Finite Group Databasebeta

← #29576556

Subgroups — #29576556

Order 15616.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456734C211yesmaximal, characteristic
2#21454563C211yesmaximal, characteristic, complemented
3#21446658C211yesmaximal, characteristic, complemented
4C2×#16839505C211yesmaximal, characteristic, complemented
5#21443714C211yesmaximal, characteristic, complemented
6#21443669C211yesmaximal, characteristic, complemented
7C61×#1547C211yesmaximal, characteristic, complemented
8C2×#14519784—21no—
9C2×#14519774C2211yescharacteristic
10#16840138—21no—
11#16840110—21no—
12#16839957—22no—
13#16839895C414yes—
14#16839840—21no—
15#16839834—22no—
16D4×#92418—21no—
17#16839815C2211yescharacteristic
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519693—21no—
20C2×#14519693—21no—
21#16839729—21no—
22C4×#11758194C2211yescharacteristic
23#16839505—21no—
24#16839505—21no—
25C61×#532C2211yescharacteristic
26C61×#423C2211yescharacteristic
27C2×C4⋊C8×C61C2211yescharacteristic
28C61×#332—21no—
29C61×#331—21no—
30C22×D244—21no—
31C22×D244—21no—
32C22×D244C2×C412yes—
33C2×#11758199—24no—
34#14519784—42no—
35#14519784—42no—
36C2×#11758194—22no—
37C2×#11758194D411yescharacteristic
38C2×#11758194D411yescharacteristic
39C22×#92418—22no—
40C22×#92418—21no—
41#14519774—21no—
42#14519774—21no—
43#14519774—21no—
44#14519774—21no—
45C4⋊#92418—21no—
46C4⋊#92418—21no—
47C4⋊#92418D411yescharacteristic
48C4⋊#92418D411yescharacteristic
49#14519697—24no—
50#14519693—42no—
51#14519693—42no—
52C22⋊#92418—22no—
53C22⋊#92418—22no—
54C22⋊#92418C2×C412yes—
55#14519688—21no—
56C4×#92418—21no—
57C4×#92418—21no—
58C2×#41×C61—21no—
59C2×M16×C61—21no—
60C22×C8×C61—21no—
61C4⋊Q8×C61D411yescharacteristic
62#127×C61D411yescharacteristic
63#124×C61—21no—
64#121×C61—21no—
65#121×C61C2×C412yes—
66C4×D4×C61—21no—
67C4×D4×C61D412yes—
68C2×C42×C61C2311yescharacteristic
69C4⋊C8×C61—21no—
70C4⋊C8×C61—21no—
71C4⋊C8×C61—21no—
72C4⋊C8×C61—21no—
73C23×D61—22no—
74#41×C61—41no—
75#41×C61—41no—
76#41×C61—41no—
77#41×C61—41no—
78C2×Q8×C61—21no—
79C2×D4×C61—22no—
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61C22⋊C412yes—
83C22×C4×C61—22no—
84C22×C4×C61C22×C411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61#4111yescharacteristic
87C2×D244—22no—
88C2×D244—22no—
89C2×D244—41no—
90C2×D244—41no—
91C2×D244—41no—
92C2×D244—41no—
93C2×D244—41no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244—41no—
97C2×D244—82no—
98C2×D244—81no—
99C2×D244—81no—
100M16×C61—41no—
101M16×C61—41no—
102C2×C8×C61—41no—
103C2×C8×C61—41no—
104C2×C8×C61—81no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61C22⋊C412yes—
107C22⋊C4×C61—22no—
108C22⋊C4×C61—22no—
109C42×C61C2×D411yescharacteristic
110C42×C61C2×D411yescharacteristic
111C42×C61C2×D411yescharacteristic
112C42×C61#4111yescharacteristic
113#11758199—84no—
114#11758194—21no—
115#11758194—21no—
116#11758194—21no—
117#11758194—44no—
118#11758194D812yescomplemented
119C2×#92418—21no—
120C2×#92418—22no—
121C2×#92418—21no—
122C2×#92418—21no—
123C2×#92418—21no—
124C2×#92418—44no—
125C2×#92418—41no—
126C2×#92418—41no—
127C23×C61—22no—
128C23×C61C4×D411yescharacteristic
129C22×D61—42no—
130C22×D61—42no—
131C22×D61—42no—
132C22×D61—82no—
133C22×D61—82no—
134Q8×C61—41no—
135Q8×C61—41no—
136D4×C61—42no—
137D4×C61—42no—
138D4×C61—42no—
139D4×C61—41no—
140D4×C61—41no—
141D4×C61—41no—
142D4×C61—41no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—22no—
146C2×C4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61C2×C22⋊C411yescharacteristic
150C2×C4×C61C4×D411yescharacteristic
151C2×C4×C61#12011yescharacteristic
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12311yescharacteristic
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61#13511yescharacteristic
157C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
159D244—44no—
160D244—82no—
161D244—82no—
162D244—81no—
163D244—81no—
164D244—81no—
165D244—81no—
166D244—81no—
167D244—81no—
168C8×C61—81no—
169C8×C61—81no—
170#92418—22no—
171#92418—42no—
172#92418—41no—
173#92418—41no—
174#92418—41no—
175#9273—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61#38611yescharacteristic
182C22×C61C4×D811yescharacteristic
183C22×C61#44211yescharacteristic
184C2×D61—82no—
185C2×D61—82no—
186C2×D61—82no—
187C2×D61—162no—
188C2×D61—162no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—42no—
193C4×C61#44711yescharacteristic
194C4×C61#44911yescharacteristic
195C4×C61#45911yescharacteristic
196C4×C61#46311yescharacteristic
197C4×C61#47111yescharacteristic
198C4×C61#47211yescharacteristic
199#3205—611no—
200#2935—611no—
201#1965—611no—
202#1547D6111yescharacteristic, complemented
203C2×#331—611no—
204#1301—611no—
205#1283—611no—
206C2×C61—21no—
207C2×C61—21no—
208C2×C61—82no—
209C2×C61#187911yescharacteristic
210C2×C61#190711yescharacteristic
211C2×C61#191411yescharacteristic
212D61—162no—
213C2×#131—1221no—
214#532C2×D6111yescharacteristic
215C2×#127—611no—
216#493—1221no—
217#486—1221no—
218#459—1222no—
219#447—614no—
220#436—1221no—
221#435—1222no—
222C8×D4—1221no—
223#431—611no—
224#423C2×D6111yescharacteristic
225C2×C4⋊C8—611no—
226C2×C4⋊C8C2×D6111yescharacteristic
227C2×#100—1221no—
228C2×#100—1221no—
229#408—1221no—
230C4×M16—611no—
231#332—21no—
232#331—21no—
233#331—1221no—
234#331—1221no—
235C61#927311yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C22×D4—612no—
238C22×D4—1221no—
239C22×D4—1221no—
240C2×D8—1224no—
241#131—2442no—
242#131—2442no—
243C2×M16—21no—
244C2×M16—611no—
245C2×M16—611no—
246C2×M16—1222no—
247C22×C8—21no—
248C22×C8—1222no—
249C22×C8—1221no—
250C4⋊Q8C22⋊D6111yescharacteristic
251#127—1221no—
252#127—1221no—
253#127—1221no—
254#127—1221no—
255#127C22⋊D6111yescharacteristic
256#124—21no—
257#121—21no—
258#121C4×D6112yes—
259C4×D4—21no—
260C4×D4D24412yes—
261C2×C42C22×D6111yescharacteristic
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—611no—
267C4⋊C8—611no—
268C4⋊C8—1221no—
269C4⋊C8—1221no—
270#102—1224no—
271#100—2442no—
272#100—2442no—
273C22⋊C8—612no—
274C22⋊C8—1222no—
275C22⋊C8—1222no—
276#97—1221no—
277C4×C8—1221no—
278C4×C8—1221no—
279C24—1222no—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284C2×Q8—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—1222no—
289C2×D4—1222no—
290C2×D4—4881no—
291C2×D4—4881no—
292C2×D4—4882no—
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#1175819812yes—
302C22×C4—22no—
303C22×C4C2×C4×D6111yescharacteristic
304C22×C4D4×D6111yescharacteristic
305C22×C4#1175822411yescharacteristic
306D8—4884no—
307M16—41no—
308M16—41no—
309M16—612no—
310M16—1221no—
311M16—1221no—
312M16—1221no—
313M16—2444no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—81no—
317C2×C8—1221no—
318C2×C8—1222no—
319C2×C8—1221no—
320C2×C8—1221no—
321C2×C8—1221no—
322C2×C8—2444no—
323C2×C8—2441no—
324C2×C8—2441no—
325C4⋊C4—21no—
326C4⋊C4#1175819812yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C42C2×D24411yescharacteristic
330C42D4×D6111yescharacteristic
331C42#1175822411yescharacteristic
332C42C2×C22⋊D6111yescharacteristic
333C23—22no—
334C23—4882no—
335C23—4882no—
336C23—2442no—
337C23—2442no—
338C23—2442no—
339C23#1451973711yescharacteristic
340Q8—41no—
341Q8—41no—
342D4—42no—
343D4—42no—
344D4—42no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—4882no—
350D4—4882no—
351D4—4881no—
352D4—4881no—
353D4—4881no—
354D4—4881no—
355D4—4881no—
356D4—4881no—
357D4—2444no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4C2×#1175819811yescharacteristic
365C2×C4#1451973711yescharacteristic
366C2×C4#1451974711yescharacteristic
367C2×C4#1451974911yescharacteristic
368C2×C4#1451974911yescharacteristic
369C2×C4#1451976111yescharacteristic
370C2×C4D8×D6111yescharacteristic
371C2×C4#1451979811yescharacteristic
372C2×C4#1451980111yescharacteristic
373C2×C4#1451980411yescharacteristic
374C8—81no—
375C8—81no—
376C8—1222no—
377C8—2442no—
378C8—2441no—
379C8—2441no—
380C8—2441no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—9762no—
387C22—9762no—
388C22—4882no—
389C22—4882no—
390C22—4882no—
391C22#1683963911yescharacteristic
392C22#1683984211yescharacteristic
393C22#1683987111yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4#1683989711yescharacteristic
399C4#1683990811yescharacteristic
400C4#1683995511yescharacteristic
401C4#1683997911yescharacteristic
402C4#1684002711yescharacteristic
403C4#1684003011yescharacteristic
404C2—21no—
405C2—21no—
406C2—82no—
407C2—9762no—
408C2#2144616511yescharacteristic
409C2#2144632511yescharacteristic
410C2#2144636411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1547C2
socleC22×C61#386
prime radical#21456734C2
cosocleC2×C42×C61C23
hypercenter#1547D61
hypocenterC61#9273
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
2. #21454563C2semidirect
3. #21446658C2semidirect
4. C2×#16839505C2semidirect
5. #21443714C2semidirect
6. #21443669C2semidirect
7. C61×#1547C2semidirect
118. #11758194D8semidirect
202. #1547D61semidirect
235. C61#9273semidirect

Maximal subgroups

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

subgroupindex
#214567342
#214545632
#214466582
C2×#168395052
#214437142
#214436692
C61×#15472
#927361