Finite Group Databasebeta

← #29732128

Subgroups — #29732128

Order 15616.

664 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455612C211yesmaximal, characteristic
2C2×#16840039C211yesmaximal, characteristic
3#21454768C211yesmaximal, characteristic, complemented
4#21447327C212yesmaximal, complemented
5#21447176C212yesmaximal
6#21446809C211yesmaximal, characteristic, complemented
7#21446601C212yesmaximal, complemented
8#21446594C211yesmaximal, characteristic, complemented
9#21446480C212yesmaximal
10C4×#14519693C211yesmaximal, characteristic
11C61×#3165C211yesmaximal, characteristic, complemented
12C2×C4×D244C2211yescharacteristic
13#16840059—22no—
14#16840049—21no—
15#16840039—21no—
16#16840039—21no—
17#16840039—22no—
18#16840039—21no—
19#16839893—21no—
20#16839893—22no—
21Q8×#92418—21no—
22C2×#14519712C2212yes—
23#16839805C2212yes—
24C2×#14519710C2212yes—
25#16839789C2212yes—
26#16839786C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28#16839766C2212yes—
29C2×#14519697C2212yes—
30C2×#14519693C2211yescharacteristic
31C2×#14519693C2211yescharacteristic
32#16839718C2212yes—
33C4×#11758194C2211yescharacteristic
34#16839531C2212yes—
35#16839531C2212yes—
36#16839530C2211yescharacteristic
37#16839530C2211yescharacteristic
38#16839526C2212yes—
39C61×#533C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C61×#501—22no—
42C61×#500—21no—
43C61×#498—21no—
44C61×#428C2212yes—
45C61×#428C2212yes—
46C4×M16×C61C2211yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C2×#11758194C2311yescharacteristic
50C2×#11758194C2312yes—
51C2×#11758194C2311yescharacteristic
52C22×#92418C2312yes—
53C4×D244—21no—
54C4×D244—21no—
55C4×D244—21no—
56C4×D244—21no—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×#11758198—21no—
59#14519712—24no—
60#14519710—22no—
61#14519710—22no—
62#14519710—22no—
63#14519708—22no—
64C4⋊#92418—21no—
65C4⋊#92418—21no—
66C4⋊#92418—22no—
67C4⋊#92418—22no—
68C4⋊#92418D411yescharacteristic
69C4⋊#92418D412yes—
70C4⋊#92418D411yescharacteristic
71#14519697—22no—
72#14519697—22no—
73#14519697—22no—
74#14519697—22no—
75#14519693—22no—
76#14519693Q814yescomplemented
77#14519688—21no—
78#14519688—22no—
79C4×#92418—21no—
80C4×#92418—22no—
81C22×Q8×C61—21no—
82C2×M16×C61C2311yescharacteristic
83C2×M16×C61C2311yescharacteristic
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61D411yescharacteristic
89C4⋊Q8×C61D411yescharacteristic
90#125×C61D412yes—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95#117×C61C2312yes—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2312yes—
98C2×C42×C61C2311yescharacteristic
99#107×C61—22no—
100#107×C61—22no—
101#106×C61—22no—
102#106×C61—22no—
103#97×C61—21no—
104C4×C8×C61—21no—
105C23×D61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61—41no—
109C2×Q8×C61—41no—
110C2×Q8×C61—82no—
111C22×C4×C61—22no—
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61C2411yescharacteristic
115C22×Dic61—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244C2×Q812yes—
122C2×C4×D61—41no—
123C2×C4×D61—41no—
124C2×C4×D61—82no—
125M16×C61—21no—
126M16×C61—21no—
127M16×C61C2×Q812yes—
128M16×C61#4112yes—
129C2×C8×C61—21no—
130C2×C8×C61—21no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—41no—
138C4⋊C4×C61—41no—
139C4⋊C4×C61C2×D412yes—
140C4⋊C4×C61C2×D412yes—
141C22⋊C4×C61—22no—
142C42×C61—22no—
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147#11758198—41no—
148#11758198—41no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151#11758194—21no—
152#11758194—21no—
153#11758194—22no—
154#11758194—21no—
155#11758194—22no—
156#11758194C2×Q812yes—
157C2×#92418—21no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—21no—
161C2×#92418—22no—
162C2×#92418C2×D412yes—
163C2×#92418C2×D412yes—
164C23×C61C2×#4111yescharacteristic
165C22×D61—22no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—81no—
169C22×D61—81no—
170Q8×C61—42no—
171Q8×C61—81no—
172Q8×C61—82no—
173Q8×C61—82no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—82no—
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#12111yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×Q811yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C2×C4×C612−1+411yescharacteristic
189C2×C4×C612−1+411yescharacteristic
190C2×Dic61—41no—
191C2×Dic61—81no—
192D244—22no—
193D244—22no—
194D244—41no—
195D244—41no—
196C4×D61—82no—
197C4×D61—82no—
198C8×C61—22no—
199C8×C61—22no—
200#92418—22no—
201#92418—22no—
202#92418—22no—
203#92418—41no—
204#92418—42no—
205#25489—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61C2×#12111yescharacteristic
209C22×C61#53311yescharacteristic
210C22×C61#54111yescharacteristic
211C2×D61—42no—
212C2×D61—82no—
213C2×D61—81no—
214C2×D61—81no—
215C2×D61—82no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—42no—
220C4×C61—42no—
221C4×C61—82no—
222C4×C61#47212yes—
223C4×C61#54711yescharacteristic
224C4×C61#54811yescharacteristic
225C4×C61D4×Q811yescharacteristic
226C4×C61D4×Q811yescharacteristic
227Dic61—81no—
228#3165D6111yescharacteristic, complemented
229#3077—611no—
230C2×#474—611no—
231#2962—611no—
232#2089—612no—
233#2063—612no—
234#1997—611no—
235#1952—612no—
236#1950—611no—
237#1931—612no—
238C4×#100—611no—
239C2×C61—21no—
240C2×C61—21no—
241C2×C61#242911yescharacteristic
242C2×C61C2×#47211yescharacteristic
243C2×C61#306911yescharacteristic
244D61—82no—
245#533C2×D6111yescharacteristic
246C2×C4⋊Q8C2×D6111yescharacteristic
247C2×C4×D4—611no—
248#501—22no—
249#500—21no—
250#498—21no—
251#478—1222no—
252#476—1221no—
253#474—1221no—
254#474—1222no—
255#474—1221no—
256#474—1221no—
257#446—1222no—
258#446—1221no—
259C8×Q8—1221no—
260C2×#108—612no—
261#428—612no—
262#428C2×D6112yes—
263#428C2×D6112yes—
264C2×#107—612no—
265#424—612no—
266#423—611no—
267C2×C4⋊C8—611no—
268#418—612no—
269C2×#102—612no—
270C2×#100—611no—
271C2×#100—611no—
272#405—612no—
273C4×M16—611no—
274C4×M16C2×D6111yescharacteristic
275#343—611no—
276#343—611no—
277#343—612no—
278#343—612no—
279#341—612no—
280C61#2548911yescharacteristic, complemented, Sylow
281C22×Q8—21no—
282C22×D4—611no—
283C23×C4—1221no—
284C2×M16—611no—
285C2×M16—612no—
286C2×M16—611no—
287C2×M16C22×D6111yescharacteristic
288C2×M16C22×D6111yescharacteristic
289C22×C8—612no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8C22⋊D6111yescharacteristic
295C4⋊Q8C22⋊D6111yescharacteristic
296#125C22⋊D6112yes—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×D4—1221no—
302C4×D4—1221no—
303C4×D4—1221no—
304C4×D4—1221no—
305#117C22×D6112yes—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4C22×D6112yes—
309C2×C22⋊C4—1221no—
310C2×C42C22×D6111yescharacteristic
311#108—1224no—
312#107—22no—
313#107—22no—
314#107—1222no—
315#107—1222no—
316#107—1222no—
317#106—22no—
318#106—22no—
319#106—1222no—
320C4⋊C8—611no—
321C4⋊C8—611no—
322C4⋊C8—612no—
323C4⋊C8—1222no—
324C4⋊C8—1222no—
325C4⋊C8—1221no—
326C4⋊C8—1221no—
327#102—1222no—
328#102—1222no—
329#102—1222no—
330#102—1222no—
331#100—614no—
332#100—1222no—
333#97—21no—
334#97—1222no—
335#97—1221no—
336C4×C8—21no—
337C4×C8—1222no—
338C4×C8—1221no—
339C24—1221no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—82no—
345C2×D4—612no—
346C2×D4—1221no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C22×C4—22no—
352C22×C4—1221no—
353C22×C4—4882no—
354C22×C4—2441no—
355C22×C4—2441no—
356C22×C4#1175822411yescharacteristic
357C22×C4#1175822411yescharacteristic
358C22×C4C23×D6111yescharacteristic
359M16—21no—
360M16—21no—
361M16—612no—
362M16—1221no—
363M16—1221no—
364M16—1222no—
365M16—1221no—
366M16—1222no—
367M16Q8×D6112yes—
368M16#1175822612yes—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—612no—
372C2×C8—612no—
373C2×C8—1221no—
374C2×C8—1222no—
375C2×C8—1222no—
376C2×C8—1221no—
377C2×C8—1222no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4C2×C22⋊D6112yes—
389C4⋊C4C2×C22⋊D6112yes—
390C22⋊C4—22no—
391C22⋊C4—2441no—
392C22⋊C4—2441no—
393C42—22no—
394C42D4×D6111yescharacteristic
395C42D4×D6111yescharacteristic
396C42C2×C22⋊D6111yescharacteristic
397C42C2×C22⋊D6111yescharacteristic
398C23—1222no—
399C23—4881no—
400C23—4881no—
401C23—2441no—
402C23—2441no—
403C23C2×#1175822411yescharacteristic
404Q8—42no—
405Q8—81no—
406Q8—82no—
407Q8—82no—
408D4—1222no—
409D4—1222no—
410D4—2441no—
411D4—2441no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—4882no—
419C2×C4—4882no—
420C2×C4—4881no—
421C2×C4—2441no—
422C2×C4#1451974911yescharacteristic
423C2×C4#1451974911yescharacteristic
424C2×C4#1451974911yescharacteristic
425C2×C4#1451974911yescharacteristic
426C2×C4C2×D4×D6111yescharacteristic
427C2×C4C22×C22⋊D6111yescharacteristic
428C2×C4C2×Q8×D6111yescharacteristic
429C2×C4C2×#1175822611yescharacteristic
430C2×C4#1451982511yescharacteristic
431C2×C4#1451982511yescharacteristic
432C8—22no—
433C8—22no—
434C8—1222no—
435C8—1222no—
436C8—1222no—
437C8—2441no—
438C8—2442no—
439C22—21no—
440C22—21no—
441C22—4882no—
442C22—4882no—
443C22—4881no—
444C22—4881no—
445C22—2442no—
446C22C2×#1451974911yescharacteristic
447C22#1684031711yescharacteristic
448C22#1684037311yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—82no—
455C4—4881no—
456C4#1684003012yes—
457C4#1684041511yescharacteristic
458C4#1684042311yescharacteristic
459C4Q8×C22⋊D6111yescharacteristic
460C4#1684042711yescharacteristic
461C2—21no—
462C2—21no—
463C2—4882no—
464C2#2144942111yescharacteristic
465C2C2×#1684003011yescharacteristic
466C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3165C2
socleC22×C61C2×#121
prime radicalC22×D244C23
hypercenter#3165D61
hypocenterC61#25489
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
3. #21454768C2semidirect
4. #21447327C2semidirect
6. #21446809C2semidirect
7. #21446601C2semidirect
8. #21446594C2semidirect
11. C61×#3165C2semidirect
76. #14519693Q8semidirect
228. #3165D61semidirect
280. C61#25489semidirect

Maximal subgroups

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

subgroupindex
#214556122
C2×#168400392
#214547682
#214473272
#214471762
#214468092
#214466012
#214465942
#214464802
C4×#145196932
C61×#31652
#2548961