Finite Group Databasebeta

← #29576558

Subgroups — #29576558

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#21446659C211yesmaximal, characteristic, complemented
3#21444496C211yesmaximal, characteristic, complemented
4C2×#16839505C211yesmaximal, characteristic, complemented
5#21443716C211yesmaximal, characteristic, complemented
6#21443670C211yesmaximal, characteristic, complemented
7C61×#2935C211yesmaximal, characteristic, complemented
8C2×#14519774C2211yescharacteristic
9#16840138—21no—
10#16840110—21no—
11#16839957—22no—
12#16839895C414yes—
13#16839815C2211yescharacteristic
14#16839786C2211yescharacteristic
15C2×C4⋊#92418C2211yescharacteristic
16C2×#14519693—21no—
17C2×#14519693—21no—
18#16839508—21no—
19#16839506—21no—
20#16839505—21no—
21#16839505—21no—
22C2×#131×C61—21no—
23C61×#532C2211yescharacteristic
24C61×#436—21no—
25C61×#435—22no—
26C8×D4×C61—21no—
27C2×C4⋊C8×C61C2211yescharacteristic
28C61×#408—21no—
29C4×M16×C61C2211yescharacteristic
30C22×D244—21no—
31C22×D244—21no—
32C22×D244C2×C412yes—
33C2×#11758199—24no—
34C2×#11758194—21no—
35C22×#92418—21no—
36#14519774—21no—
37#14519774—21no—
38#14519774—21no—
39#14519774—21no—
40C4⋊#92418—21no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44C4⋊#92418—21no—
45#14519697—24no—
46#14519693—42no—
47#14519693—42no—
48C22⋊#92418C2×C412yes—
49C4×#92418—21no—
50C2×#41×C61—21no—
51#131×C61—42no—
52#131×C61—42no—
53C2×M16×C61—22no—
54C2×M16×C61D411yescharacteristic
55C2×M16×C61D411yescharacteristic
56C22×C8×C61—22no—
57C22×C8×C61—21no—
58C4⋊Q8×C61D411yescharacteristic
59#127×C61D411yescharacteristic
60#124×C61—21no—
61#121×C61—21no—
62#121×C61C2×C412yes—
63C4×D4×C61—21no—
64C4×D4×C61D412yes—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C4⋊C8×C61D411yescharacteristic
68C4⋊C8×C61D411yescharacteristic
69C22⋊C8×C61—22no—
70C22⋊C8×C61—22no—
71#97×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—21no—
101M16×C61—21no—
102M16×C61—21no—
103M16×C61—44no—
104M16×C61D812yescomplemented
105C2×C8×C61—21no—
106C2×C8×C61—21no—
107C2×C8×C61—21no—
108C2×C8×C61—21no—
109C2×C8×C61—44no—
110C2×C8×C61—41no—
111C2×C8×C61—41no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61C22⋊C412yes—
114C22⋊C4×C61—22no—
115C22⋊C4×C61—22no—
116C42×C61C2×D411yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61#4111yescharacteristic
120#11758199—84no—
121#11758194—41no—
122#11758194—41no—
123C2×#92418—22no—
124C2×#92418—41no—
125C2×#92418—41no—
126C2×#92418—81no—
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—22no—
169C8×C61—41no—
170C8×C61—41no—
171C8×C61—41no—
172#92418—42no—
173#92418—81no—
174#92418—81no—
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#2935D6111yescharacteristic, complemented
201#1965—611no—
202#1547—611no—
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—21no—
214#532C2×D6111yescharacteristic
215C2×#127—611no—
216#493—1221no—
217#486—1221no—
218#459—1222no—
219#447—614no—
220#436—21no—
221#435—22no—
222C8×D4—21no—
223#431—611no—
224#423—611no—
225C2×C4⋊C8—611no—
226C2×C4⋊C8C2×D6111yescharacteristic
227C2×#100—1221no—
228C2×#100—1221no—
229#408—21no—
230C4×M16C2×D6111yescharacteristic
231#332—1221no—
232#331—1221no—
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—42no—
242#131—42no—
243C2×M16—22no—
244C2×M16—1221no—
245C2×M16C22⋊D6111yescharacteristic
246C2×M16C22⋊D6111yescharacteristic
247C22×C8—22no—
248C22×C8—21no—
249C22×C8—1221no—
250C4⋊Q8D24411yescharacteristic
251#127—1221no—
252#127—1221no—
253#127—1221no—
254#127—1221no—
255#127D24411yescharacteristic
256#124—21no—
257#121—21no—
258#121C4×D6112yes—
259C4×D4—21no—
260C4×D4C22⋊D6112yes—
261C2×C42C22×D6111yescharacteristic
262C4⋊C8—21no—
263C4⋊C8—1221no—
264C4⋊C8—1221no—
265C4⋊C8—1221no—
266C4⋊C8—1221no—
267C4⋊C8—1221no—
268C4⋊C8C22⋊D6111yescharacteristic
269C4⋊C8C22⋊D6111yescharacteristic
270#102—1224no—
271#100—2442no—
272#100—2442no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—612no—
276#97—21no—
277C4×C8—21no—
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×C4#1175822211yescharacteristic
305C22×C4C2×C22⋊D6111yescharacteristic
306D8—4884no—
307M16—21no—
308M16—21no—
309M16—21no—
310M16—44no—
311M16—2441no—
312M16—2441no—
313M16#1175819912yescomplemented
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—44no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—1222no—
322C2×C8—4881no—
323C2×C8—2441no—
324C2×C8—2441no—
325C4⋊C4—21no—
326C4⋊C4#1175819812yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C42C2×D24411yescharacteristic
330C42#1175822211yescharacteristic
331C42C2×C22⋊D6111yescharacteristic
332C42C2×C22⋊D6111yescharacteristic
333C23—22no—
334C23—4882no—
335C23—4882no—
336C23—2442no—
337C23—2442no—
338C23—2442no—
339C23C4×C22⋊D6111yescharacteristic
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×C4C4×C22⋊D6111yescharacteristic
366C2×C4#1451974611yescharacteristic
367C2×C4#1451975111yescharacteristic
368C2×C4#1451975111yescharacteristic
369C2×C4#1451976011yescharacteristic
370C2×C4C2×#1175819911yescharacteristic
371C2×C4#1451979711yescharacteristic
372C2×C4#1451980211yescharacteristic
373C2×C4#1451980311yescharacteristic
374C8—22no—
375C8—41no—
376C8—41no—
377C8—41no—
378C8—4881no—
379C8—4881no—
380C8—2442no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—9762no—
387C22—9762no—
388C22—4882no—
389C22—4882no—
390C22—4882no—
391C22#1683964111yescharacteristic
392C22C4×#1175819911yescharacteristic
393C22#1683987211yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4#1683989611yescharacteristic
399C4#1683990711yescharacteristic
400C4#1683995711yescharacteristic
401C4#1683997711yescharacteristic
402C4#1684002511yescharacteristic
403C4#1684003211yescharacteristic
404C2—21no—
405C2—21no—
406C2—82no—
407C2—9762no—
408C2#2144616711yescharacteristic
409C2#2144632711yescharacteristic
410C2#2144636611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2935C2
socleC22×C61#386
prime radical#21456734C2
cosocleC2×C42×C61C23
hypercenter#2935D61
hypocenterC61#9273
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
2. #21446659C2semidirect
3. #21444496C2semidirect
4. C2×#16839505C2semidirect
5. #21443716C2semidirect
6. #21443670C2semidirect
7. C61×#2935C2semidirect
104. M16×C61D8semidirect
200. #2935D61semidirect
235. C61#9273semidirect
313. M16#11758199semidirect

Maximal subgroups

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

subgroupindex
#214567342
#214466592
#214444962
C2×#168395052
#214437162
#214436702
C61×#29352
#927361