Finite Group Databasebeta

← #29729787

Subgroups — #29729787

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#21456201C211yesmaximal, characteristic, complemented
2#21455612C211yesmaximal, characteristic
3C2×#16839968C211yesmaximal, characteristic
4#21447185C212yesmaximal, complemented
5#21447082C212yesmaximal
6#21446722C212yesmaximal
7#21446663C211yesmaximal, characteristic, complemented
8#21446595C211yesmaximal, characteristic, complemented
9#21446474C212yesmaximal, complemented
10C4×#14519693C211yesmaximal, characteristic
11C61×#2962C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840132—22no—
16#16840114—21no—
17#16840114—21no—
18#16840059—22no—
19#16840049—21no—
20#16840039—21no—
21#16839968—21no—
22#16839968—21no—
23#16839968—22no—
24C2×#14519713C2212yes—
25#16839805C2212yes—
26#16839786C2211yescharacteristic
27#16839770C2212yes—
28#16839770C2212yes—
29#16839766C2212yes—
30C2×#14519702C2212yes—
31C2×#14519695C2212yes—
32C2×#14519695C2212yes—
33C2×#14519693C2211yescharacteristic
34C2×#14519693C2211yescharacteristic
35C4×#11758194C2211yescharacteristic
36#16839530C2211yescharacteristic
37#16839530C2211yescharacteristic
38#16839527C2212yes—
39C61×#533C2211yescharacteristic
40C61×#446—22no—
41C61×#446—21no—
42C8×Q8×C61—21no—
43C61×#424C2212yes—
44C2×C4⋊C8×C61C2211yescharacteristic
45C61×#405C2212yes—
46C4×M16×C61C2211yescharacteristic
47C22×Dic122C2312yes—
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C2×#11758190—24no—
51C2×#11758190—22no—
52C2×#11758190—22no—
53C2×#11758190—42no—
54C2×#11758194C2311yescharacteristic
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2312yes—
57#14519777—21no—
58#14519777—21no—
59#14519777—21no—
60#14519777—21no—
61C4×D244—21no—
62C4×D244—21no—
63C4×D244—21no—
64C4×D244—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×#11758198—21no—
69#14519713—42no—
70#14519710—22no—
71#14519708—22no—
72C4⋊#92418—21no—
73C4⋊#92418—21no—
74#14519702—22no—
75#14519702—22no—
76#14519700—22no—
77#14519700—22no—
78#14519700—22no—
79#14519700—22no—
80#14519697—22no—
81#14519697—22no—
82#14519695—24no—
83#14519695—42no—
84#14519693—22no—
85#14519693—22no—
86#14519688—21no—
87C4×#92418—21no—
88C2×M16×C61C2311yescharacteristic
89C2×M16×C61C2311yescharacteristic
90C22×C8×C61C2312yes—
91C4⋊Q8×C61D411yescharacteristic
92C4⋊Q8×C61D411yescharacteristic
93#125×C61D412yes—
94C22⋊Q8×C61—21no—
95C22⋊Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×Q8×C61—21no—
98#117×C61C2312yes—
99C2×C42×C61C2311yescharacteristic
100C4⋊C8×C61—22no—
101C4⋊C8×C61—22no—
102C4⋊C8×C61D411yescharacteristic
103C4⋊C8×C61D411yescharacteristic
104C4⋊C8×C61D412yes—
105#97×C61—22no—
106#97×C61—21no—
107C4×C8×C61—22no—
108C4×C8×C61—21no—
109C23×D61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61C2411yescharacteristic
115C22×Dic61—22no—
116C22×Dic61—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244#4112yes—
123C2×C4×D61—41no—
124C2×C4×D61—41no—
125C2×C4×D61—82no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132M16×C61—21no—
133M16×C61—21no—
134M16×C61—21no—
135M16×C61#4112yes—
136C2×C8×C61—21no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61—21no—
140C2×C8×C61C2×D412yes—
141C2×C8×C61C2×D412yes—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61C2×D412yes—
147C4⋊C4×C61C2×D412yes—
148C22⋊C4×C61—22no—
149C42×C61—22no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154#11758198—41no—
155#11758198—41no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—41no—
159C4⋊Dic61—41no—
160C4⋊Dic61—41no—
161C4⋊Dic61—41no—
162#11758194—21no—
163#11758194—21no—
164#11758194—22no—
165#11758194—22no—
166#11758194—21no—
167#11758194—21no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—22no—
171#11758190—44no—
172#11758190—44no—
173C23×C61C2×#4111yescharacteristic
174C22×D61—22no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—81no—
178C22×D61—81no—
179Q8×C61—42no—
180Q8×C61—81no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—41no—
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C2×C4×C61H(5,𝔽2)11yescharacteristic
194C2×C4×C612−1+411yescharacteristic
195C2×Dic61—41no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—81no—
200D244—22no—
201D244—22no—
202D244—41no—
203D244—41no—
204C4×D61—82no—
205C4×D61—82no—
206Dic122—42no—
207Dic122—42no—
208Dic122—42no—
209Dic122—42no—
210C8×C61—22no—
211C8×C61—22no—
212C8×C61—22no—
213C8×C61—41no—
214#92418—41no—
215#92418—41no—
216#92418—42no—
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—42no—
232C4×C61—42no—
233C4×C61#47112yes—
234C4×C61#54611yescharacteristic
235C4×C61#54811yescharacteristic
236C4×C61#54811yescharacteristic
237C4×C61#55011yescharacteristic
238Dic61—82no—
239Dic61—81no—
240#3139—611no—
241#3077—611no—
242C2×#461—611no—
243#2962D6111yescharacteristic, complemented
244#2064—612no—
245#2046—612no—
246#1977—612no—
247#1966—611no—
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#533C2×D6111yescharacteristic
259C2×C4⋊Q8—611no—
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—22no—
271#446—21no—
272C8×Q8—21no—
273C2×#108—612no—
274#428—612no—
275#424C2×D6112yes—
276#423—611no—
277C2×C4⋊C8C2×D6111yescharacteristic
278#419—612no—
279#419—612no—
280#418—612no—
281C2×#103—612no—
282C2×#101—612no—
283C2×#101—612no—
284C2×#100—611no—
285C2×#100—611no—
286#405C2×D6112yes—
287C4×M16—611no—
288C4×M16C2×D6111yescharacteristic
289#343—611no—
290#343—611no—
291#341—612no—
292C61#2531311yescharacteristic, complemented, Sylow
293C22×Q8—612no—
294C22×D4—611no—
295C23×C4—1221no—
296C2×SD16—1224no—
297C2×SD16—1222no—
298C2×SD16—1222no—
299C2×SD16—2442no—
300C2×M16—611no—
301C2×M16—612no—
302C2×M16—611no—
303C2×M16C22×D6111yescharacteristic
304C2×M16C22×D6111yescharacteristic
305C22×C8C22×D6112yes—
306C4⋊Q8—1221no—
307C4⋊Q8—1221no—
308C4⋊Q8—1221no—
309C4⋊Q8—1221no—
310C4⋊Q8D24411yescharacteristic
311C4⋊Q8D24411yescharacteristic
312#125D24412yes—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×D4—1221no—
318C4×D4—1221no—
319C4×D4—1221no—
320C4×D4—1221no—
321#117C22×D6112yes—
322C2×C4⋊C4—611no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C42C22×D6111yescharacteristic
327#108—2442no—
328#107—1222no—
329#106—1222no—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—1221no—
333C4⋊C8—1221no—
334C4⋊C8C22⋊D6111yescharacteristic
335C4⋊C8C22⋊D6111yescharacteristic
336C4⋊C8C22⋊D6112yes—
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—1224no—
346#101—2442no—
347#100—1222no—
348#100—1222no—
349#97—22no—
350#97—21no—
351#97—1221no—
352C4×C8—22no—
353C4×C8—21no—
354C4×C8—1221no—
355C24—1221no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—1222no—
359C2×Q8—1222no—
360C2×Q8—1222no—
361C2×Q8—1222no—
362C2×Q8—1222no—
363C2×Q8—1222no—
364C2×D4—612no—
365C2×D4—1221no—
366C2×D4—1221no—
367C2×D4—1221no—
368C2×D4—1221no—
369C2×D4—1221no—
370C22×C4—1221no—
371C22×C4—1222no—
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—21no—
383M16—1221no—
384M16—1221no—
385M16—1222no—
386M16—1222no—
387M16—1221no—
388M16—1221no—
389M16#1175822612yes—
390C2×C8—21no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—21no—
394C2×C8—1221no—
395C2×C8—1221no—
396C2×C8—1222no—
397C2×C8C2×C22⋊D6112yes—
398C2×C8C2×C22⋊D6112yes—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—2441no—
406C4⋊C4—2441no—
407C4⋊C4—2441no—
408C4⋊C4—2441no—
409C4⋊C4C2×D24412yes—
410C4⋊C4C2×D24412yes—
411C22⋊C4—22no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C42—22no—
415C42C2×D24411yescharacteristic
416C42C2×D24411yescharacteristic
417C42C2×C22⋊D6111yescharacteristic
418C42C2×C22⋊D6111yescharacteristic
419C23—1222no—
420C23—4881no—
421C23—4881no—
422C23—2441no—
423C23—2441no—
424C23C2×#1175822211yescharacteristic
425Q8—42no—
426Q8—81no—
427Q8—2442no—
428Q8—2442no—
429Q8—2442no—
430Q8—2442no—
431D4—1222no—
432D4—1222no—
433D4—2441no—
434D4—2441no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—4882no—
440C2×C4—4882no—
441C2×C4—4881no—
442C2×C4—2441no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4—2442no—
446C2×C4#1451975111yescharacteristic
447C2×C4#1451975111yescharacteristic
448C2×C4#1451975111yescharacteristic
449C2×C4#1451975111yescharacteristic
450C2×C4C22×C22⋊D6111yescharacteristic
451C2×C4C22×D24411yescharacteristic
452C2×C4#41×D6111yescharacteristic
453C2×C4C2×#1175822611yescharacteristic
454C2×C4#1451982311yescharacteristic
455C2×C4#1451982511yescharacteristic
456C8—22no—
457C8—22no—
458C8—22no—
459C8—41no—
460C8—2441no—
461C8—2441no—
462C8—2442no—
463C22—21no—
464C22—21no—
465C22—4882no—
466C22—4882no—
467C22—4881no—
468C22—4881no—
469C22—2442no—
470C22C2×#1451975111yescharacteristic
471C22#1684029911yescharacteristic
472C22#1684038411yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—4881no—
479C4—4882no—
480C4#1684002512yes—
481C4#1684040011yescharacteristic
482C4#1684042311yescharacteristic
483C4#1684042411yescharacteristic
484C4#1684043411yescharacteristic
485C2—21no—
486C2—21no—
487C2—4882no—
488C2#2144937911yescharacteristic
489C2C2×#1684002511yescharacteristic
490C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2962C2
socleC22×C61C2×#121
prime radicalC22×D244C23
hypercenter#2962D61
hypocenterC61#25313
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. #21456201C2semidirect
4. #21447185C2semidirect
7. #21446663C2semidirect
8. #21446595C2semidirect
9. #21446474C2semidirect
11. C61×#2962C2semidirect
243. #2962D61semidirect
292. C61#25313semidirect

Maximal subgroups

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

subgroupindex
#214562012
#214556122
C2×#168399682
#214471852
#214470822
#214467222
#214466632
#214465952
#214464742
C4×#145196932
C61×#29622
#2531361