Finite Group Databasebeta

← #29576513

Subgroups — #29576513

Order 15616.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456744C211yesmaximal, characteristic
2#21454562C211yesmaximal, characteristic
3#21446809C211yesmaximal, characteristic, complemented
4#21444494C211yesmaximal, characteristic
5#21443716C211yesmaximal, characteristic, complemented
6#21443669C211yesmaximal, characteristic, complemented
7C2×C61×#333C211yesmaximal, characteristic, complemented
8C2×#14519783—21no—
9#16840312C2211yescharacteristic
10#16840136—21no—
11#16840112—21no—
12#16839964—22no—
13#16839901C414yes—
14#16839839—21no—
15#16839832—22no—
16#16839827—21no—
17#16839815C2211yescharacteristic
18#16839786C2211yescharacteristic
19#16839728—21no—
20C4×#11758194C2211yescharacteristic
21#16839531—21no—
22#16839531—21no—
23#16839507—21no—
24#16839505—21no—
25C2×C4⋊Q8×C61C2211yescharacteristic
26C2×C4⋊C8×C61C2211yescharacteristic
27C2×C4⋊C8×C61C2211yescharacteristic
28C61×#333—21no—
29C61×#333—21no—
30C2×#11758222—21no—
31C2×#11758201—24no—
32#14519783—42no—
33#14519783—42no—
34C2×#11758194—21no—
35C2×#11758194D411yescharacteristic
36C2×#11758194D411yescharacteristic
37C2×#11758189—22no—
38C2×C8×D61—22no—
39#14519777D411yescharacteristic
40#14519774D411yescharacteristic
41#14519765—21no—
42#14519751—21no—
43#14519751C2×C412yes—
44C4×D244—21no—
45C4×D244D412yes—
46C4⋊#92418—21no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49#14519701—24no—
50C22⋊#92418C2×C412yes—
51#14519689—22no—
52#14519689—22no—
53#14519688—21no—
54C4×#92418—21no—
55C4×#92418—21no—
56C22×Q8×C61C2×C412yes—
57C22×C8×C61—21no—
58C22×C8×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61—21no—
61C4⋊Q8×C61—21no—
62C4⋊Q8×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C4⋊C8×C61—21no—
68C4⋊C8×C61—21no—
69C4⋊C8×C61D411yescharacteristic
70C4⋊C8×C61D411yescharacteristic
71C2×Q8×C61—22no—
72C2×Q8×C61—22no—
73C2×Q8×C61—82no—
74C22×C4×C61—22no—
75C22×C4×C61C22×C411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61#4111yescharacteristic
78C2×C22⋊D61—22no—
79#11758222—41no—
80#11758222—41no—
81#11758222—41no—
82#11758222—41no—
83C2×D244—21no—
84C2×D244—21no—
85C2×D244C22⋊C412yes—
86C2×C4×D61—22no—
87C2×Dic122—21no—
88C2×C8×C61—21no—
89C2×C8×C61—21no—
90C2×C8×C61—41no—
91C2×C8×C61—41no—
92C2×C8×C61—41no—
93C2×C8×C61—81no—
94C4⋊C4×C61—41no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C42×C61C2×D411yescharacteristic
99C42×C61C2×D411yescharacteristic
100C42×C61C2×D411yescharacteristic
101C42×C61#4111yescharacteristic
102#11758201—84no—
103#11758198—22no—
104#11758198—22no—
105C4⋊Dic61—21no—
106C4⋊Dic61C22⋊C412yes—
107#11758194—21no—
108#11758194—21no—
109#11758194—21no—
110#11758194—41no—
111#11758194—41no—
112#11758194SD1612yescomplemented
113C2×#92418—21no—
114C2×#92418—22no—
115C2×#92418—21no—
116C2×#92418—41no—
117#11758189—44no—
118C8×D61—44no—
119C23×C61C4×D411yescharacteristic
120C22×D61—22no—
121Q8×C61—44no—
122Q8×C61—82no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—41no—
126C2×C4×C61—42no—
127C2×C4×C61—82no—
128C2×C4×C61C2×C22⋊C411yescharacteristic
129C2×C4×C61C4×D411yescharacteristic
130C2×C4×C61#12011yescharacteristic
131C2×C4×C61#12111yescharacteristic
132C2×C4×C61#12111yescharacteristic
133C2×C4×C61#12311yescharacteristic
134C2×C4×C61C2×SD1611yescharacteristic
135C2×C4×C61#13511yescharacteristic
136C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C61#13711yescharacteristic
138C22⋊D61—42no—
139C22⋊D61—42no—
140C2×Dic61—22no—
141D244—42no—
142D244—41no—
143D244—41no—
144D244—41no—
145D244—41no—
146C4×D61—42no—
147C4×D61—42no—
148Dic122—41no—
149Dic122—41no—
150C8×C61—41no—
151C8×C61—41no—
152C8×C61—81no—
153#92418—22no—
154#92418—42no—
155#92418—41no—
156#92418—81no—
157#9267—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61#38611yescharacteristic
161C22×C61C4×SD1611yescharacteristic
162C22×C61#44011yescharacteristic
163C2×D61—42no—
164C2×D61—42no—
165C2×D61—42no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—21no—
169C4×C61—82no—
170C4×C61#45011yescharacteristic
171C4×C61#45211yescharacteristic
172C4×C61#46111yescharacteristic
173C4×C61#46311yescharacteristic
174C4×C61#47111yescharacteristic
175C4×C61#47311yescharacteristic
176Dic61—42no—
177#3206—611no—
178#2935—611no—
179#1997—611no—
180#1547—611no—
181C2×#333D6111yescharacteristic, complemented
182#1301—611no—
183#1283—611no—
184C2×C61—21no—
185C2×C61—21no—
186C2×C61#187711yescharacteristic
187C2×C61#190811yescharacteristic
188C2×C61#191411yescharacteristic
189D61—82no—
190C2×#131—1221no—
191#532—611no—
192C2×C4⋊Q8C2×D6111yescharacteristic
193#492—1221no—
194#487—1221no—
195#461—1222no—
196#448—614no—
197#436—1221no—
198#435—1222no—
199C8×D4—1221no—
200#431—611no—
201#423—611no—
202C2×C4⋊C8C2×D6111yescharacteristic
203C2×C4⋊C8C2×D6111yescharacteristic
204#408—1221no—
205C4×M16—611no—
206#343—1221no—
207#343—1221no—
208#333—21no—
209#333—21no—
210#332—1221no—
211#331—1221no—
212C61#926711yescharacteristic, complemented, Sylow
213C2×#41—1221no—
214C22×Q8C4×D6112yes—
215C2×SD16—1224no—
216#131—2442no—
217#131—2442no—
218C2×M16—611no—
219C2×M16—611no—
220C2×M16—1222no—
221C2×M16—1221no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—1222no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—611no—
230#127—611no—
231#124—1221no—
232#121—612no—
233#121—1221no—
234C4×D4—612no—
235C4×D4—1221no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C42C22×D6111yescharacteristic
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—1221no—
243C4⋊C8—1221no—
244C4⋊C8—1221no—
245C4⋊C8C22⋊D6111yescharacteristic
246C4⋊C8C22⋊D6111yescharacteristic
247#103—1224no—
248C22⋊C8—612no—
249C22⋊C8—1222no—
250C22⋊C8—1222no—
251#97—1221no—
252C4×C8—1221no—
253C4×C8—1221no—
254#41—2441no—
255#41—2441no—
256#41—2441no—
257#41—2441no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—82no—
261C2×Q8—1221no—
262C2×D4—612no—
263C2×D4—1222no—
264C2×D4—1221no—
265C2×D4—1221no—
266C22×C4—22no—
267C22×C4—1222no—
268C22×C4C2×C4×D6111yescharacteristic
269C22×C4D4×D6111yescharacteristic
270C22×C4#1175822411yescharacteristic
271SD16—4884no—
272M16—612no—
273M16—1221no—
274M16—1221no—
275M16—1221no—
276M16—2444no—
277M16—2441no—
278M16—2441no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—81no—
285C2×C8—1221no—
286C2×C8—1222no—
287C2×C8—1221no—
288C2×C8—2444no—
289C2×C8—2441no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—612no—
295C4⋊C4—1221no—
296C22⋊C4—1222no—
297C22⋊C4—1222no—
298C42#1175822211yescharacteristic
299C42D4×D6111yescharacteristic
300C42D4×D6111yescharacteristic
301C42C2×C22⋊D6111yescharacteristic
302C23—1222no—
303C23#1451973711yescharacteristic
304Q8—44no—
305Q8—82no—
306Q8—2441no—
307Q8—2441no—
308D4—2442no—
309D4—2442no—
310D4—2442no—
311D4—2441no—
312D4—2441no—
313D4—2441no—
314D4—2441no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—1222no—
321C2×C4—2442no—
322C2×C4—2442no—
323C2×C4C22⋊C4×D6111yescharacteristic
324C2×C4C4×C22⋊D6111yescharacteristic
325C2×C4#1451974511yescharacteristic
326C2×C4#1451974811yescharacteristic
327C2×C4#1451975011yescharacteristic
328C2×C4#1451975911yescharacteristic
329C2×C4SD16×D6111yescharacteristic
330C2×C4#1451980011yescharacteristic
331C2×C4#1451980411yescharacteristic
332C2×C4#1451980911yescharacteristic
333C8—41no—
334C8—41no—
335C8—81no—
336C8—1222no—
337C8—4881no—
338C8—2442no—
339C8—2441no—
340C22—21no—
341C22—21no—
342C22—2442no—
343C22—2442no—
344C22—2442no—
345C22#1683963811yescharacteristic
346C22#1683984711yescharacteristic
347C22#1683986511yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—82no—
352C4—2442no—
353C4#1683991111yescharacteristic
354C4#1683992311yescharacteristic
355C4#1683996611yescharacteristic
356C4#1683998011yescharacteristic
357C4#1684002211yescharacteristic
358C4#1684003411yescharacteristic
359C2—21no—
360C2—21no—
361C2—4882no—
362C2#2144615011yescharacteristic
363C2#2144633711yescharacteristic
364C2#2144636311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#333C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenterC2×#333D61
hypocenterC61#9267
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
3. #21446809C2semidirect
5. #21443716C2semidirect
6. #21443669C2semidirect
7. C2×C61×#333C2semidirect
112. #11758194SD16semidirect
181. C2×#333D61semidirect
212. C61#9267semidirect

Maximal subgroups

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

subgroupindex
#214567442
#214545622
#214468092
#214444942
#214437162
#214436692
C2×C61×#3332
#926761