Finite Group Databasebeta

← #29583818

Subgroups — #29583818

Order 15616.

522 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839962C211yesmaximal, characteristic
2#21446625C211yesmaximal, characteristic
3#21445824C211yesmaximal, characteristic, complemented
4C2×#16839515C211yesmaximal, characteristic, complemented
5C2×#16839505C211yesmaximal, characteristic
6#21443703C211yesmaximal, characteristic, complemented
7C61×#1984C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C2×#14519774C2211yescharacteristic
10#16839962—22no—
11#16839962—22no—
12C2×C4⋊#92418C2211yescharacteristic
13C2×C4⋊#92418C2211yescharacteristic
14C2×#14519701—21no—
15C2×#14519699—21no—
16C2×#14519699—21no—
17C2×#14519697—21no—
18#16839632C2211yescharacteristic
19#16839524—21no—
20#16839515—22no—
21#16839505—22no—
22C2×C4×Q8×C61C2211yescharacteristic
23C2×C4⋊C8×C61C2211yescharacteristic
24C2×C4×C8×C61C2211yescharacteristic
25C61×#336—21no—
26C61×#336—21no—
27C22×D244—21no—
28C22×D244—21no—
29C22×D244—21no—
30C2×#11758201—24no—
31C2×#11758201—42no—
32C2×#11758201—42no—
33C22×#92418—21no—
34C22×#92418—21no—
35#14519774—21no—
36#14519774—21no—
37#14519774—21no—
38#14519774D412yes—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44C4⋊#92418—21no—
45C4⋊#92418D412yes—
46#14519701—42no—
47#14519699—42no—
48#14519699—42no—
49#14519697—24no—
50C22×Q8×C61D411yescharacteristic
51C22×C8×C61—21no—
52C22×C8×C61D411yescharacteristic
53C22×C8×C61D411yescharacteristic
54C4×Q8×C61—21no—
55C4×Q8×C61—21no—
56C4×Q8×C61—21no—
57C4×Q8×C61—21no—
58C2×C4⋊C4×C61—21no—
59C2×C4⋊C4×C61D411yescharacteristic
60C2×C42×C61—21no—
61C2×C42×C61C2311yescharacteristic
62C4⋊C8×C61—21no—
63C4⋊C8×C61—21no—
64C4×C8×C61—21no—
65C4×C8×C61—21no—
66C23×D61—41no—
67C2×Q8×C61—21no—
68C2×Q8×C61—21no—
69C2×Q8×C61—21no—
70C2×Q8×C61—21no—
71C2×Q8×C61—21no—
72C2×Q8×C61D812yes—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61#4111yescharacteristic
78C22×Dic61—41no—
79C2×D244—22no—
80C2×D244—22no—
81C2×D244—22no—
82C2×D244—41no—
83C2×D244—41no—
84C2×D244—41no—
85C2×D244—41no—
86C2×D244—41no—
87C2×D244—41no—
88C2×D244—41no—
89C2×D244—41no—
90C2×D244—41no—
91C2×D244—41no—
92C2×D244—41no—
93C2×D244—81no—
94C2×D244—81no—
95C2×C8×C61—21no—
96C2×C8×C61—21no—
97C2×C8×C61—21no—
98C2×C8×C61—21no—
99C2×C8×C61—21no—
100C2×C8×C61—41no—
101C2×C8×C61—41no—
102C2×C8×C61—41no—
103C2×C8×C61SD1612yescomplemented
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C42×C61—41no—
109C42×C61—41no—
110C42×C61C2×D411yescharacteristic
111C42×C61C2×D411yescharacteristic
112C42×C61#4111yescharacteristic
113C42×C61#4111yescharacteristic
114#11758201—48no—
115C4⋊Dic61—41no—
116C4⋊Dic61—41no—
117C4⋊Dic61—41no—
118C4⋊Dic61—41no—
119C4⋊Dic61—81no—
120C2×#92418—22no—
121C2×#92418—41no—
122C2×#92418—41no—
123C2×#92418—41no—
124C2×#92418—41no—
125C2×#92418—41no—
126C23×C61#12111yescharacteristic
127C22×D61—42no—
128C22×D61—81no—
129C22×D61—81no—
130C22×D61—81no—
131C22×D61—81no—
132C22×D61—81no—
133C22×D61—81no—
134Q8×C61—22no—
135Q8×C61—22no—
136Q8×C61—41no—
137Q8×C61—41no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—22no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61#12011yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12311yescharacteristic
150C2×C4×C61#12311yescharacteristic
151C2×C4×C61#12411yescharacteristic
152C2×C4×C61C2×D811yescharacteristic
153C2×C4×C61C2×SD1611yescharacteristic
154C2×C4×C61#13511yescharacteristic
155C2×C4×C61#13511yescharacteristic
156C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C61#13711yescharacteristic
159C2×C4×C61#13711yescharacteristic
160C2×Dic61—81no—
161C2×Dic61—81no—
162C2×Dic61—81no—
163D244—42no—
164D244—42no—
165D244—42no—
166D244—42no—
167D244—82no—
168D244—82no—
169D244—81no—
170D244—81no—
171D244—81no—
172D244—81no—
173C8×C61—22no—
174C8×C61—41no—
175C8×C61—81no—
176#92418—42no—
177#92418—81no—
178#10474—611nomaximal, Sylow
179C22×C61#39411yescharacteristic
180C22×C61#46011yescharacteristic
181C22×C61#46411yescharacteristic
182C22×C61#46611yescharacteristic
183C22×C61#46811yescharacteristic
184C22×C61#47111yescharacteristic
185C22×C61#47211yescharacteristic
186C2×D61—82no—
187C2×D61—82no—
188C2×D61—82no—
189C2×D61—161no—
190C2×D61—161no—
191C2×D61—161no—
192C2×D61—161no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—42no—
196C4×C61—81no—
197C4×C61#44711yescharacteristic
198C4×C61#45211yescharacteristic
199C4×C61#46111yescharacteristic
200C4×C61#46311yescharacteristic
201C4×C61#46511yescharacteristic
202C4×C61#46811yescharacteristic
203C4×C61#48011yescharacteristic
204C4×C61#48211yescharacteristic
205Dic61—161no—
206C2×#460—611no—
207#1984D6111yescharacteristic, complemented
208#1957—611no—
209#1803—611no—
210C2×#335—611no—
211C2×#331—611no—
212#1296—611no—
213C2×C61#166811yescharacteristic
214C2×C61#168211yescharacteristic
215C2×C61#168611yescharacteristic
216C2×C61#169811yescharacteristic
217C2×C61#203311yescharacteristic
218C2×C61#205111yescharacteristic
219C2×C61#205511yescharacteristic
220D61—162no—
221C22×SD16—1221no—
222C2×#127—611no—
223C2×C4×Q8C2×D6111yescharacteristic
224#460—1222no—
225#460—1222no—
226C2×C4⋊C8—611no—
227C2×C4⋊C8—611no—
228C2×C4⋊C8C2×D6111yescharacteristic
229C2×#103—1221no—
230C2×#102—1221no—
231C2×#102—1221no—
232C2×#102—1221no—
233C2×C4×C8C2×D6111yescharacteristic
234#384—611no—
235#340—1221no—
236#336—21no—
237#336—21no—
238#335—1222no—
239#331—1222no—
240C61#1047411yescharacteristic, complemented, Sylow
241C22×Q8D24411yescharacteristic
242C22×D4—1221no—
243C22×D4—1221no—
244C22×D4—1221no—
245C2×SD16—1224no—
246C2×SD16—2442no—
247C2×SD16—2442no—
248C22×C8—21no—
249C22×C8—1221no—
250C22×C8—1221no—
251C22×C8C22⋊D6111yescharacteristic
252C22×C8C22⋊D6111yescharacteristic
253#127—612no—
254#127—1221no—
255#127—1221no—
256#127—1221no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4D24411yescharacteristic
266C2×C42—21no—
267C2×C42C22×D6111yescharacteristic
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—612no—
271C4⋊C8—1221no—
272C4⋊C8—1221no—
273C4⋊C8—1221no—
274#103—2442no—
275#102—1224no—
276#102—2442no—
277#102—2442no—
278C4×C8—21no—
279C4×C8—21no—
280C24—2441no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8D48812yes—
287C2×D4—1222no—
288C2×D4—1222no—
289C2×D4—1222no—
290C2×D4—4881no—
291C2×D4—4881no—
292C2×D4—2441no—
293C2×D4—2441no—
294C2×D4—2441no—
295C2×D4—2441no—
296C2×D4—2441no—
297C2×D4—2441no—
298C2×D4—2441no—
299C2×D4—2441no—
300C2×D4—2441no—
301C2×D4—2441no—
302C2×D4—2441no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—2441no—
306C22×C4C2×D24411yescharacteristic
307C22×C4#1175822211yescharacteristic
308C22×C4C2×C22⋊D6111yescharacteristic
309SD16—2448no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—1222no—
319C2×C8—2441no—
320C2×C8—2441no—
321C2×C8—2441no—
322C2×C8—2441no—
323C2×C8—2441no—
324C2×C8#1175820112yescomplemented
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—4881no—
330C4⋊C4—2441no—
331C4⋊C4—2441no—
332C4⋊C4—2441no—
333C4⋊C4—2441no—
334C42—41no—
335C42—41no—
336C42D4×D6111yescharacteristic
337C42D4×D6111yescharacteristic
338C42#1175822411yescharacteristic
339C42#1175822411yescharacteristic
340C23—4881no—
341C23—4881no—
342C23—4881no—
343C23—4881no—
344C23—4881no—
345C23—4881no—
346C23—2442no—
347C23#1451975111yescharacteristic
348Q8—22no—
349Q8—22no—
350Q8—41no—
351Q8—41no—
352D4—4882no—
353D4—4882no—
354D4—4881no—
355D4—4881no—
356D4—4881no—
357D4—4881no—
358D4—2442no—
359D4—2442no—
360D4—2442no—
361D4—2442no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—4881no—
371C2×C4—4881no—
372C2×C4—4881no—
373C2×C4#1451974711yescharacteristic
374C2×C4#1451974811yescharacteristic
375C2×C4#1451974911yescharacteristic
376C2×C4#1451975911yescharacteristic
377C2×C4#1451976111yescharacteristic
378C2×C4#1451976311yescharacteristic
379C2×C4C2×D48811yescharacteristic
380C2×C4C2×#1175820111yescharacteristic
381C2×C4#1451979711yescharacteristic
382C2×C4#1451979911yescharacteristic
383C2×C4#1451980211yescharacteristic
384C2×C4#1451980511yescharacteristic
385C2×C4#1451980811yescharacteristic
386C2×C4#1451981011yescharacteristic
387C8—22no—
388C8—41no—
389C8—81no—
390C8—4881no—
391C8—2442no—
392C22—9761no—
393C22—9761no—
394C22—9761no—
395C22—9761no—
396C22—4882no—
397C22—4882no—
398C22—4882no—
399C22#1683967811yescharacteristic
400C22#1683996211yescharacteristic
401C22#1683998411yescharacteristic
402C22#1683999811yescharacteristic
403C22#1684000811yescharacteristic
404C22#1684002511yescharacteristic
405C22#1684003211yescharacteristic
406C4—21no—
407C4—21no—
408C4—42no—
409C4—81no—
410C4—9761no—
411C4#1683989811yescharacteristic
412C4#1683992611yescharacteristic
413C4#1683996411yescharacteristic
414C4#1683997411yescharacteristic
415C4#1683999311yescharacteristic
416C4#1684001011yescharacteristic
417C4#1684007611yescharacteristic
418C4#1684008711yescharacteristic
419C2—9762no—
420C2#2144517911yescharacteristic
421C2#2144526311yescharacteristic
422C2#2144529111yescharacteristic
423C2#2144535911yescharacteristic
424C2#2144700211yescharacteristic
425C2#2144711011yescharacteristic
426C2#2144713411yescharacteristic
subgroupG/H
centerC23#14519751
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1984C2
socleC23×C61#121
prime radicalC2×#14519774C22
hypercenter#1984D61
hypocenterC61#10474
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984

Complements

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

NH ≅ G/Ntype
3. #21445824C2semidirect
4. C2×#16839515C2semidirect
6. #21443703C2semidirect
7. C61×#1984C2semidirect
103. C2×C8×C61SD16semidirect
207. #1984D61semidirect
240. C61#10474semidirect
324. C2×C8#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168399622
#214466252
#214458242
C2×#168395152
C2×#168395052
#214437032
C61×#19842
#1047461