Finite Group Databasebeta

← #29575852

Subgroups — #29575852

Order 15616.

486 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454854C211yesmaximal, characteristic
2#21445644C211yesmaximal, characteristic
3#21444434C211yesmaximal, characteristic, complemented
4#21444318C211yesmaximal, characteristic, complemented
5#21443717C211yesmaximal, characteristic, complemented
6#21443676C211yesmaximal, characteristic, complemented
7C61×#3110C211yesmaximal, characteristic
8#16839840—21no—
9#16839834—22no—
10D4×#92418—21no—
11#16839819C2211yescharacteristic
12#16839815C2211yescharacteristic
13C2×#14519698—21no—
14#16839726—21no—
15C2×C22⋊#92418—21no—
16C2×#14519688C2211yescharacteristic
17C4×C4⋊Dic61C2211yescharacteristic
18#16839524—21no—
19#16839504—22no—
20#16839499—21no—
21#16839499—21no—
22C61×#533C2211yescharacteristic
23C2×C4×D4×C61C2211yescharacteristic
24C61×#484—21no—
25C61×#474C411yescharacteristic, complemented
26C61×#474C411yescharacteristic, complemented
27C61×#461C411yescharacteristic, complemented
28C61×#461C411yescharacteristic, complemented
29C61×#450—21no—
30C61×#440—21no—
31C61×#440—21no—
32C61×#432C2211yescharacteristic
33C2×#11758194—22no—
34C22×#92418—21no—
35C22×#92418—22no—
36C2×C4⋊Dic61—21no—
37C2×C4×Dic61—21no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40#14519698—24no—
41C22⋊#92418—22no—
42C22⋊#92418—22no—
43C22⋊#92418—21no—
44C22⋊#92418D411yescharacteristic
45C22⋊#92418D411yescharacteristic
46#14519688—21no—
47#14519688—21no—
48#14519688D412yes—
49C4×#92418—21no—
50#14519684—21no—
51C22×D4×C61D411yescharacteristic
52C23×C4×C61—21no—
53C2×SD16×C61—21no—
54C2×SD16×C61—21no—
55C4⋊Q8×C61C2×C411yescharacteristic
56C4⋊Q8×C61C2×C411yescharacteristic
57#125×C61—21no—
58C22⋊Q8×C61—21no—
59C22⋊Q8×C61—21no—
60C4×Q8×C61—21no—
61C4×Q8×C61—21no—
62C4×D4×C61—21no—
63C4×D4×C61—21no—
64C4×D4×C61C2×C411yescharacteristic
65C4×D4×C61C2×C411yescharacteristic
66C4×D4×C61D411yescharacteristic
67C4×D4×C61D411yescharacteristic
68#117×C61—21no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C22⋊C4×C61—21no—
71C2×C42×C61C2311yescharacteristic
72#107×C61—21no—
73#107×C61—21no—
74C4⋊C8×C61C2×C411yescharacteristic
75C4⋊C8×C61C2×C411yescharacteristic
76#103×C61—21no—
77#103×C61—21no—
78#102×C61—21no—
79#102×C61—21no—
80C22⋊C8×C61—21no—
81#97×C61—21no—
82C24×C61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—21no—
85C2×D4×C61—22no—
86C2×D4×C61—21no—
87C2×D4×C61—41no—
88C2×D4×C61C22⋊C411yescharacteristic
89C2×D4×C61C22⋊C411yescharacteristic
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—82no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C22×Dic61—41no—
100SD16×C61—42no—
101SD16×C61—42no—
102C2×C8×C61—21no—
103C2×C8×C61—21no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61C22⋊C411yescharacteristic
112C4⋊C4×C61C22⋊C411yescharacteristic
113C22⋊C4×C61—21no—
114C22⋊C4×C61—21no—
115C22⋊C4×C61—21no—
116C22⋊C4×C61—21no—
117C22⋊C4×C61—41no—
118C42×C61—41no—
119C42×C61C22×C411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61#4111yescharacteristic
123C4⋊Dic61—22no—
124C4⋊Dic61—41no—
125C4×Dic61—81no—
126#11758194—82no—
127C2×#92418—22no—
128C2×#92418—21no—
129C2×#92418—21no—
130C2×#92418—22no—
131C2×#92418—41no—
132C2×#92418—41no—
133C2×#92418—82no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—42no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—81no—
140C23×C61#12111yescharacteristic
141Q8×C61—41no—
142Q8×C61—41no—
143Q8×C61—81no—
144D4×C61—21no—
145D4×C61—21no—
146D4×C61—21no—
147D4×C61—42no—
148D4×C61#10412yes—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—81no—
161C2×C4×C61—81no—
162C2×C4×C61—81no—
163C2×C4×C61C2×C22⋊C411yescharacteristic
164C2×C4×C61C4×D411yescharacteristic
165C2×C4×C61C4×D411yescharacteristic
166C2×C4×C61#12011yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12311yescharacteristic
169C2×Dic61—42no—
170C2×Dic61—81no—
171C2×Dic61—81no—
172C8×C61—41no—
173C8×C61—41no—
174#92418—42no—
175#92418—41no—
176#92418—42no—
177#92418—41no—
178#9165—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—22no—
182C22×C61—41no—
183C22×C61—41no—
184C22×C61—41no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—82no—
188C22×C61—82no—
189C22×C61#38611yescharacteristic
190C22×C61#47111yescharacteristic
191C22×C61#47311yescharacteristic
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—41no—
198C4×C61—41no—
199C4×C61—81no—
200C4×C61—81no—
201C4×C61C2×#10411yescharacteristic
202C4×C61#42111yescharacteristic
203Dic61—82no—
204#3110D6111yescharacteristic
205#2974—611no—
206#1762—611no—
207#1531—611no—
208#1503—611no—
209#1302—611no—
210#1285—611no—
211C2×C61—21no—
212C2×C61—21no—
213C2×C61—42no—
214C2×C61—81no—
215C2×C61#187111yescharacteristic
216C2×C61#190411yescharacteristic
217C2×C61#190811yescharacteristic
218#533C2×D6111yescharacteristic
219C2×C4×D4C2×D6111yescharacteristic
220#484—21no—
221#474Dic6111yescharacteristic, complemented
222#474Dic6111yescharacteristic, complemented
223#461Dic6111yescharacteristic, complemented
224#461Dic6111yescharacteristic, complemented
225#450—21no—
226#440—21no—
227#440—21no—
228#436—1221no—
229#435—1222no—
230C8×D4—1221no—
231#432—611no—
232#432C2×D6111yescharacteristic
233#431—611no—
234C2×#102—1221no—
235#407—1221no—
236C2×C22⋊C8—1221no—
237C2×#97—611no—
238C4×C4⋊C4—611no—
239#340—1221no—
240#330—1222no—
241#328—1221no—
242#328—1221no—
243C61#916511yescharacteristic, complemented, Sylow
244C22×D4C22⋊D6111yescharacteristic
245C23×C4—21no—
246C2×SD16—21no—
247C2×SD16—21no—
248C2×M16—1222no—
249C22×C8—1222no—
250C22×C8—1221no—
251C4⋊Q8C2×Dic6111yescharacteristic
252C4⋊Q8C2×Dic6111yescharacteristic
253#125—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4C2×Dic6111yescharacteristic
261C4×D4C2×Dic6111yescharacteristic
262C4×D4C22⋊D6111yescharacteristic
263C4×D4C22⋊D6111yescharacteristic
264#117—21no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4C22⋊D6111yescharacteristic
267C2×C22⋊C4—21no—
268C2×C42—1221no—
269C2×C42C22×D6111yescharacteristic
270#107—21no—
271#107—21no—
272C4⋊C8—1221no—
273C4⋊C8—1221no—
274C4⋊C8C2×Dic6111yescharacteristic
275C4⋊C8C2×Dic6111yescharacteristic
276#103—21no—
277#103—21no—
278#102—21no—
279#102—21no—
280#102—1224no—
281C22⋊C8—21no—
282C22⋊C8—611no—
283C22⋊C8—611no—
284C22⋊C8—1222no—
285C22⋊C8—1222no—
286C22⋊C8—1221no—
287#97—21no—
288#97—612no—
289#97—1221no—
290#97—1221no—
291C4×C8—1221no—
292#95—1221no—
293C24—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4C22⋊Dic6111yescharacteristic
300C2×D4C22⋊Dic6111yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—82no—
307C22×C4—2441no—
308C22×C4D4×D6111yescharacteristic
309C22×C4#1175822411yescharacteristic
310C22×C4C2×C22⋊D6111yescharacteristic
311SD16—42no—
312SD16—42no—
313M16—4882no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—1222no—
317C2×C8—1221no—
318C2×C8—1221no—
319C2×C8—1222no—
320C2×C8—4882no—
321C2×C8—2441no—
322C2×C8—2441no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—1222no—
331C4⋊C4—2441no—
332C4⋊C4C22⋊Dic6111yescharacteristic
333C4⋊C4C22⋊Dic6111yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C22⋊C4—41no—
339C42—41no—
340C42—4881no—
341C42D4×D6111yescharacteristic
342C42#1175822411yescharacteristic
343C42C22×Dic6111yescharacteristic
344C42C2×C22⋊D6111yescharacteristic
345C23—21no—
346C23—21no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—81no—
351C23#1451974911yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—81no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4#1451970412yes—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—4881no—
375C2×C4—4881no—
376C2×C4—2442no—
377C2×C4C2×C22⋊Dic6111yescharacteristic
378C2×C4D4×Dic6111yescharacteristic
379C2×C4D4×Dic6111yescharacteristic
380C2×C4#1451974511yescharacteristic
381C2×C4#1451974911yescharacteristic
382C2×C4#1451975811yescharacteristic
383C8—41no—
384C8—41no—
385C8—2441no—
386C8—2442no—
387C8—2442no—
388C8—2441no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—82no—
398C22—82no—
399C22#1683964211yescharacteristic
400C22#1684002711yescharacteristic
401C22#1684003511yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—81no—
410C4—81no—
411C4—4882no—
412C4C2×#1451970411yescharacteristic
413C4#1683977911yescharacteristic
414C2—21no—
415C2—21no—
416C2—42no—
417C2—81no—
418C2#2144612211yescharacteristic
419C2#2144630811yescharacteristic
420C2#2144633611yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3110C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#3110D61
hypocenterC61#9165
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
3. #21444434C2semidirect
4. #21444318C2semidirect
5. #21443717C2semidirect
6. #21443676C2semidirect
25. C61×#474C4semidirect
26. C61×#474C4semidirect
27. C61×#461C4semidirect
28. C61×#461C4semidirect
221. #474Dic61semidirect
222. #474Dic61semidirect
223. #461Dic61semidirect
224. #461Dic61semidirect
243. C61#9165semidirect

Maximal subgroups

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

subgroupindex
#214548542
#214456442
#214444342
#214443182
#214437172
#214436762
C61×#31102
#916561