Finite Group Databasebeta

← #29574824

Subgroups — #29574824

Order 15616.

592 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840043C211yesmaximal, characteristic
2#21448770C211yesmaximal, characteristic
3#21446779C211yesmaximal, characteristic, complemented
4#21446741C211yesmaximal, characteristic, complemented
5C2×#16839489C211yesmaximal, characteristic
6#21443658C211yesmaximal, characteristic, complemented
7C61×#1296C211yesmaximal, characteristic, complemented
8C2×#14519777C2211yescharacteristic
9C2×C4×D244C2211yescharacteristic
10C2×#14519735—21no—
11#16840043—22no—
12#16840043C414yescomplemented
13C2×#14519711—21no—
14C2×#14519702—21no—
15C2×#14519702—21no—
16C2×#14519699—21no—
17C2×C4×#92418C2211yescharacteristic
18#16839663—21no—
19#16839652—22no—
20#16839619—21no—
21C4⋊C4×Dic61—21no—
22C4×C4⋊Dic61C2211yescharacteristic
23#16839524—22no—
24#16839489—22no—
25C2×C4⋊C8×C61C2211yescharacteristic
26C2×C4⋊C8×C61C2211yescharacteristic
27C61×#384C2211yescharacteristic
28C22×Dic122—21no—
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31C22×#92418—21no—
32#14519777—21no—
33#14519777—21no—
34#14519777—21no—
35#14519777C2×C412yes—
36C4×D244—21no—
37C4×D244—21no—
38C4×D244C2×C412yes—
39C4×D244D412yes—
40#14519735—42no—
41#14519735—42no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×C4⋊Dic61D411yescharacteristic
46C2×C4⋊Dic61D411yescharacteristic
47C2×#11758198—21no—
48C2×#11758198—22no—
49C2×#11758198—22no—
50C2×C4×Dic61—22no—
51C2×C4×Dic61—21no—
52#14519711—24no—
53#14519702—42no—
54#14519702—42no—
55#14519699—24no—
56C4×#92418—21no—
57C4×#92418—21no—
58#14519684—22no—
59#14519684—21no—
60C22×C8×C61—21no—
61C22×C8×C61—21no—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61D412yes—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C4⋊C8×C61—21no—
68C4⋊C8×C61—21no—
69C4⋊C8×C61C2×C412yes—
70C23×D61—21no—
71C22×C4×C61—22no—
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—41no—
79C2×D244—21no—
80C2×D244—21no—
81C2×D244—21no—
82C2×D244—21no—
83C2×D244—21no—
84C2×D244C22⋊C412yes—
85C2×C4×D61—22no—
86C2×C4×D61—22no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61—41no—
90C2×C4×D61—41no—
91C2×Dic122—22no—
92C2×Dic122—41no—
93C2×Dic122—41no—
94C2×Dic122—41no—
95C2×Dic122—41no—
96C2×Dic122—41no—
97C2×C8×C61—22no—
98C2×C8×C61—41no—
99C2×C8×C61—41no—
100C2×C8×C61—41no—
101C2×C8×C61—41no—
102C2×C8×C61—41no—
103C4⋊C4×C61—42no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C42×C61C22×C411yescharacteristic
109C42×C61C2×D411yescharacteristic
110C42×C61#4111yescharacteristic
111C42×C61#4111yescharacteristic
112#11758198—22no—
113#11758198—22no—
114#11758198—44no—
115#11758198—82no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—22no—
119C4⋊Dic61—22no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122C4⋊Dic61—41no—
123C4⋊Dic61—41no—
124C4⋊Dic61C22⋊C412yes—
125C4⋊Dic61D812yes—
126C4×Dic61—44no—
127C4×Dic61—41no—
128C4×Dic61—41no—
129C2×#92418—22no—
130C2×#92418—41no—
131C2×#92418—41no—
132C23×C61#12011yescharacteristic
133C22×D61—22no—
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—41no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—41no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61C2×C22⋊C411yescharacteristic
146C2×C4×C61C4×D411yescharacteristic
147C2×C4×C61C4×D411yescharacteristic
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12311yescharacteristic
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61#13511yescharacteristic
153C2×C4×C61#13511yescharacteristic
154C2×C4×C61#13511yescharacteristic
155C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C61#13711yescharacteristic
158C2×C4×C61#13711yescharacteristic
159C2×Dic61—22no—
160C2×Dic61—22no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—42no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—81no—
170C2×Dic61—81no—
171D244—22no—
172D244—22no—
173D244—41no—
174D244—41no—
175C4×D61—42no—
176C4×D61—42no—
177C4×D61—42no—
178C4×D61—42no—
179Dic122—42no—
180Dic122—42no—
181Dic122—81no—
182Dic122—81no—
183C8×C61—42no—
184C8×C61—81no—
185#92418—42no—
186#8984—611nomaximal, Sylow
187C22×C61#38611yescharacteristic
188C22×C61#44711yescharacteristic
189C22×C61#44911yescharacteristic
190C22×C61#45211yescharacteristic
191C22×C61#45211yescharacteristic
192C22×C61#45511yescharacteristic
193C22×C61#45611yescharacteristic
194C2×D61—42no—
195C2×D61—42no—
196C2×D61—42no—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—82no—
204C4×C61#41611yescharacteristic
205C4×C61#41911yescharacteristic
206C4×C61C4×D811yescharacteristic
207C4×C61#44211yescharacteristic
208C4×C61#45911yescharacteristic
209C4×C61#46311yescharacteristic
210C4×C61#48511yescharacteristic
211C4×C61#48511yescharacteristic
212Dic61—42no—
213Dic61—42no—
214Dic61—82no—
215Dic61—81no—
216C2×#474—611no—
217#2354—611no—
218#1990—611no—
219#1981—611no—
220C2×D4⋊C8—611no—
221#1296D6111yescharacteristic, complemented
222#1279—611no—
223C2×C61#164111yescharacteristic
224C2×C61#164611yescharacteristic
225C2×C61#164711yescharacteristic
226C2×C61#165211yescharacteristic
227C2×C61#187311yescharacteristic
228C2×C61#187911yescharacteristic
229C2×C61#188211yescharacteristic
230D61—82no—
231C2×C4⋊Q8—611no—
232C2×C4×D4—611no—
233C2×#117—1221no—
234#474—614no—
235#474—1222no—
236C2×#107—1221no—
237C2×C4⋊C8C2×D6111yescharacteristic
238C2×C4⋊C8C2×D6111yescharacteristic
239C2×#103—1221no—
240C2×#103—1221no—
241C2×#102—1221no—
242C2×C4×C8—611no—
243#390—1221no—
244#388—1222no—
245#384C2×D6111yescharacteristic
246#380—1221no—
247C4×C4⋊C4—611no—
248C4×C4⋊C4—1221no—
249#340—1222no—
250D4⋊C8—1222no—
251C61#898411yescharacteristic, complemented, Sylow
252C22×Q8—1221no—
253C22×D4—611no—
254C23×C4—1221no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—1221no—
258C4⋊Q8—612no—
259C4⋊Q8—1221no—
260C4⋊Q8—1221no—
261C4⋊Q8—1221no—
262C4×D4—612no—
263C4×D4—612no—
264C4×D4—1221no—
265C4×D4—1221no—
266#117—2442no—
267#117—2442no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—611no—
271C2×C4⋊C4—611no—
272C2×C4⋊C4—611no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4D24412yes—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1222no—
278C2×C22⋊C4—1222no—
279C2×C42—1222no—
280C2×C42—1221no—
281C2×C42C22×D6111yescharacteristic
282#107—1224no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8C4×D6112yes—
287#103—2442no—
288#103—2442no—
289#102—1224no—
290C4×C8—1221no—
291C4×C8—1221no—
292#95—1222no—
293#95—1221no—
294C24—1221no—
295C2×Q8—1222no—
296C2×Q8—2441no—
297C2×Q8—2441no—
298C2×Q8—2441no—
299C2×Q8—2441no—
300C2×Q8—2441no—
301C2×D4—612no—
302C2×D4—1221no—
303C2×D4—1221no—
304C2×D4—1221no—
305C2×D4—1221no—
306C2×D4—1221no—
307C22×C4—22no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1222no—
311C22×C4—1222no—
312C22×C4—1221no—
313C22×C4—2441no—
314C22×C4—2441no—
315C22×C4—2441no—
316C22×C4—2441no—
317C22×C4—2441no—
318C22×C4C2×D24411yescharacteristic
319C22×C4D4×D6111yescharacteristic
320C22×C4D4×D6111yescharacteristic
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—1222no—
328C2×C8—2441no—
329C2×C8—2441no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—612no—
336C4⋊C4—612no—
337C4⋊C4—1221no—
338C4⋊C4—1221no—
339C4⋊C4—1222no—
340C4⋊C4—1222no—
341C4⋊C4—1221no—
342C4⋊C4—1221no—
343C4⋊C4—2441no—
344C4⋊C4—2441no—
345C22⋊C4—1222no—
346C22⋊C4—1222no—
347C22⋊C4—4882no—
348C22⋊C4—2444no—
349C42—2444no—
350C42—2441no—
351C42—2441no—
352C42C2×C4×D6111yescharacteristic
353C42#1175822211yescharacteristic
354C42D4×D6111yescharacteristic
355C42#1175822611yescharacteristic
356C23—1222no—
357C23—2441no—
358C23—2441no—
359C23—2441no—
360C23—2441no—
361C23—2441no—
362C23—2441no—
363C23#1451974711yescharacteristic
364Q8—4881no—
365Q8—4881no—
366Q8—2442no—
367Q8—2442no—
368D4—1222no—
369D4—1222no—
370D4—2441no—
371D4—2441no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—1222no—
378C2×C4—1222no—
379C2×C4—4881no—
380C2×C4—4881no—
381C2×C4—2441no—
382C2×C4—2442no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4C22⋊C4×D6111yescharacteristic
394C2×C4#1451973911yescharacteristic
395C2×C4C4×D24411yescharacteristic
396C2×C4#1451975011yescharacteristic
397C2×C4#1451975211yescharacteristic
398C2×C4#1451976211yescharacteristic
399C2×C4D8×D6111yescharacteristic
400C2×C4#1451979611yescharacteristic
401C2×C4#1451979811yescharacteristic
402C2×C4#1451980011yescharacteristic
403C2×C4#1451980411yescharacteristic
404C2×C4#1451980411yescharacteristic
405C2×C4#1451980611yescharacteristic
406C2×C4#1451980911yescharacteristic
407C8—42no—
408C8—81no—
409C8—2442no—
410C22—4881no—
411C22—4881no—
412C22—4881no—
413C22—4881no—
414C22—2442no—
415C22—2442no—
416C22—2442no—
417C22#1683964011yescharacteristic
418C22#1683989711yescharacteristic
419C22#1683991011yescharacteristic
420C22#1683992511yescharacteristic
421C22#1683992711yescharacteristic
422C22#1683994211yescharacteristic
423C22#1683994711yescharacteristic
424C4—21no—
425C4—21no—
426C4—82no—
427C4—4882no—
428C4—4881no—
429C4—2442no—
430C4—2442no—
431C4#1683975511yescharacteristic
432C4#1683976911yescharacteristic
433C4#1683984411yescharacteristic
434C4#1683987311yescharacteristic
435C4#1683995611yescharacteristic
436C4#1683998011yescharacteristic
437C4#1684010511yescharacteristic
438C4#1684010611yescharacteristic
439C2—4882no—
440C2#2144501311yescharacteristic
441C2#2144504811yescharacteristic
442C2#2144505511yescharacteristic
443C2#2144509011yescharacteristic
444C2#2144613011yescharacteristic
445C2#2144616811yescharacteristic
446C2#2144618711yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1296C2
socleC23×C61#120
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1296D61
hypocenterC61#8984
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #21446779C2semidirect
4. #21446741C2semidirect
6. #21443658C2semidirect
7. C61×#1296C2semidirect
12. #16840043C4semidirect
221. #1296D61semidirect
251. C61#8984semidirect

Maximal subgroups

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

subgroupindex
C2×#168400432
#214487702
#214467792
#214467412
C2×#168394892
#214436582
C61×#12962
#898461