Finite Group Databasebeta

← #29573909

Subgroups — #29573909

Order 15616.

440 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454937C211yesmaximal, characteristic
2C4×#14519705C211yesmaximal, characteristic
3#21444569C211yesmaximal, characteristic, complemented
4#21444204C211yesmaximal, characteristic
5#21443669C211yesmaximal, characteristic, complemented
6#21443652C211yesmaximal, characteristic, complemented
7C61×#1842C211yesmaximal, characteristic, complemented
8C4×#11758222C2211yescharacteristic
9#16839889—21no—
10#16839839—21no—
11#16839833—22no—
12#16839818—21no—
13#16839815C2211yescharacteristic
14#16839815C2211yescharacteristic
15#16839784—21no—
16C2×#14519705C411yescharacteristic
17C2×#14519705C411yescharacteristic
18C4×#11758194C2211yescharacteristic
19#16839522C411yescharacteristic
20#16839522C411yescharacteristic
21#16839514—21no—
22#16839512—21no—
23#16839492—21no—
24#16839489—21no—
25C2×C4⋊C8×C61—22no—
26C2×C4⋊C8×C61C2211yescharacteristic
27C2×C4⋊C8×C61C2211yescharacteristic
28C43×C61C2211yescharacteristic
29C2×#11758222C2×C411yescharacteristic
30C2×#11758194C2×C411yescharacteristic
31C2×#11758194C2×C411yescharacteristic
32C2×#11758189—22no—
33C4×Dic122D411yescharacteristic
34C4×Dic122Q811yescharacteristic
35C4×D244D411yescharacteristic
36C4×D244Q811yescharacteristic
37C4×C22⋊D61—21no—
38C4×C22⋊D61—21no—
39#14519736C2×C411yescharacteristic
40#14519734—21no—
41C42×D61—21no—
42C4⋊#92418—21no—
43C4⋊#92418D411yescharacteristic
44C4⋊#92418D411yescharacteristic
45#14519706—22no—
46#14519705—22no—
47#14519705—22no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50#14519689—22no—
51#14519688—21no—
52#14519687—22no—
53C4×#92418—21no—
54C4×#92418—21no—
55C4×#92418—21no—
56C22×C8×C61—21no—
57C22×C8×C61D412yes—
58C2×C42×C61—21no—
59C2×C42×C61—21no—
60C2×C42×C61C2×C411yescharacteristic
61C2×C42×C61C2×C411yescharacteristic
62C2×C42×C61C2311yescharacteristic
63C4⋊C8×C61—21no—
64C4⋊C8×C61—21no—
65C4⋊C8×C61—21no—
66C4⋊C8×C61—21no—
67C4⋊C8×C61—44no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61C22×C411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61#4111yescharacteristic
73C2×C22⋊D61—21no—
74#11758222—21no—
75#11758222—21no—
76#11758222—21no—
77#11758222—21no—
78C2×D244C4⋊C411yescharacteristic
79C2×C4×D61—21no—
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×Dic122C4⋊C411yescharacteristic
83C2×C8×C61—22no—
84C2×C8×C61—42no—
85C2×C8×C61—42no—
86C2×C8×C61—41no—
87C42×C61—22no—
88C42×C61—22no—
89C42×C61—21no—
90C42×C61—21no—
91C42×C61—21no—
92C42×C61—42no—
93C42×C61M1612yescomplemented
94C42×C61C2×D411yescharacteristic
95C42×C61C2×D411yescharacteristic
96C42×C61C2×Q811yescharacteristic
97C42×C61#4111yescharacteristic
98C22⋊Dic61—21no—
99#11758198—21no—
100#11758198—21no—
101C4⋊Dic61C4⋊C411yescharacteristic
102C4⋊Dic61C4⋊C411yescharacteristic
103#11758196—21no—
104#11758196—21no—
105C4×Dic61—21no—
106C4×Dic61—21no—
107C4×Dic61—21no—
108#11758194—21no—
109#11758194—21no—
110#11758194—21no—
111#11758194M1612yescomplemented
112C2×#92418—21no—
113C2×#92418—21no—
114C2×#92418—21no—
115C2×#92418—21no—
116C2×#92418—41no—
117#11758189—82no—
118C23×C61C4×D411yescharacteristic
119C22×D61—21no—
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—22no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61C2×C4⋊C411yescharacteristic
130C2×C4×C61C4×D411yescharacteristic
131C2×C4×C61#12011yescharacteristic
132C2×C4×C61C22⋊Q811yescharacteristic
133C2×C4×C61C22⋊Q811yescharacteristic
134C2×C4×C61#12311yescharacteristic
135C2×C4×C61C2×M1611yescharacteristic
136C2×C4×C61C2×M1611yescharacteristic
137C2×C4×C61#13111yescharacteristic
138C2×C4×C61#13111yescharacteristic
139C22⋊D61—41no—
140C22⋊D61—41no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144D244—21no—
145D244—21no—
146C4×D61—41no—
147C4×D61—41no—
148C4×D61—41no—
149C4×D61—41no—
150C4×D61—81no—
151Dic122—21no—
152Dic122—21no—
153C8×C61—42no—
154C8×C61—81no—
155#92418—22no—
156#92418—41no—
157#92418—41no—
158#92418—41no—
159#92418—81no—
160#8841—611nomaximal, Sylow
161C22×C61—21no—
162C22×C61—21no—
163C22×C61#38511yescharacteristic
164C22×C61#43511yescharacteristic
165C22×C61#43611yescharacteristic
166C2×D61—41no—
167C2×D61—41no—
168C2×D61—41no—
169C4×C61—24no—
170C4×C61—21no—
171C4×C61—21no—
172C4×C61—21no—
173C4×C61—42no—
174C4×C61#42311yescharacteristic
175C4×C61#42411yescharacteristic
176C4×C61#43511yescharacteristic
177C4×C61#43611yescharacteristic
178C4×C61#45311yescharacteristic
179C4×C61#45611yescharacteristic
180Dic61—41no—
181Dic61—41no—
182Dic61—41no—
183#2989—611no—
184#1842D6111yescharacteristic, complemented
185C4×#104—611no—
186#1565—611no—
187#1477—611no—
188#1283—611no—
189#1276—611no—
190C2×C61—21no—
191C2×C61—21no—
192C2×C61#185111yescharacteristic
193C2×C61#185911yescharacteristic
194C2×C61#186111yescharacteristic
195D61—81no—
196C4×#41—611no—
197#446—1221no—
198#436—1221no—
199#435—1222no—
200#432—1221no—
201#431—611no—
202#431—611no—
203#423—1221no—
204C2×C4⋊C8—22no—
205C2×C4⋊C8C2×D6111yescharacteristic
206C2×C4⋊C8C2×D6111yescharacteristic
207C2×#104—611no—
208C2×#104—611no—
209C4×M16—611no—
210C43C2×D6111yescharacteristic
211#339—611no—
212#339—611no—
213#335—1221no—
214#334—1221no—
215Q8⋊C8—1221no—
216D4⋊C8—1221no—
217C61#884111yescharacteristic, complemented, Sylow
218C2×#41—611no—
219C2×M16—611no—
220C2×M16—611no—
221C2×M16—1222no—
222C22×C8—21no—
223C22×C8C22⋊D6112yes—
224C4×Q8—611no—
225C4×Q8—611no—
226C4×D4—611no—
227C4×D4—611no—
228C4×D4—1221no—
229C4×D4—1221no—
230#117—611no—
231#117—1221no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—1221no—
235C2×C42C4×D6111yescharacteristic
236C2×C42C4×D6111yescharacteristic
237C2×C42C22×D6111yescharacteristic
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—44no—
243C4⋊C8—611no—
244C4⋊C8—611no—
245C4⋊C8—1222no—
246C4⋊C8—1221no—
247#104—1222no—
248#104—1222no—
249C22⋊C8—1222no—
250C22⋊C8—1221no—
251C22⋊C8—1221no—
252#97—1222no—
253#97—1221no—
254C4×C8—1221no—
255C4×C8—1221no—
256C4×C8—1221no—
257#41—1221no—
258#41—1221no—
259#41—1221no—
260#41—1221no—
261C2×Q8—611no—
262C2×D4—611no—
263C2×D4—1221no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4—1221no—
269C22×C4C2×C4×D6111yescharacteristic
270C22×C4#1175822211yescharacteristic
271C22×C4C2×C22⋊D6111yescharacteristic
272M16—612no—
273M16—1221no—
274M16—1221no—
275M16—1221no—
276M16—4882no—
277C2×C8—22no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—1221no—
282C2×C8—1221no—
283C2×C8—1221no—
284C2×C8—1221no—
285C2×C8—2441no—
286C4⋊C4—611no—
287C4⋊C4—611no—
288C4⋊C4—1221no—
289C4⋊C4—1221no—
290C22⋊C4—1221no—
291C22⋊C4—1221no—
292C22⋊C4—1221no—
293C42—22no—
294C42—22no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—42no—
299C42—1221no—
300C42—1221no—
301C42—1221no—
302C42#1175818912yescomplemented
303C42D4×D6111yescharacteristic
304C42D4×D6111yescharacteristic
305C42Q8×D6111yescharacteristic
306C42#1175822611yescharacteristic
307C23—1221no—
308C23C4×C22⋊D6111yescharacteristic
309Q8—1221no—
310Q8—1221no—
311D4—1221no—
312D4—1221no—
313D4—2441no—
314D4—2441no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—1221no—
325C2×C4—1221no—
326C2×C4—1221no—
327C2×C4—4881no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4C4⋊C4×D6111yescharacteristic
333C2×C4#1451973911yescharacteristic
334C2×C4#1451974511yescharacteristic
335C2×C4#1451975311yescharacteristic
336C2×C4#1451975511yescharacteristic
337C2×C4#1451976211yescharacteristic
338C2×C4M16×D6111yescharacteristic
339C2×C4C2×#1175818911yescharacteristic
340C2×C4#1451978311yescharacteristic
341C2×C4#1451978511yescharacteristic
342C8—42no—
343C8—81no—
344C8—1222no—
345C8—4881no—
346C8—2441no—
347C8—2441no—
348C8—2441no—
349C22—21no—
350C22—21no—
351C22—2441no—
352C22—2441no—
353C22—2441no—
354C22#1683963311yescharacteristic
355C22#1683983311yescharacteristic
356C22#1683983811yescharacteristic
357C4—24no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—2441no—
363C4—2441no—
364C4—2441no—
365C4#1683978411yescharacteristic
366C4#1683978711yescharacteristic
367C4#1683983211yescharacteristic
368C4#1683983911yescharacteristic
369C4#1683992911yescharacteristic
370C4#1683994411yescharacteristic
371C2—21no—
372C2—21no—
373C2—4881no—
374C2#2144600011yescharacteristic
375C2#2144604411yescharacteristic
376C2#2144605811yescharacteristic
subgroupG/H
centerC2×C4#14519745
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1842C2
socleC22×C61#385
prime radicalC2×#11758222C2×C4
cosocleC2×C42×C61C23
hypercenter#1842D61
hypocenterC61#8841
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1842

Complements

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

NH ≅ G/Ntype
3. #21444569C2semidirect
5. #21443669C2semidirect
6. #21443652C2semidirect
7. C61×#1842C2semidirect
93. C42×C61M16semidirect
111. #11758194M16semidirect
184. #1842D61semidirect
217. C61#8841semidirect
302. C42#11758189semidirect

Maximal subgroups

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

subgroupindex
#214549372
C4×#145197052
#214445692
#214442042
#214436692
#214436522
C61×#18422
#884161