—
78#14519768—21no—
79#14519768—21no—
80#14519757—21no—
81#14519757—21no—
82#14519753—21no—
83#14519753—21no—
84Q8×Dic61—21no—
85Q8×Dic61—21no—
86#14519742—21no—
87#14519742—21no—
88#14519735—21no—
89#14519735—21no—
90C4⋊C4×D61—21no—
91C4⋊C4×D61—21no—
92C4⋊C4×D61—21no—
93C4⋊C4×D61—21no—
94C42×D61C2311yescharacteristic
95#14519711—22no—
96#14519710—22no—
97#14519710—22no—
98#14519710—22no—
99#14519709—22no—
100#14519708—22no—
101#14519708—22no—
102#14519706D412yes—
103#14519702—22no—
104#14519702—22no—
105#14519702—22no—
106#14519702—22no—
107#14519701—22no—
108#14519701—22no—
109#14519701—22no—
110#14519700—22no—
111#14519700—22no—
112#14519700—22no—
113#14519689D412yes—
114#14519687D412yes—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C4×#92418C2311yescharacteristic
118C8×Dic61D412yes—
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123#107×C61—22no—
124#106×C61—22no—
125#103×C61—22no—
126#103×C61—22no—
127#103×C61—22no—
128#103×C61—22no—
129C4×C8×C61C2311yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C4×C8×C61C2311yescharacteristic
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136Q8×D61—44no—
137Q8×D61—44no—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×C4×D61C2×D411yescharacteristic
141C2×C4×D61C2×D411yescharacteristic
142C2×C4×D61C2×D411yescharacteristic
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—41no—
148C2×Dic122—41no—
149C2×C8×C61C2×D412yes—
150C2×C8×C61C2×D412yes—
151C2×C8×C61C2×D412yes—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C42×C61C2411yescharacteristic
161#11758202—44no—
162#11758198—41no—
163#11758198—41no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—41no—
173C4⋊Dic61—41no—
174#11758196—41no—
175#11758196—41no—
176#11758196—41no—
177#11758196—41no—
178#11758196—41no—
179#11758196—41no—
180C4×Dic61—41no—
181C4×Dic61—41no—
182C4×Dic61C2×D411yescharacteristic
183C4×Dic61C2×D411yescharacteristic
184C4×Dic61C2×D411yescharacteristic
185C2×#92418C2×D412yes—
186C2×#92418C2×D412yes—
187C2×#92418C2×D412yes—
188#11758189—24no—
189C8×D61—24no—
190C22×D61#12711yescharacteristic
191Q8×C61—42no—
192Q8×C61—42no—
193Q8×C61—42no—
194Q8×C61—42no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61#12711yescharacteristic
207C2×Dic61#12711yescharacteristic
208C2×Dic61#12711yescharacteristic
209C4×D61—22no—
210C4×D61—22no—
211C4×D61—82no—
212C4×D61—82no—
213C4×D61#13514yes—
214Dic122—42no—
215Dic122—42no—
216Dic122—42no—
217Dic122—42no—
218Dic122—82no—
219Dic122—82no—
220C8×C61—22no—
221C8×C61—22no—
222C8×C61—22no—
223#92418—22no—
224#92418—22no—
225#92418#13714yes—
226#19636—611nomaximal, Sylow
227C22×C61C2×#12711yescharacteristic
228C2×D61—21no—
229C2×D61—21no—
230C2×D61#49512yes—
231C4×C61—81no—
232C4×C61—81no—
233C4×C61—81no—
234C4×C61—81no—
235C4×C61C2×#13512yes—
236C4×C61C2×#13712yes—
237C4×C61#57812yes—
238Dic61—21no—
239Dic61—21no—
240Dic61—81no—
241Dic61—81no—
242Dic61—81no—
243Dic61—81no—
244Dic61#49512yes—
245#3229—611no—
246#3229—611no—
247#3170—611no—
248C2×#500—611no—
249#3139—611no—
250#3139—611no—
251#1718—611no—
252#1718—611no—
253#1718—611no—
254#1718D6111yescharacteristic, complemented
255#1710—611no—
256#1710—611no—
257#1707—611no—
258#1707—611no—
259C8×M16—611no—
260C2×C61C2×#49511yescharacteristic
261C2×C61#315211yescharacteristic
262C2×C61#315611yescharacteristic
263D61—22no—
264#533—611no—
265#533—611no—
266C2×C4⋊Q8—611no—
267C2×C4⋊Q8—611no—
268#500—611no—
269#500—611no—
270#500—611no—
271#500—611no—
272#500—1222no—
273#500C2×D6111yescharacteristic
274#498—611no—
275#498—611no—
276#498—611no—
277#498C2×D6111yescharacteristic
278#494—611no—
279#491—1222no—
280#491—1222no—
281#487—611no—
282#487—611no—
283#487—611no—
284#487—611no—
285#487—611no—
286#487—611no—
287#487—611no—
288#487—611no—
289#487—611no—
290#487—611no—
291#487C2×D6111yescharacteristic
292#487C2×D6111yescharacteristic
293#487C2×D6111yescharacteristic
294#487C2×D6111yescharacteristic
295#483—1222no—
296#483—1222no—
297#475—1222no—
298#475—1222no—
299#467—1222no—
300C4×Q16—1222no—
301#431—611no—
302C2×#107—1222no—
303#419—1222no—
304#419—1222no—
305C4×M16—611no—
306C2×C4×C8—611no—
307#322—611no—
308#322C2×D6111yescharacteristic
309C82—611no—
310C82—611no—
311C61#1963611yescharacteristic, complemented, Sylow
312C22×Q8—1221no—
313C22×Q8—1221no—
314C2×Q16—1222no—
315C2×M16—612no—
316C22×C8—612no—
317C4⋊Q8—611no—
318C4⋊Q8—611no—
319C4⋊Q8—611no—
320C4⋊Q8—611no—
321C4⋊Q8—1221no—
322C4⋊Q8—1221no—
323C4⋊Q8—1221no—
324C4⋊Q8—1221no—
325C4⋊Q8—1221no—
326C4⋊Q8—1221no—
327C4⋊Q8C22×D6111yescharacteristic
328C4⋊Q8C22×D6111yescharacteristic
329C4⋊Q8C22×D6111yescharacteristic
330C4⋊Q8C22×D6111yescharacteristic
331#125—1221no—
332#125—1221no—
333C22⋊Q8—1221no—
334C22⋊Q8—1221no—
335C22⋊Q8—1221no—
336C22⋊Q8—1221no—
337C4×Q8—1221no—
338C4×Q8—1221no—
339C4×Q8—1221no—
340C4×Q8—1221no—
341#117—1221no—
342#117—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C42—611no—
348#107—22no—
349#107—1222no—
350#107—1222no—
351#107—1222no—
352#107—1222no—
353#106—22no—
354#106—1222no—
355#106—1222no—
356#106—1222no—
357C4⋊C8—612no—
358#103—22no—
359#103—22no—
360#103—22no—
361#103—22no—
362#103—1222no—
363#103—1222no—
364#103—1222no—
365#103—1222no—
366#103—1222no—
367#103—1222no—
368#103—1222no—
369#103—1222no—
370#103—1222no—
371#103—1222no—
372C22⋊C8—612no—
373#97—612no—
374C4×C8—612no—
375C4×C8—611no—
376C4×C8—611no—
377C4×C8—611no—
378C4×C8C22×D6111yescharacteristic
379C4×C8C22×D6111yescharacteristic
380C4×C8C22×D6111yescharacteristic
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—1221no—
386C2×Q8—1221no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—2441no—
390C2×Q8—2441no—
391C2×Q8—2444no—
392C2×Q8—2444no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—2441no—
397C22×C4—2441no—
398Q16—2444no—
399M16—1224no—
400C2×C8—612no—
401C2×C8—612no—
402C2×C8—612no—
403C2×C8—1224no—
404C2×C8D4×D6112yes—
405C2×C8D4×D6112yes—
406C2×C8D4×D6112yes—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—1221no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4—1221no—
419C4⋊C4—1221no—
420C4⋊C4—1221no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—2441no—
424C4⋊C4—2441no—
425C4⋊C4—2441no—
426C4⋊C4—2441no—
427C4⋊C4—2441no—
428C4⋊C4—2441no—
429C4⋊C4—2441no—
430C4⋊C4—2441no—
431C22⋊C4—2441no—
432C22⋊C4—2441no—
433C42—611no—
434C42—611no—
435C42—611no—
436C42—2441no—
437C42—2441no—
438C42C23×D6111yescharacteristic
439C23—611no—
440Q8—42no—
441Q8—42no—
442Q8—42no—
443Q8—42no—
444Q8—4882no—
445Q8—4882no—
446Q8—2442no—
447Q8—2442no—
448Q8—2442no—
449Q8—2442no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—611no—
455C2×C4—614no—
456C2×C4—611no—
457C2×C4—611no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—4882no—
461C2×C4—4882no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4C2×D4×D6111yescharacteristic
467C2×C4C2×D4×D6111yescharacteristic
468C2×C4C2×D4×D6111yescharacteristic
469C8—22no—
470C8—22no—
471C8—22no—
472C8—614no—
473C8—1222no—
474C8—1222no—
475C22—612no—
476C22—1221no—
477C22—1221no—
478C22#127×D6111yescharacteristic
479C4—81no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—612no—
484C4—1221no—
485C4—1221no—
486C4—4881no—
487C4—4881no—
488C4—4881no—
489C4—4881no—
490C4#135×D6112yes—
491C4#137×D6112yes—
492C4#1684057812yes—
493C2—1222no—
494C2#495×D6111yescharacteristic
495C2#2145627511yescharacteristic
496C2#2145631111yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1718C2
socleC22×C61C2×#127
prime radicalC22×D61#127
hypercenter#1718D61
hypocenterC61#19636
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1718

Complements

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

NH ≅ G/Ntype
7. #21445468C2semidirect
8. #21445465C2semidirect
9. #21445464C2semidirect
10. #21445427C2semidirect
11. #21445427C2semidirect
12. #21445415C2semidirect
13. #21445414C2semidirect
15. C61×#1718C2semidirect
254. #1718D61semidirect
311. C61#19636semidirect

Maximal subgroups

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

subgroupindex
#214569642
#214569592
#214564042
#500×D612
#214561962
#214454682
#214454652
#214454642
#214454272
#214454152
#214454142
#214440982
C61×#17182
#1963661