Finite Group Databasebeta

← #29730421

Subgroups — #29730421

Order 15616.

732 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456215C211yesmaximal, characteristic
2#21455291C211yesmaximal, characteristic
3#21454562C211yesmaximal, characteristic
4#21447265C212yesmaximal, complemented
5#21447119C212yesmaximal, complemented
6#21446809C211yesmaximal, characteristic, complemented
7#21446595C211yesmaximal, characteristic, complemented
8#21446337C212yesmaximal, complemented
9#21445879C212yesmaximal, complemented
10C4×#14519696C211yesmaximal, characteristic, complemented
11C2×C61×#475C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519783C2212yes—
14#16840312C2211yescharacteristic
15#16840124—22no—
16#16840114—21no—
17#16840110—21no—
18#16839984—22no—
19#16839972—21no—
20#16839962—21no—
21#16839839—21no—
22#16839832—22no—
23#16839827—21no—
24C2×#14519713C2212yes—
25#16839786C2211yescharacteristic
26#16839761C2212yes—
27#16839761C2212yes—
28#16839761C2212yes—
29C2×#14519696C2211yescharacteristic
30C2×#14519696C2211yescharacteristic
31#16839728C2212yes—
32C4×#11758194C2211yescharacteristic
33C4×#11758194C2211yescharacteristic
34#16839531C2211yescharacteristic
35#16839531C2212yes—
36#16839531C2212yes—
37#16839531C2211yescharacteristic
38#16839527C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#475—21no—
42C61×#475—22no—
43C61×#475—21no—
44C2×#106×C61C2212yes—
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×#103×C61C2212yes—
47C2×#11758222C2312yes—
48#14519808—24no—
49#14519808—42no—
50C2×#11758202—22no—
51C2×#11758201—22no—
52#14519783—24no—
53#14519783—42no—
54C2×#11758194C2311yescharacteristic
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2312yes—
58C2×#11758194C2311yescharacteristic
59C2×#11758189—22no—
60C2×C8×D61—22no—
61#14519777D411yescharacteristic
62#14519774D411yescharacteristic
63#14519765D412yes—
64#14519751—21no—
65#14519751—21no—
66C4×D244—21no—
67C4×D244—21no—
68#14519713—42no—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71#14519700—22no—
72#14519700—22no—
73#14519700—22no—
74#14519697—22no—
75#14519697—22no—
76#14519697—22no—
77#14519696—22no—
78#14519696—22no—
79#14519689—22no—
80#14519689—22no—
81#14519688—21no—
82#14519688—21no—
83C4×#92418—21no—
84C4×#92418—21no—
85C22×Q8×C61—21no—
86C22×Q8×C61C2311yescharacteristic
87C22×C8×C61C2312yes—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#106×C61—22no—
103#106×C61—22no—
104C4⋊C8×C61D411yescharacteristic
105C4⋊C8×C61D412yes—
106C4⋊C8×C61D411yescharacteristic
107#103×C61—22no—
108#103×C61—22no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—41no—
115C2×Q8×C61—41no—
116C2×Q8×C61—82no—
117C2×Q8×C61#4112yes—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61#4111yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—22no—
125#11758222—22no—
126#11758222—22no—
127#11758222—22no—
128#11758222—22no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244C2×D412yes—
132C2×C4×D61—22no—
133C2×Dic122C2×D412yes—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61C2×D412yes—
137C2×C8×C61C2×D412yes—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C42×C61—41no—
147C42×C61—41no—
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152#11758202—44no—
153#11758201—44no—
154#11758198—22no—
155#11758198—22no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158#11758194—21no—
159#11758194—22no—
160#11758194—21no—
161#11758194—21no—
162#11758194—21no—
163#11758194—22no—
164#11758194—21no—
165#11758194—21no—
166#11758194—21no—
167#11758194#4112yes—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—22no—
173#11758189—44no—
174C8×D61—44no—
175C23×C61C2×#4111yescharacteristic
176C22×D61—22no—
177Q8×C61—22no—
178Q8×C61—22no—
179Q8×C61—41no—
180Q8×C61—41no—
181Q8×C61—82no—
182Q8×C61—82no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—81no—
191C2×C4×C61—82no—
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61H(5,𝔽2)11yescharacteristic
201C2×C4×C612−1+411yescharacteristic
202C22⋊D61—42no—
203C22⋊D61—42no—
204C2×Dic61—22no—
205D244—22no—
206D244—22no—
207D244—42no—
208D244—81no—
209C4×D61—42no—
210C4×D61—42no—
211Dic122—22no—
212Dic122—22no—
213C8×C61—22no—
214C8×C61—22no—
215#92418—22no—
216#92418—41no—
217#92418—42no—
218#92418—41no—
219#92418—41no—
220#25361—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61C2×#12111yescharacteristic
224C22×C61#52911yescharacteristic
225C22×C61#54211yescharacteristic
226C2×D61—42no—
227C2×D61—42no—
228C2×D61—42no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—42no—
233C4×C61—82no—
234C4×C61—81no—
235C4×C61#47112yes—
236C4×C61#54611yescharacteristic
237C4×C61#54611yescharacteristic
238C4×C61#54711yescharacteristic
239C4×C61#54811yescharacteristic
240Dic61—42no—
241#3141—611no—
242C2×#475D6111yescharacteristic, complemented
243#3037—611no—
244#2935—611no—
245#2079—612no—
246#2052—612no—
247#1997—611no—
248#1950—611no—
249#1908—612no—
250#1815—612no—
251C4×#101—611no—
252C2×C61—21no—
253C2×C61—21no—
254C2×C61#242711yescharacteristic
255C2×C61C2×#47111yescharacteristic
256C2×C61#306911yescharacteristic
257D61—82no—
258C2×#137—612no—
259C2×#131—612no—
260#532—611no—
261C2×C4⋊Q8C2×D6111yescharacteristic
262C2×C4×Q8C2×D6111yescharacteristic
263#489—1222no—
264#487—1221no—
265#486—1221no—
266#475—21no—
267#475—22no—
268#475—21no—
269#464—1222no—
270#462—1221no—
271#460—1221no—
272#436—1221no—
273#435—1222no—
274C8×D4—1221no—
275C2×#108—612no—
276C2×#106C2×D6112yes—
277#423—611no—
278C2×C4⋊C8C2×D6111yescharacteristic
279#417—612no—
280#417—612no—
281#417—612no—
282C2×#103C2×D6112yes—
283C2×#101—611no—
284C2×#101—611no—
285#408—612no—
286C4×M16—611no—
287C4×M16—611no—
288#343—612no—
289#343—611no—
290#343—612no—
291#343—611no—
292#341—612no—
293C61#2536111yescharacteristic, complemented, Sylow
294C2×#41—612no—
295C22×Q8—21no—
296C22×Q8C22×D6111yescharacteristic
297#137—1224no—
298#137—2442no—
299C2×Q16—1222no—
300C2×SD16—1222no—
301#131—1224no—
302#131—2442no—
303C2×M16—611no—
304C2×M16—612no—
305C2×M16—611no—
306C2×M16—611no—
307C2×M16—611no—
308C2×M16—1222no—
309C22×C8—1222no—
310C22×C8C22×D6112yes—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—611no—
316#127—611no—
317#124—612no—
318#121—1221no—
319#121—1221no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×D4—1221no—
325C4×D4—1221no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4C22×D6111yescharacteristic
329C2×C4⋊C4C22×D6112yes—
330C2×C42—21no—
331C2×C42C22×D6111yescharacteristic
332#108—2442no—
333#106—22no—
334#106—22no—
335C4⋊C8—1221no—
336C4⋊C8—1221no—
337C4⋊C8C22⋊D6111yescharacteristic
338C4⋊C8C22⋊D6112yes—
339C4⋊C8C22⋊D6111yescharacteristic
340#103—22no—
341#103—22no—
342#103—1222no—
343#103—1222no—
344#103—1222no—
345#102—1222no—
346#102—1222no—
347#102—1222no—
348#101—1222no—
349#101—1222no—
350C22⋊C8—1222no—
351C22⋊C8—1222no—
352#97—1221no—
353#97—1221no—
354C4×C8—1221no—
355C4×C8—1221no—
356#41—1222no—
357#41—1222no—
358#41—1222no—
359#41—1222no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—82no—
368C2×Q8—612no—
369C2×Q8#1175822612yes—
370C2×D4—612no—
371C2×D4—1222no—
372C2×D4—1221no—
373C2×D4—1221no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—1222no—
378C22×C4#1175822411yescharacteristic
379C22×C4#1175822411yescharacteristic
380C22×C4C23×D6111yescharacteristic
381Q16—2444no—
382SD16—2444no—
383M16—612no—
384M16—1221no—
385M16—1221no—
386M16—1221no—
387M16—1221no—
388M16—1222no—
389M16—1221no—
390M16—1222no—
391M16—1221no—
392M16—1221no—
393M16—2444no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—1221no—
397C2×C8—1221no—
398C2×C8—1221no—
399C2×C8—1222no—
400C2×C8—1221no—
401C2×C8—2444no—
402C2×C8C2×C22⋊D6112yes—
403C2×C8C2×C22⋊D6112yes—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C22⋊C4—1222no—
415C22⋊C4—1222no—
416C42—41no—
417C42—41no—
418C42D4×D6111yescharacteristic
419C42D4×D6111yescharacteristic
420C42C2×C22⋊D6111yescharacteristic
421C42C2×C22⋊D6111yescharacteristic
422C23—1222no—
423C23C2×#1175822411yescharacteristic
424Q8—22no—
425Q8—22no—
426Q8—41no—
427Q8—41no—
428Q8—82no—
429Q8—82no—
430Q8—1222no—
431Q8—1222no—
432D4—1222no—
433D4—1222no—
434D4—4881no—
435D4—2442no—
436D4—2442no—
437D4—2442no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—1222no—
448C2×C4—2442no—
449C2×C4—2442no—
450C2×C4#1451974811yescharacteristic
451C2×C4#1451974811yescharacteristic
452C2×C4#1451974811yescharacteristic
453C2×C4#1451974811yescharacteristic
454C2×C4C2×D4×D6111yescharacteristic
455C2×C4C22×C22⋊D6111yescharacteristic
456C2×C4#41×D6111yescharacteristic
457C2×C4C2×#1175822611yescharacteristic
458C2×C4#1451982311yescharacteristic
459C2×C4#1451982511yescharacteristic
460C8—22no—
461C8—22no—
462C8—1222no—
463C8—2441no—
464C8—2442no—
465C8—2441no—
466C8—2441no—
467C22—21no—
468C22—21no—
469C22—2442no—
470C22—2442no—
471C22—2442no—
472C22C2×#1451974811yescharacteristic
473C22#1684029811yescharacteristic
474C22#1684038511yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—42no—
479C4—82no—
480C4—81no—
481C4—2442no—
482C4#1684002212yes—
483C4#1684040011yescharacteristic
484C4#1684040311yescharacteristic
485C4#1684041411yescharacteristic
486C4#1684042311yescharacteristic
487C2—21no—
488C2—21no—
489C2—4882no—
490C2#2144939211yescharacteristic
491C2C2×#1684002211yescharacteristic
492C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#475C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenterC2×#475D61
hypocenterC61#25361
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
4. #21447265C2semidirect
5. #21447119C2semidirect
6. #21446809C2semidirect
7. #21446595C2semidirect
8. #21446337C2semidirect
9. #21445879C2semidirect
10. C4×#14519696C2semidirect
11. C2×C61×#475C2semidirect
242. C2×#475D61semidirect
293. C61#25361semidirect

Maximal subgroups

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

subgroupindex
#214562152
#214552912
#214545622
#214472652
#214471192
#214468092
#214465952
#214463372
#214458792
C4×#145196962
C2×C61×#4752
#2536161