Finite Group Databasebeta

← #29729794

Subgroups — #29729794

Order 15616.

712 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456199C211yesmaximal, characteristic, complemented
2#21455618C211yesmaximal, characteristic
3C2×#16839964C211yesmaximal, characteristic
4#21454766C211yesmaximal, characteristic, complemented
5#21447182C212yesmaximal, complemented
6#21447078C212yesmaximal
7#21446723C212yesmaximal
8#21446592C211yesmaximal, characteristic, complemented
9#21446476C212yesmaximal, complemented
10C4×#14519693C211yesmaximal, characteristic
11C61×#1966C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13#16840319C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840130—22no—
16#16840112—21no—
17#16840112—21no—
18#16840063—22no—
19#16840053—21no—
20#16840043—21no—
21#16839964—21no—
22#16839964—22no—
23#16839964—21no—
24#16839892—21no—
25#16839890—22no—
26C8×Dic122—21no—
27C2×#14519712C2212yes—
28#16839804C2212yes—
29#16839790C2212yes—
30C2×C4⋊#92418C2211yescharacteristic
31#16839771C2212yes—
32#16839771C2212yes—
33#16839768C2212yes—
34C2×#14519701C2212yes—
35C2×#14519693C2211yescharacteristic
36C2×#14519693C2211yescharacteristic
37#16839720C2212yes—
38C4×#11758194C2211yescharacteristic
39#16839530C2211yescharacteristic
40#16839530C2211yescharacteristic
41#16839526C2212yes—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#423C2211yescharacteristic
44C2×#101×C61C2212yes—
45C2×#101×C61C2212yes—
46C4×M16×C61C2211yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C2×#11758201—24no—
50C2×#11758201—22no—
51C2×#11758201—22no—
52C2×#11758201—42no—
53C2×#11758194C2311yescharacteristic
54C2×#11758194C2311yescharacteristic
55C22×#92418C2312yes—
56#14519777D411yescharacteristic
57#14519777D411yescharacteristic
58#14519769D412yes—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244—21no—
66C4×D244—21no—
67#14519736C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758198—21no—
70#14519712—42no—
71#14519711—22no—
72#14519709—22no—
73C4⋊#92418D411yescharacteristic
74C4⋊#92418D412yes—
75C4⋊#92418D411yescharacteristic
76#14519706—22no—
77#14519706—22no—
78#14519701—22no—
79#14519701—22no—
80#14519701—22no—
81#14519701—22no—
82#14519701—22no—
83#14519701—22no—
84#14519699—22no—
85#14519699—22no—
86#14519693—22no—
87#14519693—22no—
88#14519688—21no—
89#14519687—22no—
90C4×#92418—21no—
91C8×Dic61—22no—
92C22×Q8×C61C2312yes—
93C2×M16×C61C2311yescharacteristic
94C2×M16×C61C2312yes—
95C2×M16×C61C2311yescharacteristic
96C4⋊Q8×C61—21no—
97C4⋊Q8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—21no—
104C4⋊C8×C61—21no—
105#101×C61—24no—
106#101×C61—42no—
107#97×C61—21no—
108C4×C8×C61—21no—
109C23×D61—21no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61#4111yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C22×C4×C61C2411yescharacteristic
120C22×Dic61—21no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244#4112yes—
127C2×C4×D61—41no—
128C2×C4×D61—41no—
129C2×C4×D61—82no—
130C2×Dic122—21no—
131C2×Dic122—21no—
132M16×C61—21no—
133M16×C61—22no—
134M16×C61—21no—
135M16×C61—21no—
136M16×C61—22no—
137M16×C61#4112yes—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—22no—
141C4⋊C4×C61—41no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C22⋊Dic61—22no—
150#11758201—44no—
151#11758201—44no—
152#11758198—41no—
153#11758198—41no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61C2×D412yes—
159C4⋊Dic61C2×D412yes—
160#11758196—22no—
161#11758196—22no—
162C4×Dic61—22no—
163#11758194—21no—
164#11758194—21no—
165#11758194—21no—
166#11758194—21no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418C2×D412yes—
172C2×#92418C2×D412yes—
173C23×C61C2×#4111yescharacteristic
174C22×D61—22no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—81no—
178C22×D61—81no—
179Q8×C61—42no—
180Q8×C61—42no—
181Q8×C61—42no—
182Q8×C61—42no—
183C2×C4×C61—22no—
184C2×C4×C61—41no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61H(5,𝔽2)11yescharacteristic
197C2×C4×C612−1+411yescharacteristic
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—41no—
201C2×Dic61—81no—
202D244—22no—
203D244—22no—
204D244—41no—
205D244—41no—
206C4×D61—82no—
207C4×D61—82no—
208Dic122—42no—
209Dic122—81no—
210C8×C61—22no—
211C8×C61—41no—
212C8×C61—42no—
213#92418—22no—
214#92418—22no—
215#92418—41no—
216#92418—41no—
217#25313—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61C2×#12111yescharacteristic
221C22×C61#52911yescharacteristic
222C22×C61#54211yescharacteristic
223C2×D61—42no—
224C2×D61—82no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—82no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—82no—
232C4×C61#47112yes—
233C4×C61#54611yescharacteristic
234C4×C61#54811yescharacteristic
235C4×C61#54811yescharacteristic
236C4×C61#55011yescharacteristic
237Dic61—42no—
238Dic61—42no—
239Dic61—81no—
240#3139—611no—
241#3077—611no—
242C2×#461—611no—
243#2962—611no—
244#2064—612no—
245#2046—612no—
246#1977—612no—
247#1966D6111yescharacteristic, complemented
248#1950—611no—
249#1930—612no—
250C4×#100—611no—
251C2×C61—21no—
252C2×C61—21no—
253C2×C61#242611yescharacteristic
254C2×C61C2×#47111yescharacteristic
255C2×C61#306911yescharacteristic
256D61—82no—
257C22×SD16—612no—
258#533—611no—
259C2×C4⋊Q8C2×D6111yescharacteristic
260C2×C4×D4—611no—
261#491—1222no—
262#487—1221no—
263#487—1221no—
264#478—1222no—
265#476—1221no—
266#474—1221no—
267#461—1221no—
268#461—1222no—
269#461—1221no—
270#446—1221no—
271#446—1222no—
272C8×Q8—1221no—
273C2×#108—612no—
274#428—612no—
275#424—612no—
276#423C2×D6111yescharacteristic
277C2×C4⋊C8—611no—
278#419—612no—
279#419—612no—
280#418—612no—
281C2×#103—612no—
282C2×#101C2×D6112yes—
283C2×#101C2×D6112yes—
284C2×#100—611no—
285C2×#100—611no—
286#405—612no—
287C4×M16—611no—
288C4×M16C2×D6111yescharacteristic
289#343—611no—
290#343—611no—
291#341—612no—
292C61#2531311yescharacteristic, complemented, Sylow
293C22×Q8C22×D6112yes—
294C22×D4—611no—
295C23×C4—1221no—
296C2×SD16—1222no—
297C2×SD16—1222no—
298C2×SD16—1224no—
299C2×SD16—2442no—
300C2×M16—611no—
301C2×M16—611no—
302C2×M16C22×D6112yes—
303C2×M16C22×D6111yescharacteristic
304C2×M16C22×D6111yescharacteristic
305C22×C8—612no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—611no—
311C4⋊Q8—611no—
312#125—612no—
313C22⋊Q8—1221no—
314C22⋊Q8—1221no—
315C4×Q8—1221no—
316C4×Q8—1221no—
317C4×D4—1221no—
318C4×D4—1221no—
319C4×D4—1221no—
320C4×D4—1221no—
321#117—612no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—611no—
325C2×C22⋊C4—1221no—
326C2×C42C22×D6111yescharacteristic
327#108—2442no—
328#107—1222no—
329#106—1222no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—611no—
333C4⋊C8—611no—
334C4⋊C8—612no—
335C4⋊C8—1222no—
336C4⋊C8—1222no—
337#103—1222no—
338#103—1222no—
339#103—1222no—
340#103—1222no—
341#103—1222no—
342#103—1222no—
343#102—1222no—
344#102—1222no—
345#101—24no—
346#101—42no—
347#100—1222no—
348#100—1222no—
349#97—21no—
350#97—1222no—
351#97—1221no—
352C4×C8—21no—
353C4×C8—1222no—
354C4×C8—1221no—
355C24—1221no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—1221no—
363C2×Q8—1221no—
364C2×D4—612no—
365C2×D4—1221no—
366C2×D4—1221no—
367C2×D4—1221no—
368C2×D4—1221no—
369C2×D4—1221no—
370C22×C4—22no—
371C22×C4—1221no—
372C22×C4—4882no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4#1175822211yescharacteristic
376C22×C4#1175822211yescharacteristic
377C22×C4C23×D6111yescharacteristic
378SD16—2444no—
379SD16—2444no—
380M16—21no—
381M16—21no—
382M16—22no—
383M16—21no—
384M16—22no—
385M16—1221no—
386M16—1221no—
387M16—1221no—
388M16—1221no—
389M16#1175822412yes—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—22no—
393C2×C8—612no—
394C2×C8—612no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—1221no—
398C2×C8—1221no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—612no—
404C4⋊C4—612no—
405C4⋊C4—1222no—
406C4⋊C4—1222no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C22⋊C4—1222no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C42—1222no—
415C42D4×D6111yescharacteristic
416C42D4×D6111yescharacteristic
417C42D4×D6111yescharacteristic
418C42D4×D6111yescharacteristic
419C23—1222no—
420C23—4881no—
421C23—4881no—
422C23—2441no—
423C23—2441no—
424C23C2×#1175822211yescharacteristic
425Q8—42no—
426Q8—42no—
427Q8—42no—
428Q8—42no—
429Q8—4881no—
430Q8—2442no—
431D4—1222no—
432D4—1222no—
433D4—2441no—
434D4—2441no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—1222no—
441C2×C4—1222no—
442C2×C4—4882no—
443C2×C4—4882no—
444C2×C4—4881no—
445C2×C4—2441no—
446C2×C4#1451975011yescharacteristic
447C2×C4#1451975011yescharacteristic
448C2×C4#1451975011yescharacteristic
449C2×C4#1451975011yescharacteristic
450C2×C4C2×D4×D6111yescharacteristic
451C2×C4C2×D4×D6111yescharacteristic
452C2×C4#41×D6111yescharacteristic
453C2×C4C2×#1175822411yescharacteristic
454C2×C4#1451982211yescharacteristic
455C2×C4#1451982411yescharacteristic
456C8—22no—
457C8—41no—
458C8—42no—
459C8—1222no—
460C8—1222no—
461C8—2441no—
462C8—2441no—
463C22—21no—
464C22—21no—
465C22—4882no—
466C22—4882no—
467C22—4881no—
468C22—4881no—
469C22—2442no—
470C22C2×#1451975011yescharacteristic
471C22#1684029411yescharacteristic
472C22#1684037711yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—82no—
477C4—4881no—
478C4—2442no—
479C4—2442no—
480C4#1684002812yes—
481C4#1684040911yescharacteristic
482C4#1684041811yescharacteristic
483C4#1684042011yescharacteristic
484C4#1684043211yescharacteristic
485C2—21no—
486C2—21no—
487C2—4882no—
488C2#2144938611yescharacteristic
489C2C2×#1684002811yescharacteristic
490C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1966C2
socleC22×C61C2×#121
prime radicalC22×D244C23
hypercenter#1966D61
hypocenterC61#25313
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1966

Complements

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

NH ≅ G/Ntype
1. #21456199C2semidirect
4. #21454766C2semidirect
5. #21447182C2semidirect
8. #21446592C2semidirect
9. #21446476C2semidirect
11. C61×#1966C2semidirect
247. #1966D61semidirect
292. C61#25313semidirect

Maximal subgroups

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

subgroupindex
#214561992
#214556182
C2×#168399642
#214547662
#214471822
#214470782
#214467232
#214465922
#214464762
C4×#145196932
C61×#19662
#2531361