Finite Group Databasebeta

← #29653997

Subgroups — #29653997

Order 15616.

1032 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#21459224C211yesmaximal, characteristic, complemented
2#21456760C212yesmaximal, complemented
3#21444758C214yesmaximal, complemented
4#21444741C214yesmaximal, complemented
5#21444382C212yesmaximal, complemented
6#21444133C211yesmaximal, characteristic, complemented
7C61×#2619C211yesmaximal, characteristic
8#16840393—21no—
9#16840393C2212yes—
10#16840390—22no—
11#16840390—22no—
12#16840388—21no—
13#16840388C2212yes—
14#16840319—21no—
15#16840319C2212yes—
16#16840315—24no—
17C2×#14519736—21no—
18#16840134—22no—
19#16840108—22no—
20#16839959—24no—
21#16839913C2218yes—
22#16839815C2212yes—
23#16839498C2218yes—
24#16839488C2212yes—
25#16839486C2212yes—
26C2×#127×C61C2211yescharacteristic
27C2×#121×C61—21no—
28C2×#117×C61C2211yescharacteristic
29C61×#394—21no—
30C61×#394C2212yes—
31C61×#392—21no—
32C61×#392C2212yes—
33C61×#390—21no—
34C61×#382C2211yescharacteristic
35C2×#11758200—28no—
36#14519777—21no—
37#14519777—21no—
38#14519777D412yescomplemented
39#14519777D412yescomplemented
40#14519776—24no—
41#14519776—24no—
42#14519769—22no—
43#14519769—22no—
44#14519768—24no—
45#14519768—24no—
46#14519756—22no—
47#14519756—22no—
48#14519756—22no—
49#14519756C2314yes—
50#14519754—24no—
51#14519754—24no—
52#14519754—24no—
53#14519754—24no—
54C4×Dic122—22no—
55C4×Dic122—22no—
56C4×Dic122—22no—
57C4×Dic122D414yescomplemented
58#14519736—22no—
59#14519736—22no—
60#14519736—22no—
61#14519736—22no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C22⋊Dic61—22no—
65C2×C4×Dic61—22no—
66C4⋊#92418—22no—
67#14519698—28no—
68C22⋊#92418C2314yes—
69C4×#92418—22no—
70C22×D4×C61—22no—
71C22×D4×C61—21no—
72C22×D4×C61—21no—
73C22×D4×C61C2312yes—
74C23×C4×C61D411yescharacteristic
75#127×C61—22no—
76#127×C61—22no—
77#121×C61—44no—
78#117×C61—22no—
79#117×C61—22no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61D411yescharacteristic
82C2×C4⋊C4×C61C2311yescharacteristic
83C2×C22⋊C4×C61—22no—
84C2×C22⋊C4×C61—22no—
85C2×C22⋊C4×C61—22no—
86C2×C22⋊C4×C61D411yescharacteristic
87C2×C22⋊C4×C61C2311yescharacteristic
88C2×C42×C61D411yescharacteristic
89C2×C42×C61C2311yescharacteristic
90#95×C61—21no—
91#95×C61C2312yes—
92C24×C61—22no—
93C24×C61C2×D411yescharacteristic
94C2×D4×C61—24no—
95C2×D4×C61—44no—
96C2×D4×C61—42no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C2×D4×C61—42no—
100C2×D4×C61—42no—
101C2×D4×C61—82no—
102C2×D4×C61—82no—
103C2×D4×C61—82no—
104C2×D4×C61—82no—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61SD1614yescomplemented
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2411yescharacteristic
116C22×Dic61—22no—
117C22×Dic61—22no—
118C2×Dic122—22no—
119C2×Dic122—22no—
120C2×Dic122—22no—
121C2×Dic122C2×D414yes—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—42no—
125C22⋊C4×C61—22no—
126C22⋊C4×C61—22no—
127C22⋊C4×C61—44no—
128C22⋊C4×C61—82no—
129C22⋊C4×C61—82no—
130C42×C61—21no—
131C42×C61—21no—
132C42×C61C2×D412yes—
133C42×C61C2×D412yes—
134C22⋊Dic61—24no—
135C22⋊Dic61—24no—
136#11758200—88no—
137C4⋊Dic61—22no—
138C4⋊Dic61—22no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61C2×D414yes—
145#11758196—24no—
146#11758196—24no—
147#11758196—24no—
148#11758196—24no—
149C4×Dic61—24no—
150C4×Dic61—24no—
151C2×#92418—24no—
152C23×C61—21no—
153C23×C61—21no—
154C23×C61—41no—
155C23×C61—41no—
156C23×C61—42no—
157C23×C61—42no—
158C23×C61—42no—
159C23×C61—42no—
160C23×C61—42no—
161C23×C61—42no—
162C23×C61—42no—
163C23×C61C2×SD1612yes—
164C23×C61C22×D411yescharacteristic
165D4×C61—48no—
166D4×C61—84no—
167D4×C61—88no—
168D4×C61—82no—
169D4×C61—82no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—24no—
173C2×C4×C61—24no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61#12012yes—
184C2×C4×C61#12011yescharacteristic
185C2×C4×C61#12011yescharacteristic
186C2×C4×C61C2×SD1612yes—
187C2×C4×C61C2×SD1612yes—
188C2×C4×C61#13712yes—
189C2×C4×C61C22×D412yes—
190C2×Dic61—24no—
191C2×Dic61—24no—
192C2×Dic61—82no—
193C2×Dic61—82no—
194Dic122—44no—
195Dic122—82no—
196Dic122—82no—
197Dic122—82no—
198#92418—44no—
199#18302—611nomaximal, Sylow
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—22no—
203C22×C61—42no—
204C22×C61—41no—
205C22×C61—41no—
206C22×C61—81no—
207C22×C61—82no—
208C22×C61—82no—
209C22×C61—82no—
210C22×C61—82no—
211C22×C61—82no—
212C22×C61—82no—
213C22×C61—82no—
214C22×C61C2×#12011yescharacteristic
215C22×C61C22×SD1611yescharacteristic
216C22×C61C2×#13711yescharacteristic
217C4×C61—22no—
218C4×C61—22no—
219C4×C61—22no—
220C4×C61—41no—
221C4×C61—41no—
222C4×C61—82no—
223C4×C61#44812yes—
224C4×C61#44812yes—
225Dic61—44no—
226Dic61—44no—
227#3458—611no—
228#3208—612no—
229#2619D6111yescharacteristic
230#1599—614no—
231#1596—614no—
232#1518—612no—
233#1458—611no—
234C2×C61—21no—
235C2×C61—21no—
236C2×C61—42no—
237C2×C61—162no—
238C2×C61C2×#44811yescharacteristic
239C2×C61#302111yescharacteristic
240C2×C61#302711yescharacteristic
241#544—612no—
242#544—1222no—
243#544—1222no—
244#544—1221no—
245#543—612no—
246#543—1221no—
247#533—612no—
248#533—1224no—
249#533—1221no—
250C2×#127C2×D6111yescharacteristic
251C2×#121—21no—
252C2×#117—1221no—
253C2×#117C2×D6111yescharacteristic
254#492—1222no—
255#486—1222no—
256#460—1224no—
257#450—618no—
258#431—612no—
259#394—21no—
260#394C2×D6112yes—
261#392—21no—
262#392C2×D6112yes—
263#390—21no—
264#382C2×D6111yescharacteristic
265#328—618no—
266#324—612no—
267#323—612no—
268C61#1830211yescharacteristic, complemented, Sylow
269C22×D4—22no—
270C22×D4—21no—
271C22×D4—21no—
272C22×D4C22×D6112yes—
273C23×C4C22⋊D6111yescharacteristic
274C2×SD16—1228no—
275C4⋊Q8—612no—
276C4⋊Q8—612no—
277C4⋊Q8—1221no—
278C4⋊Q8—1221no—
279C4⋊Q8—1224no—
280C4⋊Q8—1224no—
281#127—22no—
282#127—22no—
283#125—1222no—
284#125—1224no—
285#125—1224no—
286#125—1222no—
287C22⋊Q8—614no—
288C22⋊Q8—1224no—
289C22⋊Q8—1224no—
290C22⋊Q8—1224no—
291C22⋊Q8—1224no—
292C22⋊Q8—1222no—
293C22⋊Q8—1222no—
294C22⋊Q8—1222no—
295#121—44no—
296C4×Q8—614no—
297C4×Q8—1222no—
298C4×Q8—1222no—
299C4×Q8—1222no—
300#117—22no—
301#117—22no—
302#117—1222no—
303#117—1222no—
304#117—1222no—
305#117—1222no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—1221no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4C22⋊D6111yescharacteristic
310C2×C4⋊C4C22×D6111yescharacteristic
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—1222no—
315C2×C22⋊C4C22⋊D6111yescharacteristic
316C2×C22⋊C4C22×D6111yescharacteristic
317C2×C42—1222no—
318C2×C42C22⋊D6111yescharacteristic
319C2×C42C22×D6111yescharacteristic
320C4⋊C8—1222no—
321#102—1228no—
322C22⋊C8—614no—
323C4×C8—1222no—
324#95—21no—
325#95C22×D6112yes—
326C24—22no—
327C24C2×C22⋊D6111yescharacteristic
328C2×Q8—614no—
329C2×Q8—1222no—
330C2×Q8—1222no—
331C2×Q8—1222no—
332C2×D4—24no—
333C2×D4—44no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—82no—
340C2×D4—82no—
341C2×D4—82no—
342C2×D4—82no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—1222no—
348C22×C4—1222no—
349C22×C4#1175820014yescomplemented
350C22×C4C2×C22⋊D6111yescharacteristic
351C22×C4C2×C22⋊D6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353C22×C4C2×C22⋊D6111yescharacteristic
354C22×C4C2×C22⋊D6111yescharacteristic
355C22×C4C23×D6111yescharacteristic
356SD16—4888no—
357C2×C8—1224no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—614no—
362C4⋊C4—1222no—
363C4⋊C4—1222no—
364C4⋊C4—1224no—
365C4⋊C4—1224no—
366C4⋊C4—1222no—
367C4⋊C4—1222no—
368C4⋊C4—1224no—
369C4⋊C4—1222no—
370C4⋊C4—1222no—
371C4⋊C4—1224no—
372C4⋊C4—1222no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—44no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—1224no—
379C22⋊C4—1224no—
380C42—21no—
381C42—21no—
382C42—1224no—
383C42—1224no—
384C42D4×D6112yes—
385C42D4×D6112yes—
386C23—21no—
387C23—21no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23C2×#1175820012yes—
398C23C22×C22⋊D6111yescharacteristic
399Q8—4882no—
400Q8—4882no—
401Q8—4882no—
402Q8—2444no—
403D4—48no—
404D4—84no—
405D4—88no—
406D4—82no—
407D4—82no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—1224no—
422C2×C4—1224no—
423C2×C4—4882no—
424C2×C4—4882no—
425C2×C4#1451974512yes—
426C2×C4#1451974511yescharacteristic
427C2×C4#1451974511yescharacteristic
428C2×C4C2×#1175820012yes—
429C2×C4C2×#1175820012yes—
430C2×C4#1451980712yes—
431C2×C4C2×D4×D6112yes—
432C8—2444no—
433C22—21no—
434C22—21no—
435C22—22no—
436C22—42no—
437C22—41no—
438C22—41no—
439C22—81no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22C2×#1451974511yescharacteristic
448C22C22×#1175820011yescharacteristic
449C22C2×#1451980711yescharacteristic
450C4—22no—
451C4—22no—
452C4—22no—
453C4—41no—
454C4—41no—
455C4—82no—
456C4—2444no—
457C4—2444no—
458C4#1683990012yes—
459C4#1683990012yes—
460C2—21no—
461C2—21no—
462C2—42no—
463C2—162no—
464C2C2×#1683990011yescharacteristic
465C2#2145519011yescharacteristic
466C2#2145522011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2619C2
socleC22×C61C2×#120
hypercenter#2619D61
hypocenterC61#18302
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #21459224C2semidirect
2. #21456760C2semidirect
3. #21444758C2semidirect
4. #21444741C2semidirect
5. #21444382C2semidirect
6. #21444133C2semidirect
38. #14519777D4semidirect
39. #14519777D4semidirect
57. C4×Dic122D4semidirect
109. C22×C4×C61SD16semidirect
268. C61#18302semidirect
349. C22×C4#11758200semidirect

Maximal subgroups

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

subgroupindex
#214592242
#214567602
#214447582
#214447412
#214443822
#214441332
C61×#26192
#1830261