Finite Group Databasebeta

← #29597016

Subgroups — #29597016

Order 15616.

508 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840004C211yesmaximal, characteristic
2C2×#108×D61C211yesmaximal, characteristic
3C2×#16839579C211yesmaximal, characteristic, complemented
4C2×#16839577C211yesmaximal, characteristic, complemented
5#21447429C211yesmaximal, characteristic
6#21443927C211yesmaximal, characteristic, complemented
7C61×#1387C211yesmaximal, characteristic, complemented
8C22×Dic244C2211yescharacteristic
9C2×M16×D61—21no—
10C22×C8×D61C2211yescharacteristic
11C2×#14519757—21no—
12C2×#14519717C2211yescharacteristic
13#16840004—21no—
14#16840004—21no—
15#16840004C412yes—
16#16840004C412yes—
17C2×#14519713C2211yescharacteristic
18C2×#14519712—21no—
19#108×D61—22no—
20#108×D61—22no—
21C2×#14519700—21no—
22#16839579—22no—
23#16839577—22no—
24#16839541—21no—
25#16839541—21no—
26C2×M32×C61C2211yescharacteristic
27C2×#108×C61C2211yescharacteristic
28C2×#107×C61C2211yescharacteristic
29C22×Dic122—21no—
30C22×C4×D61D411yescharacteristic
31C2×Dic244—21no—
32C2×Dic244—21no—
33C2×Dic244—21no—
34C2×Dic244—21no—
35C2×Dic244—21no—
36C2×Dic244C2×C412yes—
37C2×#11758194—21no—
38C2×#11758189—41no—
39M16×D61—24no—
40M16×D61—42no—
41C22×#92418D411yescharacteristic
42C2×C8×D61—21no—
43C2×C8×D61—21no—
44C2×C8×D61—21no—
45C2×C8×D61—21no—
46C2×C8×D61—41no—
47C2×C8×D61C2×C411yescharacteristic
48C2×C8×D61C2×C411yescharacteristic
49C2×C8×D61D411yescharacteristic
50C2×C8×D61D411yescharacteristic
51#14519757—22no—
52#14519757—22no—
53#14519757—41no—
54#14519757—41no—
55C2×C4⋊Dic61—41no—
56C2×#11758198—41no—
57#14519717—21no—
58#14519717—21no—
59C2×#11758185—21no—
60#14519713—21no—
61#14519713D412yes—
62#14519712—24no—
63#14519700—22no—
64#14519700—22no—
65C2×M16×C61—21no—
66C22×C8×C61C2311yescharacteristic
67C2×C4⋊C4×C61—21no—
68M32×C61—21no—
69M32×C61—21no—
70C2×C16×C61—21no—
71#108×C61—21no—
72#108×C61D412yes—
73#107×C61—21no—
74#107×C61C2×C412yes—
75C23×D61SD1611yescharacteristic
76C22×C4×C61—41no—
77C22×C4×C61C2×D411yescharacteristic
78C22×Dic61—41no—
79C22×Dic61SD1611yescharacteristic
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61C22⋊C411yescharacteristic
85C2×C4×D61C22⋊C411yescharacteristic
86C2×C4×D61D811yescharacteristic
87C2×C4×D61D811yescharacteristic
88C2×Dic122—22no—
89C2×Dic122—41no—
90C2×Dic122—41no—
91C2×Dic122—41no—
92C2×Dic122—41no—
93C2×Dic122—41no—
94M16×C61—22no—
95M16×C61—41no—
96C2×C8×C61—41no—
97C2×C8×C61C22×C411yescharacteristic
98C2×C8×C61C2×D411yescharacteristic
99C2×C8×C61C2×D411yescharacteristic
100C2×C8×C61C2×D411yescharacteristic
101C2×C8×C61#4111yescharacteristic
102C2×C8×C61#4111yescharacteristic
103C4⋊C4×C61—22no—
104C4⋊C4×C61—41no—
105#11758198—42no—
106#11758198—42no—
107C4⋊Dic61—42no—
108C4⋊Dic61—42no—
109#11758194—22no—
110#11758194—41no—
111C2×#92418—21no—
112C2×#92418—21no—
113C2×#92418—41no—
114C2×#92418C22⋊C411yescharacteristic
115C2×#92418C22⋊C411yescharacteristic
116Dic244—22no—
117Dic244—22no—
118Dic244—41no—
119Dic244—41no—
120#11758189—44no—
121C8×D61—21no—
122C8×D61—21no—
123C8×D61—21no—
124C8×D61—21no—
125C8×D61—21no—
126C8×D61—21no—
127C8×D61—21no—
128C8×D61—21no—
129C8×D61—44no—
130C16×C61—41no—
131#11758185—41no—
132C23×C61C2×SD1611yescharacteristic
133C22×D61—21no—
134C22×D61—21no—
135C22×D61—22no—
136C22×D61—41no—
137C22×D61#10211yescharacteristic
138C22×D61#10211yescharacteristic
139C2×C4×C61—42no—
140C2×C4×C61—82no—
141C2×C4×C61C2×C22⋊C411yescharacteristic
142C2×C4×C61#12011yescharacteristic
143C2×C4×C61#12311yescharacteristic
144C2×C4×C61C2×D811yescharacteristic
145C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C61#13711yescharacteristic
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—42no—
150C2×Dic61—81no—
151C2×Dic61—81no—
152C2×Dic61#10211yescharacteristic
153C2×Dic61#10211yescharacteristic
154C4×D61—21no—
155C4×D61—21no—
156C4×D61—21no—
157C4×D61—21no—
158C4×D61—21no—
159C4×D61—21no—
160C4×D61—21no—
161C4×D61—21no—
162Dic122—42no—
163Dic122—42no—
164Dic122—81no—
165Dic122—81no—
166C8×C61—42no—
167C8×C61C4×D411yescharacteristic
168C8×C61C4×D411yescharacteristic
169C8×C61#12111yescharacteristic
170C8×C61#12111yescharacteristic
171#92418—21no—
172#92418—21no—
173#92418—42no—
174#13003—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61C2×#10211yescharacteristic
178C22×C61#44811yescharacteristic
179C22×C61#48111yescharacteristic
180C2×D61—21no—
181C2×D61—21no—
182C2×D61—21no—
183C2×D61—21no—
184C2×D61—21no—
185C2×D61—42no—
186C2×D61—42no—
187C2×D61#36212yescomplemented
188C4×C61—82no—
189C4×C61#38611yescharacteristic
190C4×C61#41711yescharacteristic
191C4×C61#44711yescharacteristic
192C4×C61#48011yescharacteristic
193Dic61—21no—
194Dic61—21no—
195Dic61—82no—
196C2×#467—611no—
197C22×#108—611no—
198C2×#362—611no—
199C2×#362—611no—
200#2112—611no—
201#1387—611no—
202#1387D6111yescharacteristic, complemented
203C2×C61—21no—
204C2×C61—21no—
205C2×C61#187411yescharacteristic
206C2×C61C2×#36211yescharacteristic
207C2×C61#214811yescharacteristic
208D61—22no—
209D61—41no—
210C22×Q16—611no—
211C22×M16—1221no—
212C23×C8—611no—
213C2×C22⋊Q8—1221no—
214C2×M32—611no—
215C2×M32C2×D6111yescharacteristic
216#467—612no—
217#467—612no—
218#467—1221no—
219#467—1221no—
220C2×#108—611no—
221C2×#108—1222no—
222C2×#108—1221no—
223C2×#108—1222no—
224C2×#108C2×D6111yescharacteristic
225C2×#107C2×D6111yescharacteristic
226C2×#103—1221no—
227#362—1222no—
228#362—1222no—
229C22⋊C16—1221no—
230C22⋊C16—1221no—
231C61#1300311yescharacteristic, complemented, Sylow
232C22×Q8—1221no—
233C23×C4—611no—
234C2×Q16—612no—
235C2×Q16—1221no—
236C2×Q16—1221no—
237C2×Q16—1221no—
238C2×Q16—1221no—
239C2×Q16—1221no—
240C2×M16—21no—
241C2×M16—1224no—
242C2×M16—1221no—
243C2×M16—2441no—
244C2×M16—2442no—
245C22×C8—611no—
246C22×C8—611no—
247C22×C8—611no—
248C22×C8—611no—
249C22×C8—611no—
250C22×C8—1221no—
251C22×C8—1221no—
252C22×C8—1221no—
253C22×C8—1221no—
254C22×C8—2441no—
255C22×C8C22×D6111yescharacteristic
256C22⋊Q8—1222no—
257C22⋊Q8—1222no—
258C22⋊Q8—2441no—
259C22⋊Q8—2441no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—2441no—
262C2×C22⋊C4—2441no—
263M32—21no—
264M32—21no—
265M32—1221no—
266M32—1221no—
267C2×C16—21no—
268C2×C16—1221no—
269#108—21no—
270#108—612no—
271#108—1224no—
272#108—1221no—
273#108D24412yes—
274#107—21no—
275#107C4×D6112yes—
276#103—1222no—
277#103—1222no—
278C24—611no—
279C2×Q8—1222no—
280C2×Q8—2441no—
281C2×Q8—2441no—
282C2×Q8—2441no—
283C2×Q8—2441no—
284C2×Q8—2441no—
285C22×C4—41no—
286C22×C4—611no—
287C22×C4—611no—
288C22×C4—611no—
289C22×C4—611no—
290C22×C4—611no—
291C22×C4—1221no—
292C22×C4—1221no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—2441no—
296C22×C4D4×D6111yescharacteristic
297Q16—1222no—
298Q16—1222no—
299Q16—2441no—
300Q16—2441no—
301M16—22no—
302M16—41no—
303M16—1222no—
304M16—2444no—
305M16—2441no—
306C2×C8—41no—
307C2×C8—611no—
308C2×C8—611no—
309C2×C8—1221no—
310C2×C8—1221no—
311C2×C8—1221no—
312C2×C8—1221no—
313C2×C8—1221no—
314C2×C8—1221no—
315C2×C8—1221no—
316C2×C8—1221no—
317C2×C8—1221no—
318C2×C8—1221no—
319C2×C8—2444no—
320C2×C8—2441no—
321C2×C8C2×C4×D6111yescharacteristic
322C2×C8C2×D24411yescharacteristic
323C2×C8#1175822211yescharacteristic
324C2×C8D4×D6111yescharacteristic
325C2×C8D4×D6111yescharacteristic
326C2×C8#1175822611yescharacteristic
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—2442no—
330C4⋊C4—2442no—
331C22⋊C4—2442no—
332C22⋊C4—2442no—
333C16—41no—
334C16—2441no—
335C23—611no—
336C23—611no—
337C23—1222no—
338C23—1221no—
339C23—1221no—
340C23—2441no—
341C23SD16×D6111yescharacteristic
342Q8—4881no—
343Q8—4881no—
344Q8—2442no—
345Q8—2442no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—611no—
349C2×C4—611no—
350C2×C4—1221no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—1221no—
354C2×C4—1221no—
355C2×C4—1221no—
356C2×C4—1221no—
357C2×C4—1221no—
358C2×C4—1221no—
359C2×C4—1221no—
360C2×C4—4881no—
361C2×C4—4881no—
362C2×C4—2442no—
363C2×C4C22⋊C4×D6111yescharacteristic
364C2×C4#1451974711yescharacteristic
365C2×C4#1451976211yescharacteristic
366C2×C4D8×D6111yescharacteristic
367C2×C4#1451980411yescharacteristic
368C2×C4#1451980911yescharacteristic
369C8—42no—
370C8—1221no—
371C8—1221no—
372C8—2442no—
373C8#1451973911yescharacteristic
374C8C4×D24411yescharacteristic
375C8#1451975011yescharacteristic
376C8#1451975211yescharacteristic
377C22—21no—
378C22—21no—
379C22—612no—
380C22—1221no—
381C22—1221no—
382C22—1221no—
383C22—1221no—
384C22—1221no—
385C22—2442no—
386C22—2442no—
387C22#102×D6111yescharacteristic
388C22#1683990211yescharacteristic
389C22#1684008211yescharacteristic
390C4—82no—
391C4—1221no—
392C4—1221no—
393C4—4882no—
394C4#1683964011yescharacteristic
395C4#1683976211yescharacteristic
396C4#1683989711yescharacteristic
397C4#1684007711yescharacteristic
398C2—21no—
399C2—21no—
400C2—1222no—
401C2—2441no—
402C2#2144613511yescharacteristic
403C2#362×D6111yescharacteristic
404C2#2144756511yescharacteristic
subgroupG/H
centerC22#102×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC61×#1387C2
socleC22×C61C2×#102
prime radicalC23×D61SD16
cosocleC22×C8×C61C23
hypercenter#1387D61
hypocenterC61#13003
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1387

Complements

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

NH ≅ G/Ntype
3. C2×#16839579C2semidirect
4. C2×#16839577C2semidirect
6. #21443927C2semidirect
7. C61×#1387C2semidirect
187. C2×D61#362semidirect
202. #1387D61semidirect
231. C61#13003semidirect

Maximal subgroups

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

subgroupindex
C2×#168400042
C2×#108×D612
C2×#168395792
C2×#168395772
#214474292
#214439272
C61×#13872
#1300361