Finite Group Databasebeta

← #29576591

Subgroups — #29576591

Order 15616.

564 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456572C211yesmaximal, characteristic
2#21448904C211yesmaximal, characteristic, complemented
3#21446654C211yesmaximal, characteristic
4C2×#16839497C211yesmaximal, characteristic
5#21443697C211yesmaximal, characteristic, complemented
6#21443666C211yesmaximal, characteristic, complemented
7C61×#1520C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9#16840090C412yescomplemented
10#16840080—21no—
11#16839993—22no—
12#16839912C412yescomplemented
13#16839901—21no—
14C2×#14519691—21no—
15C2×#14519691—21no—
16C2×C22⋊#92418C2211yescharacteristic
17#16839669—21no—
18#16839648—21no—
19#16839648—21no—
20#16839637—21no—
21C4⋊C4×Dic61—21no—
22C22⋊C4×Dic61—21no—
23C4×C22⋊Dic61C2211yescharacteristic
24C2×#14519684C2211yescharacteristic
25#16839497—21no—
26#16839497—21no—
27C2×C22⋊Q8×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C61×#380C2211yescharacteristic
30C61×#328—22no—
31C22×D244C2×C411yescharacteristic
32C22×C22⋊D61—21no—
33C23×Dic61—21no—
34C2×#11758201—22no—
35C22×#92418—21no—
36#14519751—21no—
37#14519751—21no—
38#14519751—21no—
39#14519751—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×C22⋊Dic61D411yescharacteristic
44C2×C22⋊Dic61D411yescharacteristic
45C2×#11758198—21no—
46C2×C4×Dic61—21no—
47C2×C4×Dic61—21no—
48C2×C4×Dic61—21no—
49#14519708—22no—
50#14519701—22no—
51#14519697—22no—
52#14519691—42no—
53#14519691—42no—
54C22⋊#92418—21no—
55C22⋊#92418C2×C412yes—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62#14519684D411yescharacteristic
63#14519684D411yescharacteristic
64C22×Q8×C61D411yescharacteristic
65C23×C4×C61C2311yescharacteristic
66C22×C8×C61—21no—
67C22⋊Q8×C61—21no—
68C22⋊Q8×C61—21no—
69C22⋊Q8×C61C2×C412yes—
70C22⋊Q8×C61D412yes—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—21no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61—21no—
78#95×C61—21no—
79C24×C61C2×D411yescharacteristic
80C23×D61—21no—
81C2×Q8×C61—21no—
82C2×Q8×C61—81no—
83C2×Q8×C61C22⋊C412yes—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—41no—
87C22×C4×C61—81no—
88C22×C4×C61C22×C411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C2×C22⋊D61—41no—
95C2×C22⋊D61—41no—
96C2×C22⋊D61—41no—
97C2×C22⋊D61—41no—
98C2×C22⋊D61—41no—
99C2×C22⋊D61—41no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61—42no—
106C22×Dic61—41no—
107C22×Dic61—41no—
108C22×Dic61—81no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—81no—
112C2×C8×C61—41no—
113C2×C8×C61—81no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—81no—
118C4⋊C4×C61C22⋊C412yes—
119C22⋊C4×C61—22no—
120C22⋊C4×C61—22no—
121C22⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C22⋊Dic61—21no—
124C22⋊Dic61—21no—
125C22⋊Dic61—21no—
126C22⋊Dic61SD1612yescomplemented
127#11758201—82no—
128#11758198—41no—
129#11758198—41no—
130C4⋊Dic61—21no—
131C4⋊Dic61—21no—
132C4⋊Dic61—42no—
133C4⋊Dic61—81no—
134C4×Dic61—42no—
135C4×Dic61—81no—
136C4×Dic61—81no—
137C2×#92418—22no—
138C2×#92418—81no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61C4×D411yescharacteristic
142C23×C61#12011yescharacteristic
143C23×C61#12111yescharacteristic
144C23×C61C2×SD1611yescharacteristic
145C23×C61#13511yescharacteristic
146C23×C61AGL(1,ℤ/8)11yescharacteristic
147C23×C61#13711yescharacteristic
148C22×D61—41no—
149C22×D61—41no—
150C22×D61—41no—
151C22×D61—81no—
152C22×D61—81no—
153Q8×C61—42no—
154Q8×C61—81no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—81no—
168C2×C4×C61—81no—
169C2×C4×C61—81no—
170C2×C4×C61C2×C22⋊C411yescharacteristic
171C2×C4×C61C4×D411yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12311yescharacteristic
174C22⋊D61—81no—
175C22⋊D61—81no—
176C22⋊D61—81no—
177C22⋊D61—81no—
178C2×Dic61—22no—
179C2×Dic61—22no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—82no—
194C2×Dic61—82no—
195D244—42no—
196D244—81no—
197C8×C61—81no—
198#92418—42no—
199#9278—611nomaximal, Sylow
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—41no—
207C22×C61—41no—
208C22×C61—41no—
209C22×C61—41no—
210C22×C61#38611yescharacteristic
211C22×C61C4×SD1611yescharacteristic
212C22×C61#44011yescharacteristic
213C22×C61#44811yescharacteristic
214C22×C61#44911yescharacteristic
215C22×C61#46011yescharacteristic
216C22×C61#46411yescharacteristic
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—81no—
220C2×D61—161no—
221C2×D61—161no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—42no—
227C4×C61—42no—
228C4×C61—161no—
229Dic61—42no—
230Dic61—42no—
231Dic61—81no—
232Dic61—81no—
233Dic61—81no—
234#3187—611no—
235#2368—611no—
236#1964—611no—
237#1520D6111yescharacteristic, complemented
238C2×#327—611no—
239#1294—611no—
240#1282—611no—
241C2×C61—21no—
242C2×C61—21no—
243C2×C61—21no—
244C2×C61—21no—
245C2×C61—21no—
246C2×C61—21no—
247C2×C61#188011yescharacteristic
248C2×C61#190611yescharacteristic
249C2×C61#191311yescharacteristic
250D61—161no—
251C2×C22⋊Q8C2×D6111yescharacteristic
252C2×#121—611no—
253#483—612no—
254#481—1221no—
255#465—1222no—
256#450—612no—
257#448—1221no—
258C2×#99—1221no—
259C2×#99—1221no—
260C2×C22⋊C8—611no—
261C2×C22⋊C8C2×D6111yescharacteristic
262#391—1221no—
263#387—1221no—
264#387—1221no—
265#385—1221no—
266#380C2×D6111yescharacteristic
267C4×C4⋊C4—1221no—
268C4×C22⋊C4—611no—
269C4×C22⋊C4—1221no—
270C2×#95—611no—
271#328—22no—
272#327—1221no—
273#327—1221no—
274C61#927811yescharacteristic, complemented, Sylow
275C22×Q8D24411yescharacteristic
276C22×D4—611no—
277C22×D4—1221no—
278C23×C4—1221no—
279C23×C4C22×D6111yescharacteristic
280C2×SD16—1222no—
281C22×C8—21no—
282C22×C8—1221no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8C4×D6112yes—
286C22⋊Q8C22⋊D6112yes—
287#121—1221no—
288#121—1221no—
289#121—1221no—
290#121—1221no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—611no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4D24411yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—611no—
300C2×C22⋊C4—611no—
301C2×C22⋊C4—1221no—
302C2×C42—1221no—
303C2×C42—1221no—
304C2×C42—1221no—
305#106—1222no—
306#103—1222no—
307#102—1222no—
308#99—2442no—
309#99—2442no—
310C22⋊C8—21no—
311C22⋊C8—21no—
312C22⋊C8—612no—
313C22⋊C8—1221no—
314#95—21no—
315#95—611no—
316#95—611no—
317#95—1221no—
318#95—1221no—
319#95—1221no—
320#95—1221no—
321#95—1221no—
322#95—1221no—
323C24—1221no—
324C24D4×D6111yescharacteristic
325C2×Q8—21no—
326C2×Q8—81no—
327C2×Q8#1175819812yes—
328C2×D4—1221no—
329C2×D4—1221no—
330C2×D4—4881no—
331C2×D4—2441no—
332C2×D4—2441no—
333C2×D4—2441no—
334C2×D4—2441no—
335C2×D4—2441no—
336C2×D4—2441no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—81no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—4881no—
347C22×C4—2442no—
348C22×C4—2441no—
349C22×C4—2441no—
350C22×C4C2×C4×D6111yescharacteristic
351C22×C4C2×D24411yescharacteristic
352C22×C4D4×D6111yescharacteristic
353C22×C4#1175822411yescharacteristic
354C22×C4#1175822411yescharacteristic
355C22×C4C2×C22⋊D6111yescharacteristic
356SD16—4882no—
357C2×C8—41no—
358C2×C8—81no—
359C2×C8—1222no—
360C2×C8—4881no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—81no—
365C4⋊C4—1221no—
366C4⋊C4—1221no—
367C4⋊C4—4881no—
368C4⋊C4—2442no—
369C4⋊C4#1175819812yes—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—612no—
375C22⋊C4—1221no—
376C22⋊C4—1221no—
377C22⋊C4—1221no—
378C22⋊C4—2441no—
379C22⋊C4—2441no—
380C42—4881no—
381C42—4881no—
382C42—2442no—
383C23—41no—
384C23—41no—
385C23—4881no—
386C23—4881no—
387C23—2441no—
388C23—2441no—
389C23—2441no—
390C23#1451973711yescharacteristic
391C23#1451974711yescharacteristic
392C23#1451974911yescharacteristic
393C23SD16×D6111yescharacteristic
394C23#1451980011yescharacteristic
395C23#1451980411yescharacteristic
396C23#1451980911yescharacteristic
397Q8—42no—
398Q8—81no—
399D4—4881no—
400D4—4881no—
401D4—4881no—
402D4—4881no—
403D4—4881no—
404D4—2442no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—4882no—
423C2×C4—4882no—
424C2×C4—2441no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4C2×#1175819811yescharacteristic
438C2×C4#1451973711yescharacteristic
439C2×C4#1451974911yescharacteristic
440C2×C4#1451976111yescharacteristic
441C8—81no—
442C8—2442no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—9761no—
454C22—9761no—
455C22—4881no—
456C22—4881no—
457C22—4881no—
458C22#1683963911yescharacteristic
459C22#1683984711yescharacteristic
460C22#1683986511yescharacteristic
461C22#1683990211yescharacteristic
462C22#1683991011yescharacteristic
463C22#1683996011yescharacteristic
464C22#1683998611yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—42no—
470C4—42no—
471C4—161no—
472C4—4881no—
473C4—4881no—
474C4—4881no—
475C4—2442no—
476C4—2442no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—9761no—
484C2#2144617211yescharacteristic
485C2#2144631811yescharacteristic
486C2#2144635911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1520C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1520D61
hypocenterC61#9278
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
2. #21448904C2semidirect
5. #21443697C2semidirect
6. #21443666C2semidirect
7. C61×#1520C2semidirect
9. #16840090C4semidirect
12. #16839912C4semidirect
126. C22⋊Dic61SD16semidirect
237. #1520D61semidirect
274. C61#9278semidirect

Maximal subgroups

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

subgroupindex
#214565722
#214489042
#214466542
C2×#168394972
#214436972
#214436662
C61×#15202
#927861