Finite Group Databasebeta

← #29577165

Subgroups — #29577165

Order 15616.

538 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
1#21455299C211yesmaximal, characteristic
2#21454951C211yesmaximal, characteristic, complemented
3#21446659C211yesmaximal, characteristic, complemented
4#21444488C211yesmaximal, characteristic, complemented
5#21443732C211yesmaximal, characteristic, complemented
6#21443709C211yesmaximal, characteristic, complemented
7C61×#1489C211yesmaximal, characteristic, complemented
8C2×#14519802C412yes—
9C2×#14519774C2211yescharacteristic
10#16839962—21no—
11#16839962—21no—
12#16839957—21no—
13#16839957—21no—
14Q8×#92418—21no—
15#16839840—22no—
16#16839840—21no—
17#16839815—21no—
18#16839815C2211yescharacteristic
19#16839786—21no—
20#16839786C2211yescharacteristic
21#16839786C2211yescharacteristic
22#16839766C412yes—
23C2×#14519693—22no—
24#16839505—21no—
25#16839505—21no—
26C4×#41×C61C2211yescharacteristic
27C61×#423C2211yescharacteristic
28C4×M16×C61C2211yescharacteristic
29C61×Q8⋊C8—21no—
30C61×D4⋊C8—21no—
31C22×D244—21no—
32C22×D244—21no—
33C22×D244C2×C412yes—
34#14519802—44no—
35C2×#11758201—22no—
36C2×#11758199—22no—
37C2×#11758194—22no—
38C2×#11758194C2×C412yes—
39C2×#11758194D412yes—
40#14519774—21no—
41#14519774—21no—
42#14519774—21no—
43#14519774—21no—
44C4⋊#92418—22no—
45C4⋊#92418—21no—
46C4⋊#92418—21no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49C4⋊#92418D411yescharacteristic
50C4⋊#92418D411yescharacteristic
51#14519697—22no—
52#14519697—22no—
53#14519693—44no—
54C22⋊#92418—22no—
55C22⋊#92418—21no—
56C4×#92418—22no—
57C4×#92418—21no—
58C2×#41×C61C2×C411yescharacteristic
59C2×M16×C61—21no—
60C2×M16×C61—21no—
61C4×Q8×C61D411yescharacteristic
62C4×Q8×C61D411yescharacteristic
63C4×D4×C61—21no—
64C4×D4×C61—21no—
65C4×D4×C61D411yescharacteristic
66C4×D4×C61D411yescharacteristic
67#117×C61—21no—
68#117×C61C2×C411yescharacteristic
69C2×C42×C61—21no—
70C2×C42×C61C2311yescharacteristic
71C4⋊C8×C61—21no—
72C4⋊C8×C61—21no—
73#97×C61—21no—
74C4×C8×C61—21no—
75C23×D61—22no—
76#41×C61—22no—
77#41×C61—41no—
78C2×Q8×C61C22⋊C411yescharacteristic
79C2×D4×C61—21no—
80C2×D4×C61C22⋊C411yescharacteristic
81C22×C4×C61—21no—
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—42no—
92C2×D244—42no—
93C2×D244—41no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244—41no—
97C2×D244—81no—
98C2×D244—81no—
99M16×C61—22no—
100M16×C61—41no—
101M16×C61—81no—
102C2×C8×C61—41no—
103C2×C8×C61—41no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61C22⋊C411yescharacteristic
107C4⋊C4×C61C22⋊C411yescharacteristic
108C22⋊C4×C61—21no—
109C22⋊C4×C61—21no—
110C22⋊C4×C61—21no—
111C42×C61—21no—
112C42×C61—21no—
113C42×C61—21no—
114C42×C61C2×D411yescharacteristic
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61#4111yescharacteristic
118#11758201—44no—
119#11758199—44no—
120#11758194—42no—
121#11758194—42no—
122#11758194—82no—
123C2×#92418—22no—
124C2×#92418—21no—
125C2×#92418—22no—
126C2×#92418—21no—
127C23×C61—21no—
128C23×C61C4×D411yescharacteristic
129C22×D61—42no—
130C22×D61—44no—
131C22×D61—82no—
132C22×D61—82no—
133Q8×C61—21no—
134Q8×C61#10012yescomplemented
135D4×C61—21no—
136D4×C61—21no—
137D4×C61—42no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—42no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
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
158D244—42no—
159D244—42no—
160D244—82no—
161D244—82no—
162D244—82no—
163D244—81no—
164D244—81no—
165D244—81no—
166D244—81no—
167C8×C61—42no—
168C8×C61—81no—
169#92418—42no—
170#92418—42no—
171#92418—41no—
172#92418—41no—
173#9360—611nomaximal, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—41no—
178C22×C61—41no—
179C22×C61—41no—
180C22×C61#38611yescharacteristic
181C2×D61—82no—
182C2×D61—82no—
183C2×D61—82no—
184C2×D61—164no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—22no—
188C4×C61—21no—
189C4×C61—41no—
190C4×C61—41no—
191C4×C61C2×#10011yescharacteristic
192C4×C61#41311yescharacteristic
193C4×C61#45311yescharacteristic
194C4×C61#45411yescharacteristic
195C4×C61#47111yescharacteristic
196C4×C61#47211yescharacteristic
197#3038—611no—
198#2991—611no—
199#1965—611no—
200#1545—611no—
201#1489D6111yescharacteristic, complemented
202#1309—611no—
203#1299—611no—
204C2×C61—41no—
205C2×C61—81no—
206C2×C61#188311yescharacteristic
207C2×C61#190011yescharacteristic
208C2×C61#190711yescharacteristic
209D61—162no—
210C2×AGL(1,ℤ/8)—612no—
211C2×#127—611no—
212C4×#41C2×D6111yescharacteristic
213#460—1221no—
214#460—1221no—
215#459—1221no—
216#459—1221no—
217C8×Q8—1221no—
218#436—1222no—
219#436—1221no—
220#431—611no—
221#431—1221no—
222#423—611no—
223#423—611no—
224#423—1221no—
225#423C2×D6111yescharacteristic
226#418—612no—
227C2×#100—1222no—
228C4×M16C2×D6111yescharacteristic
229#331—1221no—
230#331—1221no—
231Q8⋊C8—21no—
232D4⋊C8—21no—
233C61#936011yescharacteristic, complemented, Sylow
234C2×#41C4×D6111yescharacteristic
235C22×D4—612no—
236C22×D4—1221no—
237C22×D4—1221no—
238AGL(1,ℤ/8)—2444no—
239C2×SD16—1222no—
240C2×D8—1222no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—612no—
244C2×M16—612no—
245C2×M16—1222no—
246#127—1221no—
247#127—1221no—
248#127—1221no—
249#127—1221no—
250C4×Q8D24411yescharacteristic
251C4×Q8C22⋊D6111yescharacteristic
252C4×D4—21no—
253C4×D4—21no—
254C4×D4D24411yescharacteristic
255C4×D4C22⋊D6111yescharacteristic
256#117—21no—
257#117C4×D6111yescharacteristic
258C2×C42—21no—
259C2×C42C22×D6111yescharacteristic
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—611no—
263C4⋊C8—611no—
264C4⋊C8—1222no—
265C4⋊C8—1221no—
266C4⋊C8—1221no—
267C4⋊C8—1221no—
268C4⋊C8—1221no—
269#102—1222no—
270#102—1222no—
271#100—2444no—
272C22⋊C8—1222no—
273C22⋊C8—1221no—
274#97—21no—
275C4×C8—21no—
276C4×C8—1222no—
277C4×C8—1221no—
278C24—1222no—
279#41—22no—
280#41—41no—
281C2×Q8#1175819811yescharacteristic
282C2×D4—21no—
283C2×D4—1222no—
284C2×D4—1222no—
285C2×D4—4881no—
286C2×D4—4881no—
287C2×D4—2442no—
288C2×D4—2442no—
289C2×D4—2442no—
290C2×D4—2442no—
291C2×D4—2441no—
292C2×D4—2441no—
293C2×D4—2441no—
294C2×D4—2441no—
295C2×D4#1175819811yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4C2×C4×D6111yescharacteristic
300C22×C4D4×D6111yescharacteristic
301C22×C4#1175822411yescharacteristic
302SD16—2444no—
303D8—2444no—
304M16—22no—
305M16—41no—
306M16—81no—
307M16—4882no—
308M16—2442no—
309M16—2442no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—1222no—
313C2×C8—1221no—
314C2×C8—1221no—
315C2×C8—1222no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4#1175819811yescharacteristic
319C4⋊C4#1175819811yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42C2×D24411yescharacteristic
327C42D4×D6111yescharacteristic
328C42#1175822411yescharacteristic
329C42C2×C22⋊D6111yescharacteristic
330C23—21no—
331C23—4882no—
332C23—4882no—
333C23—2444no—
334C23—2442no—
335C23#1451973711yescharacteristic
336Q8—21no—
337Q8#1451969312yescomplemented
338D4—21no—
339D4—21no—
340D4—42no—
341D4—4882no—
342D4—4882no—
343D4—4882no—
344D4—4881no—
345D4—4881no—
346D4—4881no—
347D4—4881no—
348D4—2442no—
349D4—2442no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4C2×#1175819811yescharacteristic
365C2×C4#1451973711yescharacteristic
366C2×C4#1451974711yescharacteristic
367C2×C4#1451974911yescharacteristic
368C2×C4#1451974911yescharacteristic
369C2×C4#1451976111yescharacteristic
370C8—42no—
371C8—81no—
372C8—2441no—
373C8—2442no—
374C8—2441no—
375C8—2442no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—9764no—
383C22—4882no—
384C22—4882no—
385C22—4882no—
386C22#1683963911yescharacteristic
387C4—21no—
388C4—21no—
389C4—22no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4C2×#1451969311yescharacteristic
394C4#1683974411yescharacteristic
395C4#1683993111yescharacteristic
396C4#1683993611yescharacteristic
397C4#1684002711yescharacteristic
398C4#1684003011yescharacteristic
399C2—41no—
400C2—81no—
401C2—9762no—
402C2#2144619311yescharacteristic
403C2#2144628011yescharacteristic
404C2#2144632511yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1489C2
socleC22×C61#386
prime radical#21455299C2
cosocleC2×C42×C61C23
hypercenter#1489D61
hypocenterC61#9360
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1489

Complements

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

NH ≅ G/Ntype
2. #21454951C2semidirect
3. #21446659C2semidirect
4. #21444488C2semidirect
5. #21443732C2semidirect
6. #21443709C2semidirect
7. C61×#1489C2semidirect
134. Q8×C61#100semidirect
201. #1489D61semidirect
233. C61#9360semidirect
337. Q8#14519693semidirect

Maximal subgroups

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

subgroupindex
#214552992
#214549512
#214466592
#214444882
#214437322
#214437092
C61×#14892
#936061