Finite Group Databasebeta

← #29663594

Subgroups — #29663594

Order 15616.

658 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457136C211yesmaximal, characteristic
2#21456963C211yesmaximal, characteristic, complemented
3#21456276C211yesmaximal, characteristic
4#21454762C211yesmaximal, characteristic, complemented
5#21445312C212yesmaximal, complemented
6#21445294C212yesmaximal, complemented
7#21445188C212yesmaximal, complemented
8#21445170C212yesmaximal, complemented
9#21444569C211yesmaximal, characteristic, complemented
10#21444469C211yesmaximal, characteristic
11C2×C61×#332C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13C2×#14519808—21no—
14#16840319C2211yescharacteristic
15#16840312C2211yescharacteristic
16#16840158—21no—
17#16840154—22no—
18#16840141C2212yes—
19#16840136C2212yes—
20#16840114—21no—
21#16840092—22no—
22#16840053—21no—
23#16840048C2212yes—
24#16840044C2212yes—
25#16840044C2212yes—
26#16839991—22no—
27#16839968—21no—
28#16839963C2212yes—
29#16839891—22no—
30#16839889C2212yes—
31#16839886C2212yes—
32#16839815C2211yescharacteristic
33#16839788—21no—
34#16839717—21no—
35C4×#11758194C2211yescharacteristic
36#16839514C2212yes—
37#16839512C2212yes—
38#16839509C2211yescharacteristic
39#16839509C2211yescharacteristic
40#16839505C2211yescharacteristic
41#16839505C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×C4⋊C8×C61C2211yescharacteristic
45C61×#332C2212yes—
46C61×#332C2212yes—
47C2×#11758222—21no—
48#14519808—44no—
49#14519808—44no—
50C2×#11758202—22no—
51C2×#11758202—22no—
52C2×#11758190—22no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C2×#11758194D411yescharacteristic
56C2×#11758194D411yescharacteristic
57#14519777D411yescharacteristic
58#14519777D411yescharacteristic
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519769—21no—
62#14519765—21no—
63#14519756—21no—
64#14519756D412yes—
65#14519751—21no—
66#14519751—21no—
67C4×Dic122—21no—
68C4×Dic122C2312yes—
69C4×D244—21no—
70C4×D244—21no—
71#14519736—21no—
72#14519709—22no—
73#14519708—22no—
74#14519708—22no—
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2311yescharacteristic
77#14519706—22no—
78#14519706—22no—
79#14519702—22no—
80#14519700—22no—
81#14519700—22no—
82#14519699—22no—
83C22⋊#92418—21no—
84#14519688D412yes—
85#14519687—22no—
86C4×#92418—21no—
87C4×#92418C2312yes—
88C8×Dic61—22no—
89C22×Q8×C61—21no—
90C22×C8×C61—21no—
91C22×C8×C61—21no—
92C4⋊Q8×C61—21no—
93C4⋊Q8×C61—21no—
94C4⋊Q8×C61—21no—
95C4⋊Q8×C61C2312yes—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—21no—
100C4⋊C8×C61—21no—
101C4⋊C8×C61C2312yes—
102C4⋊C8×C61C2311yescharacteristic
103C4⋊C8×C61C2311yescharacteristic
104C2×Q8×C61—22no—
105C2×Q8×C61—41no—
106C2×Q8×C61—42no—
107C2×Q8×C61—42no—
108C22×C4×C61—41no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×C22⋊D61—41no—
113#11758222—42no—
114#11758222—42no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×C4×D61—41no—
119C2×Dic122—21no—
120C2×Dic122—21no—
121C2×Dic122C2×D412yes—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—22no—
125C2×C8×C61—42no—
126C2×C8×C61—81no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—41no—
130C4⋊C4×C61—41no—
131C42×C61C2×D411yescharacteristic
132C42×C61C2×D411yescharacteristic
133C42×C61#4111yescharacteristic
134C42×C61C2411yescharacteristic
135C22⋊Dic61—22no—
136#11758202—44no—
137#11758202—44no—
138#11758201—44no—
139#11758201—44no—
140#11758198—41no—
141#11758198—41no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61C2×D412yes—
148#11758196—22no—
149#11758196—22no—
150C4×Dic61—22no—
151#11758194—22no—
152#11758194SD1614yescomplemented
153C2×#92418—21no—
154C2×#92418—21no—
155C2×#92418C2×D412yes—
156C2×#92418C2×D412yes—
157#11758190—82no—
158C23×C61#12111yescharacteristic
159C22×D61—41no—
160Q8×C61—42no—
161Q8×C61—42no—
162Q8×C61—82no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—41no—
166C2×C4×C61—42no—
167C2×C4×C61—82no—
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C2×SD1612yes—
172C2×C4×C61AGL(1,ℤ/8)12yes—
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C22⋊D61—82no—
177C2×Dic61—22no—
178C2×Dic61—22no—
179C2×Dic61—41no—
180D244—41no—
181D244—41no—
182D244—41no—
183D244—41no—
184D244—81no—
185C4×D61—82no—
186Dic122—22no—
187Dic122—22no—
188Dic122—41no—
189Dic122—41no—
190Dic122—81no—
191C8×C61—41no—
192C8×C61—41no—
193C8×C61—42no—
194#92418—22no—
195#92418—41no—
196#92418—41no—
197#92418C2×SD1614yes—
198#19239—611nomaximal, Sylow
199C22×C61—21no—
200C22×C61—21no—
201C22×C61#46012yes—
202C22×C61C2×#12111yescharacteristic
203C2×D61—81no—
204C2×D61—81no—
205C2×D61—81no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—82no—
210C4×C61#46012yes—
211C4×C61C22×SD1611yescharacteristic
212C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
213C4×C61#57711yescharacteristic
214C4×C61#57811yescharacteristic
215Dic61—42no—
216Dic61—42no—
217Dic61—81no—
218#3246—611no—
219#3229—611no—
220#3152—611no—
221#2962—611no—
222#1690—612no—
223#1686—612no—
224#1670—612no—
225#1666—612no—
226#1565—611no—
227#1540—611no—
228C2×#332D6111yescharacteristic, complemented
229C2×C61—22no—
230C2×C61C2×#46011yescharacteristic
231C2×C61#306311yescharacteristic
232C2×C61#306911yescharacteristic
233D61—161no—
234C2×#137—1221no—
235C2×#137—1221no—
236#533—611no—
237#532—611no—
238C2×C4⋊Q8C2×D6111yescharacteristic
239#498—1221no—
240#497—1222no—
241#494—612no—
242#492—612no—
243#487—1221no—
244#483—1222no—
245#476—1221no—
246#475—612no—
247#475—612no—
248#475—612no—
249#465—1222no—
250#461—1221no—
251#460—612no—
252#446—612no—
253#446—1222no—
254C8×Q8—612no—
255#431—611no—
256#424—1221no—
257C2×C4⋊C8C2×D6111yescharacteristic
258C2×C4⋊C8C2×D6111yescharacteristic
259#405—1221no—
260C4×M16—611no—
261#335—612no—
262#334—612no—
263#332—611no—
264#332—611no—
265#332C2×D6112yes—
266#332C2×D6112yes—
267#331—611no—
268#331—611no—
269C61#1923911yescharacteristic, complemented, Sylow
270C2×#41—1221no—
271C22×Q8—21no—
272#137—2444no—
273#137—2444no—
274C2×Q16—1222no—
275C2×Q16—1222no—
276C2×SD16—1222no—
277C2×SD16—1222no—
278C2×SD16—1222no—
279C2×M16—611no—
280C2×M16—611no—
281C22×C8—21no—
282C22×C8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—611no—
287C4⋊Q8—611no—
288C4⋊Q8—611no—
289C4⋊Q8C22×D6112yes—
290#127—611no—
291#125—1221no—
292#124—1221no—
293C22⋊Q8—612no—
294C22⋊Q8—1221no—
295#121—1221no—
296#121—1221no—
297C4×Q8—612no—
298C4×Q8—1221no—
299C4×D4—1221no—
300C4×D4—1221no—
301#117—1221no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C42C22×D6111yescharacteristic
305#106—1222no—
306#106—1222no—
307#106—1222no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—611no—
311C4⋊C8—611no—
312C4⋊C8—1222no—
313C4⋊C8—1222no—
314C4⋊C8C22×D6111yescharacteristic
315C4⋊C8C22×D6111yescharacteristic
316C4⋊C8C22×D6112yes—
317#103—1222no—
318#103—1222no—
319#103—1222no—
320#102—1222no—
321C22⋊C8—1221no—
322#97—612no—
323#97—1222no—
324C4×C8—612no—
325C4×C8—1222no—
326C4×C8—1221no—
327#41—2442no—
328#41—2442no—
329C2×Q8—22no—
330C2×Q8—41no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—612no—
334C2×Q8—1221no—
335C2×Q8—1221no—
336C2×D4—1221no—
337C2×D4—1221no—
338C2×D4—1221no—
339C2×D4—2441no—
340C22×C4—41no—
341C22×C4—2441no—
342C22×C4#1175822211yescharacteristic
343C22×C4D4×D6111yescharacteristic
344C22×C4D4×D6111yescharacteristic
345Q16—2444no—
346Q16—2444no—
347SD16—4882no—
348SD16—2444no—
349SD16—2444no—
350M16—614no—
351M16—1222no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—42no—
356C2×C8—81no—
357C2×C8—612no—
358C2×C8—612no—
359C2×C8—1221no—
360C2×C8—1221no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—612no—
366C4⋊C4—1222no—
367C4⋊C4—1222no—
368C4⋊C4—1221no—
369C4⋊C4—1221no—
370C4⋊C4—1221no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C22⋊C4—1222no—
374C22⋊C4—2441no—
375C22⋊C4—2441no—
376C42—1222no—
377C42#1175822211yescharacteristic
378C42D4×D6111yescharacteristic
379C42D4×D6111yescharacteristic
380C42C23×D6111yescharacteristic
381C23—2441no—
382C23#1451975011yescharacteristic
383Q8—42no—
384Q8—42no—
385Q8—82no—
386Q8—1222no—
387Q8—1222no—
388Q8—4881no—
389Q8—2441no—
390Q8—2441no—
391D4—4881no—
392D4—4882no—
393D4—2441no—
394D4—2441no—
395D4—2441no—
396D4—2441no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—82no—
402C2×C4—1222no—
403C2×C4—1222no—
404C2×C4—4882no—
405C2×C4—2441no—
406C2×C4#1451975011yescharacteristic
407C2×C4#1451975011yescharacteristic
408C2×C4#1451975011yescharacteristic
409C2×C4SD16×D6112yes—
410C2×C4#1451980112yes—
411C2×C4C2×D4×D6111yescharacteristic
412C2×C4C2×D4×D6111yescharacteristic
413C2×C4C2×#1175822211yescharacteristic
414C8—41no—
415C8—41no—
416C8—42no—
417C8—614no—
418C8—1222no—
419C8—2441no—
420C8—2441no—
421C22—21no—
422C22—21no—
423C22—4881no—
424C22—4881no—
425C22—4881no—
426C22#1683996112yes—
427C22C2×#1451975011yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—82no—
432C4—4881no—
433C4—2442no—
434C4—2442no—
435C4#1683996112yes—
436C4C2×SD16×D6111yescharacteristic
437C4C2×#1451980111yescharacteristic
438C4#1684056811yescharacteristic
439C4#1684057511yescharacteristic
440C2—22no—
441C2—9761no—
442C2C2×#1683996111yescharacteristic
443C2#2145550811yescharacteristic
444C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#332C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenterC2×#332D61
hypocenterC61#19239
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. #21456963C2semidirect
4. #21454762C2semidirect
5. #21445312C2semidirect
6. #21445294C2semidirect
7. #21445188C2semidirect
8. #21445170C2semidirect
9. #21444569C2semidirect
11. C2×C61×#332C2semidirect
152. #11758194SD16semidirect
228. C2×#332D61semidirect
269. C61#19239semidirect

Maximal subgroups

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

subgroupindex
#214571362
#214569632
#214562762
#214547622
#214453122
#214452942
#214451882
#214451702
#214445692
#214444692
C2×C61×#3322
#1923961