Finite Group Databasebeta

← #29576425

Subgroups — #29576425

Order 15616.

534 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839962C211yesmaximal, characteristic
2#21446659C211yesmaximal, characteristic
3#21444488C211yesmaximal, characteristic
4#21444248C211yesmaximal, characteristic, complemented
5#21443693C211yesmaximal, characteristic, complemented
6#21443670C211yesmaximal, characteristic, complemented
7Q8×M16×C61C211yesmaximal, characteristic, complemented
8C22×#11758201C412yes—
9C2×#14519774C2211yescharacteristic
10#16839962—21no—
11#16839962—21no—
12#16839962—21no—
13#16839962—21no—
14#16839815C2211yescharacteristic
15#16839786C2211yescharacteristic
16C2×C4⋊#92418C2211yescharacteristic
17C2×#14519697C412yes—
18C2×#14519693—21no—
19C2×#14519693—21no—
20#16839505—22no—
21Q8⋊#92418—21no—
22Q8⋊#92418—21no—
23C2×C4×Q8×C61C2211yescharacteristic
24C61×#446—21no—
25C61×#446—22no—
26C8×Q8×C61—21no—
27C61×#423—21no—
28C61×#423C2211yescharacteristic
29C4×M16×C61—21no—
30C4×M16×C61C2211yescharacteristic
31C22×D244—21no—
32C22×D244—21no—
33C22×D244C2×C412yes—
34C2×#11758201—22no—
35C2×#11758201—22no—
36C2×#11758201—44no—
37C2×#11758194—21no—
38C22×#92418C2×C412yes—
39#14519774—21no—
40#14519774—21no—
41#14519774—21no—
42#14519774—21no—
43C4⋊#92418—21no—
44C4⋊#92418—21no—
45C4⋊#92418—21no—
46C4⋊#92418—21no—
47C4⋊#92418—21no—
48#14519697—22no—
49#14519697—22no—
50#14519693—42no—
51#14519693—42no—
52C22⋊#92418—21no—
53C4×#92418—21no—
54C22×Q8×C61C2×C411yescharacteristic
55C2×M16×C61—21no—
56C2×M16×C61D411yescharacteristic
57C2×M16×C61D411yescharacteristic
58C4×Q8×C61—21no—
59C4×Q8×C61—21no—
60C4×Q8×C61D411yescharacteristic
61C4×Q8×C61D411yescharacteristic
62C4×Q8×C61D411yescharacteristic
63C4×Q8×C61D411yescharacteristic
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61C2×C411yescharacteristic
66C2×C42×C61—21no—
67C2×C42×C61C2311yescharacteristic
68C4⋊C8×C61—22no—
69C4⋊C8×C61—22no—
70C4⋊C8×C61—21no—
71C4⋊C8×C61D412yes—
72#97×C61—22no—
73#97×C61—21no—
74C4×C8×C61—22no—
75C4×C8×C61—21no—
76C23×D61—22no—
77C2×Q8×C61—21no—
78C2×Q8×C61—22no—
79C2×Q8×C61—41no—
80C2×Q8×C61C22⋊C411yescharacteristic
81C2×Q8×C61C22⋊C411yescharacteristic
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61C22×C411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61#4111yescharacteristic
87C2×D244—22no—
88C2×D244—22no—
89C2×D244—42no—
90C2×D244—42no—
91C2×D244—41no—
92C2×D244—41no—
93C2×D244—41no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244—41no—
97C2×D244—41no—
98C2×D244—41no—
99C2×D244—82no—
100M16×C61—21no—
101M16×C61—21no—
102M16×C61—21no—
103M16×C61—41no—
104M16×C61—41no—
105M16×C61SD1612yescomplemented
106C2×C8×C61—21no—
107C2×C8×C61—21no—
108C2×C8×C61—22no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61C22⋊C411yescharacteristic
115C4⋊C4×C61C22⋊C411yescharacteristic
116C42×C61—21no—
117C42×C61—21no—
118C42×C61—21no—
119C42×C61—21no—
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61#4111yescharacteristic
124#11758201—44no—
125#11758201—44no—
126#11758194—41no—
127#11758194—41no—
128C2×#92418—22no—
129C2×#92418—42no—
130C2×#92418—41no—
131C2×#92418—41no—
132C23×C61C4×D411yescharacteristic
133C22×D61—42no—
134C22×D61—42no—
135C22×D61—42no—
136C22×D61—82no—
137C22×D61—82no—
138Q8×C61—21no—
139Q8×C61—21no—
140Q8×C61—21no—
141Q8×C61—42no—
142Q8×C61#10012yescomplemented
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—81no—
152C2×C4×C61C2×C22⋊C411yescharacteristic
153C2×C4×C61C4×D411yescharacteristic
154C2×C4×C61#12011yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12311yescharacteristic
158C2×C4×C61C2×SD1611yescharacteristic
159C2×C4×C61#13511yescharacteristic
160C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C61#13711yescharacteristic
162D244—42no—
163D244—42no—
164D244—82no—
165D244—82no—
166D244—82no—
167D244—81no—
168D244—81no—
169D244—81no—
170D244—81no—
171C8×C61—22no—
172C8×C61—41no—
173C8×C61—42no—
174#92418—42no—
175#92418—81no—
176#92418—81no—
177#9254—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61#38611yescharacteristic
181C22×C61C4×SD1611yescharacteristic
182C22×C61#44011yescharacteristic
183C2×D61—82no—
184C2×D61—82no—
185C2×D61—82no—
186C2×D61—162no—
187C2×D61—162no—
188C4×C61—21no—
189C4×C61—22no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—41no—
193C4×C61—41no—
194C4×C61—41no—
195C4×C61C2×#10011yescharacteristic
196C4×C61#41311yescharacteristic
197C4×C61#44811yescharacteristic
198C4×C61#44911yescharacteristic
199C4×C61#46011yescharacteristic
200C4×C61#46411yescharacteristic
201C2×#460—611no—
202Q8×M16D6111yescharacteristic, complemented
203#1965—611no—
204#1545—611no—
205#1487—611no—
206#1292—611no—
207#1283—611no—
208C2×C61—21no—
209C2×C61—21no—
210C2×C61#188011yescharacteristic
211C2×C61#190011yescharacteristic
212C2×C61#190311yescharacteristic
213D61—162no—
214C22×SD16—612no—
215C2×#127—611no—
216C2×C4×Q8C2×D6111yescharacteristic
217#460—1221no—
218#460—1221no—
219#460—1221no—
220#460—1221no—
221#446—21no—
222#446—22no—
223C8×Q8—21no—
224#431—611no—
225#423—21no—
226#423—611no—
227#423C2×D6111yescharacteristic
228C2×C4⋊C8—611no—
229C2×#102—612no—
230C2×#100—1221no—
231C2×#100—1221no—
232C4×M16—21no—
233C4×M16C2×D6111yescharacteristic
234#331—1222no—
235Q8⋊C8—1221no—
236Q8⋊C8—1221no—
237C61#925411yescharacteristic, complemented, Sylow
238C22×Q8C4×D6111yescharacteristic
239C22×D4—612no—
240C22×D4—1221no—
241C22×D4—1221no—
242C2×SD16—1222no—
243C2×SD16—1222no—
244C2×SD16—2444no—
245C2×M16—21no—
246C2×M16—1221no—
247C2×M16C22⋊D6111yescharacteristic
248C2×M16C22⋊D6111yescharacteristic
249C22×C8—612no—
250#127—1221no—
251#127—1221no—
252#127—1221no—
253#127—1221no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8D24411yescharacteristic
257C4×Q8D24411yescharacteristic
258C4×Q8C22⋊D6111yescharacteristic
259C4×Q8C22⋊D6111yescharacteristic
260C2×C4⋊C4—21no—
261C2×C4⋊C4C4×D6111yescharacteristic
262C2×C42—21no—
263C2×C42C22×D6111yescharacteristic
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—21no—
267C4⋊C8—1221no—
268C4⋊C8—1221no—
269C4⋊C8—1221no—
270C4⋊C8—1221no—
271C4⋊C8—1221no—
272C4⋊C8C22⋊D6112yes—
273#102—1222no—
274#102—1222no—
275#100—2442no—
276#100—2442no—
277C22⋊C8—1221no—
278#97—22no—
279#97—21no—
280C4×C8—22no—
281C4×C8—21no—
282C4×C8—1221no—
283C24—1222no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8#1175819811yescharacteristic
288C2×Q8#1175819811yescharacteristic
289C2×D4—1222no—
290C2×D4—1222no—
291C2×D4—4882no—
292C2×D4—2442no—
293C2×D4—2442no—
294C2×D4—2441no—
295C2×D4—2441no—
296C2×D4—2441no—
297C2×D4—2441no—
298C2×D4—2441no—
299C2×D4—2441no—
300C2×D4—2441no—
301C2×D4—2441no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4C2×C4×D6111yescharacteristic
305C22×C4#1175822211yescharacteristic
306C22×C4C2×C22⋊D6111yescharacteristic
307SD16—2444no—
308SD16—2444no—
309M16—21no—
310M16—21no—
311M16—21no—
312M16—41no—
313M16—41no—
314M16—2441no—
315M16—2441no—
316M16#1175820112yescomplemented
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—1222no—
321C2×C8—2441no—
322C2×C8—2441no—
323C2×C8—2442no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4#1175819811yescharacteristic
330C4⋊C4#1175819811yescharacteristic
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42C2×D24411yescharacteristic
336C42#1175822211yescharacteristic
337C42C2×C22⋊D6111yescharacteristic
338C42C2×C22⋊D6111yescharacteristic
339C23—4882no—
340C23—4882no—
341C23—2442no—
342C23—2442no—
343C23—2442no—
344C23C4×C22⋊D6111yescharacteristic
345Q8—21no—
346Q8—21no—
347Q8—21no—
348Q8—42no—
349Q8#1451969312yescomplemented
350D4—4882no—
351D4—4882no—
352D4—4882no—
353D4—4881no—
354D4—4881no—
355D4—4881no—
356D4—4881no—
357D4—2442no—
358D4—2442no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4C2×#1175819811yescharacteristic
369C2×C4C4×C22⋊D6111yescharacteristic
370C2×C4#1451974611yescharacteristic
371C2×C4#1451975111yescharacteristic
372C2×C4#1451975111yescharacteristic
373C2×C4#1451976011yescharacteristic
374C2×C4C2×#1175820111yescharacteristic
375C2×C4#1451979711yescharacteristic
376C2×C4#1451980211yescharacteristic
377C2×C4#1451980811yescharacteristic
378C8—22no—
379C8—41no—
380C8—42no—
381C8—4881no—
382C8—4881no—
383C8—2442no—
384C22—21no—
385C22—21no—
386C22—9762no—
387C22—9762no—
388C22—4882no—
389C22—4882no—
390C22—4882no—
391C22#1683964111yescharacteristic
392C22C4×#1175820111yescharacteristic
393C22#1683986011yescharacteristic
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4C2×#1451969311yescharacteristic
402C4#1683974411yescharacteristic
403C4#1683990111yescharacteristic
404C4#1683990711yescharacteristic
405C4#1683996211yescharacteristic
406C4#1683998411yescharacteristic
407C2—21no—
408C2—21no—
409C2—9762no—
410C2#2144617411yescharacteristic
411C2#2144628211yescharacteristic
412C2#2144629911yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingQ8×M16×C61C2
socleC22×C61#386
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenterQ8×M16D61
hypocenterC61#9254
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
4. #21444248C2semidirect
5. #21443693C2semidirect
6. #21443670C2semidirect
7. Q8×M16×C61C2semidirect
105. M16×C61SD16semidirect
142. Q8×C61#100semidirect
202. Q8×M16D61semidirect
237. C61#9254semidirect
316. M16#11758201semidirect
349. Q8#14519693semidirect

Maximal subgroups

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

subgroupindex
C2×#168399622
#214466592
#214444882
#214442482
#214436932
#214436702
Q8×M16×C612
#925461