Finite Group Databasebeta

← #29662876

Subgroups — #29662876

Order 15616.

682 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456276C211yesmaximal, characteristic
2#21455850C211yesmaximal, characteristic
3Q8×#11758194C211yesmaximal, characteristic, complemented
4#21445311C212yesmaximal, complemented
5#21445293C212yesmaximal, complemented
6#21445187C212yesmaximal, complemented
7#21445169C212yesmaximal, complemented
8#21444569C211yesmaximal, characteristic, complemented
9#21444494C211yesmaximal, characteristic
10#21444491C211yesmaximal, characteristic, complemented
11C2×C61×#475C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13C2×#14519808—21no—
14#16840312C2211yescharacteristic
15#16840154—22no—
16#16840141C2212yes—
17#16840136C2212yes—
18#16840080—21no—
19#16840047C2211yescharacteristic
20#16840047C2211yescharacteristic
21#16839962C2211yescharacteristic
22#16839962C2211yescharacteristic
23#16839901—21no—
24#16839893—22no—
25#16839893C2212yes—
26Q8×#92418C2212yes—
27C4×#11758201—21no—
28C4×#11758201—21no—
29#16839815C2211yescharacteristic
30#16839786—21no—
31#16839786C2211yescharacteristic
32C4×#11758194—21no—
33C4×#11758194C2211yescharacteristic
34#16839514C2212yes—
35#16839512C2212yes—
36#16839508C2212yes—
37#16839508C2212yes—
38#16839507C2212yes—
39#16839505C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4×Q8×C61C2211yescharacteristic
42C61×#475—21no—
43C61×#475—21no—
44C61×#475C2212yes—
45C61×#475C2212yes—
46C2×#106×C61—21no—
47C2×C4⋊C8×C61C2211yescharacteristic
48C2×#103×C61—21no—
49C2×#11758222—21no—
50#14519808—44no—
51#14519808—44no—
52C2×#11758202—22no—
53C2×#11758202—22no—
54C2×#11758201—21no—
55C2×#11758201—21no—
56C2×#11758201—22no—
57C2×#11758201—22no—
58C2×#11758194—21no—
59C2×#11758194D411yescharacteristic
60C2×#11758194D411yescharacteristic
61#14519777C2311yescharacteristic
62#14519774C2311yescharacteristic
63#14519765—21no—
64#14519751—21no—
65#14519751—21no—
66C4×D244—21no—
67C4×D244—21no—
68#14519708—21no—
69#14519708—21no—
70C4⋊#92418—21no—
71C4⋊#92418—22no—
72C4⋊#92418—22no—
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2312yes—
76#14519701—21no—
77#14519701—21no—
78#14519697—21no—
79#14519697—21no—
80C22⋊#92418—21no—
81#14519688—22no—
82#14519688D412yes—
83C4×#92418—21no—
84C4×#92418—22no—
85C4×#92418C2312yes—
86C22×Q8×C61—21no—
87C22×Q8×C61D411yescharacteristic
88C22×C8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61C2312yes—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61D411yescharacteristic
96C4×Q8×C61D411yescharacteristic
97C4×Q8×C61C2311yescharacteristic
98C4×Q8×C61C2311yescharacteristic
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105#106×C61—22no—
106#106×C61—22no—
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109#103×C61—22no—
110#103×C61—22no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—41no—
115C2×Q8×C61—41no—
116C2×Q8×C61—42no—
117C2×Q8×C61—42no—
118C2×Q8×C61C2×D411yescharacteristic
119C2×Q8×C61C2×D411yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—41no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C2×C22⋊D61—41no—
127#11758222—42no—
128#11758222—42no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×C4×D61—41no—
133C2×Dic122—21no—
134C2×C8×C61—22no—
135C2×C8×C61—42no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61C2×D411yescharacteristic
146C4⋊C4×C61C2×D411yescharacteristic
147C42×C61—21no—
148C42×C61—21no—
149C42×C61—21no—
150C42×C61—21no—
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61#4111yescharacteristic
154C42×C61C2411yescharacteristic
155#11758202—44no—
156#11758202—44no—
157#11758201—44no—
158#11758201—42no—
159#11758201—42no—
160#11758201—44no—
161#11758198—41no—
162#11758198—41no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165#11758194—22no—
166#11758194—42no—
167#11758194SD1614yescomplemented
168C2×#92418—22no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418C2×D412yes—
172C2×#92418C2×D412yes—
173C23×C61#12111yescharacteristic
174C22×D61—41no—
175Q8×C61—21no—
176Q8×C61—21no—
177Q8×C61—21no—
178Q8×C61—42no—
179Q8×C61—42no—
180Q8×C61—42no—
181Q8×C61—82no—
182Q8×C61#13712yescomplemented
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—81no—
193C2×C4×C61—82no—
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61C2×SD1612yes—
198C2×C4×C61AGL(1,ℤ/8)12yes—
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C22⋊D61—82no—
203C2×Dic61—41no—
204D244—41no—
205D244—41no—
206D244—41no—
207D244—41no—
208D244—81no—
209C4×D61—82no—
210Dic122—41no—
211Dic122—41no—
212C8×C61—42no—
213#92418—22no—
214#92418—42no—
215#92418—41no—
216#92418—41no—
217#92418C2×SD1614yes—
218#19171—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61#46012yes—
222C22×C61C2×#12111yescharacteristic
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—81no—
226C4×C61—21no—
227C4×C61—22no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—41no—
231C4×C61—41no—
232C4×C61—41no—
233C4×C61—82no—
234C4×C61#46012yes—
235C4×C61C22×SD1611yescharacteristic
236C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C4×C61C2×#13711yescharacteristic
238C4×C61#57511yescharacteristic
239Dic61—81no—
240#3152—611no—
241#3103—611no—
242C2×#475D6111yescharacteristic, complemented
243Q8×M16—611no—
244#1690—612no—
245#1686—612no—
246#1670—612no—
247#1666—612no—
248#1565—611no—
249#1547—611no—
250#1546—611no—
251C2×C61—22no—
252C2×C61C2×#46011yescharacteristic
253C2×C61#305611yescharacteristic
254C2×C61#306911yescharacteristic
255D61—161no—
256C2×#137—1221no—
257C2×#137—1221no—
258#532—611no—
259C2×C4⋊Q8C2×D6111yescharacteristic
260C2×C4×Q8C2×D6111yescharacteristic
261#497—1222no—
262#494—612no—
263#492—612no—
264#481—1221no—
265#475—21no—
266#475—21no—
267#475—611no—
268#475—611no—
269#475C2×D6112yes—
270#475C2×D6112yes—
271#460—611no—
272#460—611no—
273#448—1221no—
274#446—612no—
275#446—1222no—
276C8×Q8—612no—
277C4×SD16—1221no—
278C4×SD16—1221no—
279#431—611no—
280C2×#106—21no—
281#423—611no—
282#423—1221no—
283C2×C4⋊C8C2×D6111yescharacteristic
284C2×#103—21no—
285C4×M16—611no—
286C4×M16—1221no—
287#335—612no—
288#334—612no—
289#332—612no—
290#332—612no—
291#332—612no—
292#331—612no—
293C61#1917111yescharacteristic, complemented, Sylow
294C2×#41—1221no—
295C22×Q8—21no—
296C22×Q8C22⋊D6111yescharacteristic
297#137—2444no—
298#137—2444no—
299C2×Q16—1222no—
300C2×Q16—1222no—
301C2×SD16—1222no—
302C2×SD16—1222no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×M16—611no—
306C2×M16—611no—
307C2×M16—1221no—
308C22×C8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—611no—
313C4⋊Q8C22×D6112yes—
314#127—611no—
315#124—1221no—
316#121—1221no—
317#121—1221no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8C22⋊D6111yescharacteristic
321C4×Q8C22⋊D6111yescharacteristic
322C4×Q8C22×D6111yescharacteristic
323C4×Q8C22×D6111yescharacteristic
324C4×D4—1221no—
325C4×D4—1221no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4C22⋊D6111yescharacteristic
330C2×C42—21no—
331C2×C42C22×D6111yescharacteristic
332#106—22no—
333#106—22no—
334#106—1221no—
335#106—1221no—
336C4⋊C8—21no—
337C4⋊C8—612no—
338C4⋊C8—611no—
339C4⋊C8—611no—
340C4⋊C8—1222no—
341C4⋊C8—1222no—
342C4⋊C8—1221no—
343C4⋊C8C22×D6112yes—
344#103—22no—
345#103—22no—
346#103—1221no—
347#103—1221no—
348#102—1221no—
349#102—1221no—
350C22⋊C8—1221no—
351#97—612no—
352#97—1222no—
353C4×C8—612no—
354C4×C8—1222no—
355C4×C8—1221no—
356#41—2442no—
357#41—2442no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—1221no—
366C2×Q8C2×C22⋊D6111yescharacteristic
367C2×Q8C2×C22⋊D6111yescharacteristic
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—2441no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—41no—
375C22×C4—2441no—
376C22×C4D4×D6111yescharacteristic
377C22×C4#1175822411yescharacteristic
378C22×C4C2×C22⋊D6111yescharacteristic
379Q16—2444no—
380Q16—2444no—
381SD16—2444no—
382SD16—2444no—
383SD16—2442no—
384SD16—2442no—
385M16—614no—
386M16—1222no—
387M16—2442no—
388C2×C8—22no—
389C2×C8—42no—
390C2×C8—612no—
391C2×C8—612no—
392C2×C8—1222no—
393C2×C8—1221no—
394C2×C8—1221no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4C2×C22⋊D6111yescharacteristic
407C4⋊C4C2×C22⋊D6111yescharacteristic
408C22⋊C4—2441no—
409C22⋊C4—2441no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42D4×D6111yescharacteristic
415C42#1175822411yescharacteristic
416C42C2×C22⋊D6111yescharacteristic
417C42C23×D6111yescharacteristic
418C23—2441no—
419C23#1451974911yescharacteristic
420Q8—21no—
421Q8—21no—
422Q8—21no—
423Q8—42no—
424Q8—42no—
425Q8—42no—
426Q8—82no—
427Q8—2441no—
428Q8—2441no—
429Q8#1451980812yescomplemented
430D4—4881no—
431D4—4882no—
432D4—2441no—
433D4—2441no—
434D4—2441no—
435D4—2441no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—4882no—
448C2×C4—2441no—
449C2×C4#1451974911yescharacteristic
450C2×C4#1451974911yescharacteristic
451C2×C4#1451974911yescharacteristic
452C2×C4SD16×D6112yes—
453C2×C4#1451980412yes—
454C2×C4C2×D4×D6111yescharacteristic
455C2×C4C22×C22⋊D6111yescharacteristic
456C2×C4C2×#1175822411yescharacteristic
457C8—42no—
458C8—614no—
459C8—1222no—
460C8—2442no—
461C8—2441no—
462C8—2441no—
463C22—21no—
464C22—21no—
465C22—4881no—
466C22—4881no—
467C22—4881no—
468C22#1683996012yes—
469C22C2×#1451974911yescharacteristic
470C4—21no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—82no—
478C4—4881no—
479C4#1683996012yes—
480C4C2×SD16×D6111yescharacteristic
481C4C2×#1451980411yescharacteristic
482C4C2×#1451980811yescharacteristic
483C4#1684056011yescharacteristic
484C2—22no—
485C2—9761no—
486C2C2×#1683996011yescharacteristic
487C2#2145544511yescharacteristic
488C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#475C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenterC2×#475D61
hypocenterC61#19171
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. Q8×#11758194C2semidirect
4. #21445311C2semidirect
5. #21445293C2semidirect
6. #21445187C2semidirect
7. #21445169C2semidirect
8. #21444569C2semidirect
10. #21444491C2semidirect
11. C2×C61×#475C2semidirect
167. #11758194SD16semidirect
182. Q8×C61#137semidirect
242. C2×#475D61semidirect
293. C61#19171semidirect
429. Q8#14519808semidirect

Maximal subgroups

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

subgroupindex
#214562762
#214558502
Q8×#117581942
#214453112
#214452932
#214451872
#214451692
#214445692
#214444942
#214444912
C2×C61×#4752
#1917161