Finite Group Databasebeta

← #29665182

Subgroups — #29665182

Order 15616.

708 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456746C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3#21455283C211yesmaximal, characteristic
4#21445242C214yesmaximal, complemented
5#21445160C214yesmaximal, complemented
6#21444590C211yesmaximal, characteristic, complemented
7#21444503C211yesmaximal, characteristic, complemented
8#21444466C211yesmaximal, characteristic, complemented
9C61×#2972C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13#16840135C2212yescomplemented
14#16840113C2212yescomplemented
15#16840042—22no—
16#16840042C2214yes—
17#16839977—22no—
18#16839968—22no—
19#16839967C2212yes—
20#16839957C2212yes—
21#16839903—22no—
22C2×#14519710—21no—
23#16839786C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25#16839761—21no—
26C2×#14519698—21no—
27#16839513C2214yescomplemented
28#16839509C2214yescomplemented
29#16839507C2212yescomplemented
30#16839505C2212yescomplemented
31C2×C4×D4×C61C2211yescharacteristic
32C8×D4×C61—22no—
33C8×D4×C61C2214yes—
34C61×#431C2211yescharacteristic
35C61×#431C2211yescharacteristic
36C2×C22⋊C8×C61—22no—
37C2×#11758222—21no—
38C22×Dic122—21no—
39#14519803—44no—
40C2×#11758190—24no—
41C2×#11758200—22no—
42C2×#11758199—22no—
43C2×#11758194—21no—
44C22×#92418—21no—
45#14519777—21no—
46#14519777—21no—
47#14519777—21no—
48#14519777C2311yescharacteristic
49#14519777C2312yes—
50#14519774C2311yescharacteristic
51#14519765—21no—
52#14519751—21no—
53#14519751—21no—
54C4×D244—21no—
55C4×D244—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58#14519710—24no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61C4⋊#92418C2312yes—
62C4⋊#92418C2312yes—
63#14519702—24no—
64#14519700—22no—
65#14519698—24no—
66#14519697—22no—
67C22×D4×C61D411yescharacteristic
68C23×C4×C61—21no—
69C22×C8×C61—24no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61D412yes—
73C4×D4×C61C2312yes—
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C22⋊C4×C61—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61D412yes—
78C4⋊C8×C61C2312yes—
79C22⋊C8×C61—21no—
80C22⋊C8×C61—24no—
81C22⋊C8×C61D412yes—
82C4×C8×C61—21no—
83C4×C8×C61C2312yes—
84C24×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—22no—
87C2×D4×C61SD1614yescomplemented
88C2×D4×C61C2×D412yes—
89C22×C4×C61—21no—
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—82no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61#4111yescharacteristic
96C2×C22⋊D61—41no—
97C22×Dic61—41no—
98#11758222—42no—
99#11758222—42no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244—21no—
103C2×C4×D61—41no—
104C2×Dic122—22no—
105C2×Dic122—21no—
106C2×Dic122—41no—
107C2×Dic122—82no—
108C2×C8×C61—22no—
109C2×C8×C61—84no—
110C2×C8×C61C2×D414yes—
111C4⋊C4×C61—21no—
112C4⋊C4×C61C2×D412yes—
113C22⋊C4×C61—22no—
114C22⋊C4×C61—22no—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61#4111yescharacteristic
118C42×C61C2411yescharacteristic
119#11758200—44no—
120#11758199—44no—
121#11758198—41no—
122#11758198—41no—
123C4⋊Dic61—22no—
124C4⋊Dic61—21no—
125C4⋊Dic61—22no—
126C4⋊Dic61—21no—
127C4⋊Dic61—41no—
128C4⋊Dic61—41no—
129#11758194—42no—
130C2×#92418—22no—
131C2×#92418—22no—
132C2×#92418—42no—
133#11758190—84no—
134C23×C61—22no—
135C23×C61—24no—
136C23×C61—81no—
137C23×C61—81no—
138C23×C61#12111yescharacteristic
139C22×D61—41no—
140D4×C61—22no—
141D4×C61—24no—
142D4×C61C2×SD1614yes—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—41no—
146C2×C4×C61—44no—
147C2×C4×C61—82no—
148C2×C4×C61—81no—
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61C2×SD1612yes—
153C2×C4×C61#13512yes—
154C2×C4×C61C22×D411yescharacteristic
155C2×C4×C61C22×D411yescharacteristic
156C2×C4×C61C2×#4111yescharacteristic
157C22⋊D61—82no—
158C2×Dic61—42no—
159C2×Dic61—41no—
160C2×Dic61—82no—
161D244—41no—
162D244—41no—
163D244—42no—
164D244—81no—
165C4×D61—82no—
166Dic122—41no—
167Dic122—41no—
168Dic122—44no—
169Dic122—82no—
170C8×C61—24no—
171C8×C61—42no—
172#92418—42no—
173#92418—42no—
174#19385—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—24no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—84no—
181C22×C61#46512yes—
182C22×C61C2×#12111yescharacteristic
183C2×D61—81no—
184C2×D61—81no—
185C2×D61—81no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—42no—
190C4×C61#46512yes—
191C4×C61C22×SD1611yescharacteristic
192C4×C61C2×#13511yescharacteristic
193C4×C61#57612yes—
194Dic61—82no—
195Dic61—81no—
196#3206—611no—
197C2×#474—611no—
198#3036—611no—
199#2972D6111yescharacteristic, complemented
200#1679—614no—
201#1665—614no—
202#1570—611no—
203#1549—611no—
204#1540—611no—
205C2×C61—22no—
206C2×C61—44no—
207C2×C61#304111yescharacteristic
208C2×C61C2×#46511yescharacteristic
209C2×C61#306911yescharacteristic
210D61—161no—
211C2×AGL(1,ℤ/8)—1221no—
212#532—611no—
213C2×C4⋊Q8—611no—
214C2×C4×D4C2×D6111yescharacteristic
215#492—612no—
216#487—612no—
217#474—614no—
218#474—1222no—
219#463—1222no—
220#461—612no—
221#461—1222no—
222#459—612no—
223#448—1222no—
224C8×D4—22no—
225C8×D4C2×D6114yes—
226#431C2×D6111yescharacteristic
227#431C2×D6111yescharacteristic
228C2×#107—1221no—
229#423—611no—
230C2×C4⋊C8—611no—
231#417—1221no—
232C2×#102—1221no—
233C2×C22⋊C8—22no—
234#334—614no—
235#332—612no—
236#332—614no—
237#331—612no—
238C61#1938511yescharacteristic, complemented, Sylow
239C2×#41—1221no—
240C22×Q8—1221no—
241C22×D4D24411yescharacteristic
242C23×C4—21no—
243AGL(1,ℤ/8)—2444no—
244C2×SD16—1222no—
245C2×SD16—1224no—
246C2×D8—1222no—
247C2×M16—1221no—
248C22×C8—24no—
249C22×C8—1221no—
250C4⋊Q8—612no—
251C4⋊Q8—611no—
252C4⋊Q8—1221no—
253C4⋊Q8—1221no—
254C4⋊Q8—1221no—
255#127—611no—
256#124—1221no—
257#121—1221no—
258#121—1221no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—1221no—
262C4×D4—1221no—
263C4×D4D24412yes—
264C4×D4C22×D6112yes—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4D24411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C42C22×D6111yescharacteristic
270#107—1224no—
271C4⋊C8—612no—
272C4⋊C8—612no—
273C4⋊C8—1221no—
274C4⋊C8—1221no—
275C4⋊C8C22⋊D6112yes—
276C4⋊C8C22×D6112yes—
277#103—1224no—
278#103—1222no—
279#102—1224no—
280#102—1222no—
281C22⋊C8—21no—
282C22⋊C8—24no—
283C22⋊C8C22⋊D6112yes—
284C4×C8—21no—
285C4×C8C22×D6112yes—
286C24—21no—
287#41—2442no—
288#41—2442no—
289C2×Q8—1222no—
290C2×Q8—1221no—
291C2×Q8—4882no—
292C2×Q8—2441no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—1221no—
296C2×D4—1221no—
297C2×D4—1221no—
298C2×D4—2441no—
299C2×D4#1175819014yescomplemented
300C2×D4C2×D24412yes—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—82no—
305C22×C4—2441no—
306C22×C4—2441no—
307C22×C4C2×D24411yescharacteristic
308C22×C4#1175822211yescharacteristic
309C22×C4C2×C22⋊D6111yescharacteristic
310SD16—4884no—
311SD16—2444no—
312D8—2444no—
313M16—2442no—
314C2×C8—22no—
315C2×C8—84no—
316C2×C8—1222no—
317C2×C8—1222no—
318C2×C8—2442no—
319C2×C8C2×C22⋊D6114yes—
320C4⋊C4—21no—
321C4⋊C4—1221no—
322C4⋊C4—1222no—
323C4⋊C4—1222no—
324C4⋊C4—1221no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C4⋊C4C2×D24412yes—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—2441no—
331C22⋊C4—2441no—
332C42C2×D24411yescharacteristic
333C42#1175822211yescharacteristic
334C42C2×C22⋊D6111yescharacteristic
335C42C23×D6111yescharacteristic
336C23—22no—
337C23—24no—
338C23—81no—
339C23—81no—
340C23—2441no—
341C23#1451975111yescharacteristic
342Q8—4882no—
343Q8—2441no—
344Q8—2441no—
345Q8—2444no—
346D4—24no—
347D4—22no—
348D4—4881no—
349D4—4882no—
350D4—2441no—
351D4—2441no—
352D4—2442no—
353D4C2×#1175819014yes—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—4882no—
361C2×C4—4882no—
362C2×C4—2441no—
363C2×C4—2442no—
364C2×C4#1451975111yescharacteristic
365C2×C4#1451975111yescharacteristic
366C2×C4#1451975111yescharacteristic
367C2×C4C2×#1175819012yes—
368C2×C4#1451979912yes—
369C2×C4C22×C22⋊D6111yescharacteristic
370C2×C4C22×D24411yescharacteristic
371C2×C4C2×#1175822211yescharacteristic
372C8—24no—
373C8—42no—
374C8—2442no—
375C8—2442no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—42no—
380C22—42no—
381C22—84no—
382C22—4881no—
383C22—4881no—
384C22—4881no—
385C22#1683999112yes—
386C22C2×#1451975111yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—4881no—
392C4—4882no—
393C4#1683999112yes—
394C4C22×#1175819011yescharacteristic
395C4C2×#1451979911yescharacteristic
396C4#1684056512yes—
397C2—22no—
398C2—44no—
399C2—9761no—
400C2#2145532211yescharacteristic
401C2C2×#1683999111yescharacteristic
402C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2972C2
socleC22×C61C2×#121
prime radical#21455283C2
hypercenter#2972D61
hypocenterC61#19385
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #21456746C2semidirect
2. C2×#16840042C2semidirect
4. #21445242C2semidirect
5. #21445160C2semidirect
6. #21444590C2semidirect
7. #21444503C2semidirect
8. #21444466C2semidirect
9. C61×#2972C2semidirect
13. #16840135C22semidirect
14. #16840113C22semidirect
27. #16839513C22semidirect
28. #16839509C22semidirect
29. #16839507C22semidirect
30. #16839505C22semidirect
87. C2×D4×C61SD16semidirect
199. #2972D61semidirect
238. C61#19385semidirect
299. C2×D4#11758190semidirect

Maximal subgroups

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

subgroupindex
#214567462
C2×#168400422
#214552832
#214452422
#214451602
#214445902
#214445032
#214444662
C61×#29722
#1938561