Finite Group Databasebeta

← #29716232

Subgroups — #29716232

Order 15616.

1170 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455612C212yesmaximal, complemented
2C4×#14519773C211yesmaximal, characteristic
3#21447161C212yesmaximal, complemented
4#21446972C212yesmaximal, complemented
5#21446797C212yesmaximal, complemented
6#21446217C212yesmaximal
7#21446029C212yesmaximal, complemented
8#21445811C211yesmaximal, characteristic, complemented
9C61×#3501C211yesmaximal, characteristic, complemented
10C2×#14519773C2211yescharacteristic
11C2×C4×D244C2211yescharacteristic
12C2×C4×C22⋊D61—22no—
13#16840059—24no—
14#16840049—22no—
15#16840039—22no—
16#16839805C2214yes—
17#16839786C2212yes—
18#16839766C2214yes—
19C2×#14519694C2212yes—
20#16839665C2211yescharacteristic
21#16839663C2211yescharacteristic
22#16839662—22no—
23#16839631C2211yescharacteristic
24C42×Dic61C2211yescharacteristic
25#16839531C2212yes—
26#16839521C2214yes—
27#16839521C2214yes—
28C61×#554—22no—
29C61×#554—24no—
30D4×Q8×C61—22no—
31C61×#544C2212yes—
32C61×#543—22no—
33C61×#533C2212yes—
34C2×C22⋊Q8×C61—22no—
35C61×#518C2212yes—
36C2×C4×D4×C61C2211yescharacteristic
37C22×D244C2311yescharacteristic
38C22×C22⋊D61—22no—
39C22×C4×D61—21no—
40C2×#11758194C2314yes—
41#14519773—22no—
42#14519773Q814yescomplemented
43C4×D244—22no—
44C4×D244—22no—
45C4×C22⋊D61—48no—
46C2×C4⋊Dic61C2311yescharacteristic
47C2×#11758196—22no—
48C2×C22⋊Dic61—22no—
49C2×#11758198—22no—
50C2×#11758198—21no—
51C2×C4×Dic61—21no—
52C2×C4×Dic61—21no—
53C2×C4×Dic61C2311yescharacteristic
54C2×C4×Dic61C2311yescharacteristic
55#14519710—24no—
56#14519708—24no—
57C4⋊#92418—22no—
58C4⋊#92418—22no—
59#14519697—24no—
60#14519697—24no—
61#14519694—24no—
62C22×Q8×C61—22no—
63C22×D4×C61C2311yescharacteristic
64C23×C4×C61D412yes—
65C4⋊Q8×C61—24no—
66C4⋊Q8×C61D412yescomplemented
67C4⋊Q8×C61D412yescomplemented
68#125×C61—24no—
69#125×C61D414yescomplemented
70C22⋊Q8×C61—28no—
71C22⋊Q8×C61—28no—
72C22⋊Q8×C61—24no—
73C22⋊Q8×C61—24no—
74C22⋊Q8×C61—22no—
75C22⋊Q8×C61—22no—
76C22⋊Q8×C61—24no—
77C4×Q8×C61—22no—
78C4×Q8×C61—22no—
79C4×D4×C61—24no—
80C4×D4×C61—24no—
81C4×D4×C61—22no—
82C4×D4×C61—22no—
83#117×C61C2314yes—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61—24no—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C22⋊C4×C61—24no—
88C2×C22⋊C4×C61D412yes—
89C2×C42×C61C2311yescharacteristic
90C24×C61C2×D412yes—
91C23×D61—21no—
92C2×Q8×C61—24no—
93C2×Q8×C61—22no—
94C2×Q8×C61—22no—
95C2×Q8×C61—84no—
96C2×D4×C61—22no—
97C2×D4×C61—22no—
98C2×D4×C61C2×Q812yes—
99C2×D4×C61#4112yes—
100C22×C4×C61—24no—
101C22×C4×C61—24no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—44no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C2×C22⋊D61—28no—
110C2×C22⋊D61—44no—
111C2×C22⋊D61—44no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—21no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—21no—
118C2×D244C2×Q812yescomplemented
119C2×C4×D61—44no—
120C2×C4×D61—41no—
121C2×C4×D61—41no—
122C4⋊C4×C61—24no—
123C4⋊C4×C61—28no—
124C4⋊C4×C61—24no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—24no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61C2×D414yes—
130C4⋊C4×C61C2×D414yes—
131C22⋊C4×C61—28no—
132C22⋊C4×C61—24no—
133C22⋊C4×C61—24no—
134C42×C61—24no—
135C42×C61C2×D412yes—
136C42×C61C2×D412yes—
137C22⋊Dic61—44no—
138#11758198—44no—
139#11758198—42no—
140C4⋊Dic61—22no—
141#11758196—44no—
142C4×Dic61—24no—
143C4×Dic61—21no—
144C4×Dic61—22no—
145C4×Dic61—42no—
146C4×Dic61C2×Q812yescomplemented
147#11758194—24no—
148#11758194—24no—
149C2×#92418—24no—
150C23×C61—22no—
151C23×C61—22no—
152C23×C61—42no—
153C23×C61—42no—
154C23×C61C22⋊Q814yes—
155C23×C61C22×D411yescharacteristic
156C22×D61—22no—
157C22×D61—42no—
158C22×D61—42no—
159C22×D61—41no—
160C22×D61—41no—
161Q8×C61—44no—
162Q8×C61—82no—
163Q8×C61—84no—
164D4×C61—24no—
165D4×C61—24no—
166C2×C4×C61—22no—
167C2×C4×C61—24no—
168C2×C4×C61—24no—
169C2×C4×C61—22no—
170C2×C4×C61—41no—
171C2×C4×C61—44no—
172C2×C4×C61—42no—
173C2×C4×C61—82no—
174C2×C4×C61—84no—
175C2×C4×C61—84no—
176C2×C4×C61#12012yes—
177C2×C4×C61#12011yescharacteristic
178C2×C4×C61#12011yescharacteristic
179C2×C4×C61C22×D412yes—
180C2×C4×C61C22×Q811yescharacteristic
181C2×C4×C61C2×#4111yescharacteristic
182C2×C4×C612−1+412yes—
183C22⋊D61—416no—
184C2×Dic61—22no—
185C2×Dic61—22no—
186C2×Dic61—22no—
187C2×Dic61—22no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192D244—24no—
193D244—42no—
194C4×D61—82no—
195C4×D61—82no—
196#92418—44no—
197#24312—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—24no—
201C22×C61—44no—
202C22×C61—42no—
203C22×C61—42no—
204C22×C61—82no—
205C22×C61C2×#12011yescharacteristic
206C22×C61C2×C22⋊Q811yescharacteristic
207C22×C61#53611yescharacteristic
208C2×D61—44no—
209C2×D61—42no—
210C2×D61—82no—
211C2×D61—81no—
212C2×D61—81no—
213C4×C61—21no—
214C4×C61—22no—
215C4×C61—42no—
216C4×C61—44no—
217C4×C61—44no—
218C4×C61#45312yes—
219C4×C61#54812yes—
220C4×C61D4×Q812yes—
221Dic61—24no—
222Dic61—42no—
223Dic61—81no—
224#3501D6111yescharacteristic, complemented
225#3077—612no—
226C4×#127—611no—
227#2060—612no—
228#2027—612no—
229#1994—612no—
230#1887—612no—
231#1856—612no—
232#1800—611no—
233C2×C61—21no—
234C2×C61—21no—
235C2×C61—44no—
236C2×C61#240911yescharacteristic
237C2×C61C2×#45311yescharacteristic
238C2×C61#301811yescharacteristic
239D61—82no—
240#554—22no—
241#554—24no—
242D4×Q8—22no—
243#544C2×D6112yes—
244#543—22no—
245#533C2×D6112yes—
246C2×#127—611no—
247C2×C22⋊Q8—22no—
248#518C2×D6112yes—
249C2×C4×D4—611no—
250C2×C4×D4—1222no—
251C2×C4×D4C2×D6111yescharacteristic
252#478—1224no—
253#476—1222no—
254#474—1222no—
255#428—614no—
256#423—612no—
257#418—614no—
258C2×#100—612no—
259#390—611no—
260#390—611no—
261#390—1222no—
262#384—611no—
263C43—611no—
264#343—612no—
265#339—614no—
266#339—614no—
267C61#2431211yescharacteristic, complemented, Sylow
268C22×Q8—22no—
269C22×D4—611no—
270C22×D4—1222no—
271C22×D4C22×D6111yescharacteristic
272C23×C4—1221no—
273C23×C4C22⋊D6112yes—
274C2×M16—614no—
275C4⋊Q8—24no—
276C4⋊Q8C22⋊D6112yescomplemented
277C4⋊Q8C22⋊D6112yescomplemented
278#127—614no—
279#127—1222no—
280#125—24no—
281#125C22⋊D6114yescomplemented
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—28no—
285C22⋊Q8—28no—
286C22⋊Q8—22no—
287C22⋊Q8—24no—
288C22⋊Q8—22no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—1222no—
296C4×D4—1222no—
297C4×D4—2448no—
298#117C22×D6114yes—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—611no—
302C2×C4⋊C4—1222no—
303C2×C4⋊C4C22×D6111yescharacteristic
304C2×C22⋊C4—24no—
305C2×C22⋊C4—1222no—
306C2×C22⋊C4—1221no—
307C2×C22⋊C4—1222no—
308C2×C22⋊C4C22⋊D6112yes—
309C2×C42—611no—
310C2×C42—611no—
311C2×C42—1221no—
312C2×C42—1221no—
313C2×C42C22×D6111yescharacteristic
314#107—1224no—
315#106—1224no—
316C4⋊C8—1222no—
317C4⋊C8—1222no—
318#102—1224no—
319#102—1224no—
320#100—1224no—
321C24—1221no—
322C24C2×C22⋊D6112yes—
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—84no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—612no—
330C2×D4—1228no—
331C2×D4—1222no—
332C2×D4—1221no—
333C2×D4—1222no—
334C2×D4—2444no—
335C2×D4—2444no—
336C2×D4Q8×D6112yes—
337C2×D4#1175822612yes—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—44no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—2444no—
347C22×C4—2441no—
348C22×C4—2441no—
349C22×C4C2×C22⋊D6111yescharacteristic
350C22×C4C2×C22⋊D6112yes—
351C22×C4C2×C22⋊D6111yescharacteristic
352C22×C4C23×D6111yescharacteristic
353M16—1224no—
354M16—1224no—
355C2×C8—1224no—
356C4⋊C4—24no—
357C4⋊C4—28no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—1222no—
364C4⋊C4—2444no—
365C4⋊C4C2×C22⋊D6114yes—
366C4⋊C4C2×C22⋊D6114yes—
367C22⋊C4—28no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—2444no—
371C22⋊C4—2444no—
372C22⋊C4—2442no—
373C42—24no—
374C42—612no—
375C42—1224no—
376C42—1221no—
377C42—1222no—
378C42—2442no—
379C42C2×C22⋊D6112yes—
380C42C2×C22⋊D6112yes—
381C23—22no—
382C23—22no—
383C23—42no—
384C23—42no—
385C23—1222no—
386C23—2442no—
387C23—2442no—
388C23—2441no—
389C23—2441no—
390C23#1451975314yes—
391C23C22×C22⋊D6111yescharacteristic
392Q8—44no—
393Q8—82no—
394Q8—84no—
395D4—24no—
396D4—24no—
397D4—1224no—
398D4—24416no—
399D4—2442no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—1222no—
411C2×C4—1222no—
412C2×C4—1222no—
413C2×C4—1222no—
414C2×C4—4882no—
415C2×C4—4882no—
416C2×C4—2442no—
417C2×C4—2441no—
418C2×C4—2441no—
419C2×C4—2441no—
420C2×C4#1451974612yes—
421C2×C4#1451974611yescharacteristic
422C2×C4#1451974611yescharacteristic
423C2×C4C22×C22⋊D6112yes—
424C2×C4C2×Q8×D6111yescharacteristic
425C2×C4C2×#1175822611yescharacteristic
426C2×C4#1451982512yes—
427C8—2444no—
428C22—21no—
429C22—21no—
430C22—24no—
431C22—44no—
432C22—42no—
433C22—42no—
434C22—82no—
435C22—4882no—
436C22—4881no—
437C22—4881no—
438C22—2444no—
439C22—2442no—
440C22C2×#1451974611yescharacteristic
441C22C2×#1451975311yescharacteristic
442C22#1684033611yescharacteristic
443C4—21no—
444C4—22no—
445C4—42no—
446C4—44no—
447C4—44no—
448C4—1224no—
449C4—4881no—
450C4—2442no—
451C4#1683993012yes—
452C4#1684042312yes—
453C4Q8×C22⋊D6112yes—
454C2—21no—
455C2—21no—
456C2—44no—
457C2—4882no—
458C2#2144917811yescharacteristic
459C2C2×#1683993011yescharacteristic
460C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC2×#14519773C22
hypercenter#3501D61
hypocenterC61#24312
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21455612C2semidirect
3. #21447161C2semidirect
4. #21446972C2semidirect
5. #21446797C2semidirect
7. #21446029C2semidirect
8. #21445811C2semidirect
9. C61×#3501C2semidirect
42. #14519773Q8semidirect
66. C4⋊Q8×C61D4semidirect
67. C4⋊Q8×C61D4semidirect
69. #125×C61D4semidirect
118. C2×D244C2×Q8semidirect
146. C4×Dic61C2×Q8semidirect
224. #3501D61semidirect
267. C61#24312semidirect
276. C4⋊Q8C22⋊D61semidirect
277. C4⋊Q8C22⋊D61semidirect
281. #125C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214556122
C4×#145197732
#214471612
#214469722
#214467972
#214462172
#214460292
#214458112
C61×#35012
#2431261