Finite Group Databasebeta

← #29574174

Subgroups — #29574174

Order 15616.

618 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448719C211yesmaximal, characteristic
2#21446578C211yesmaximal, characteristic, complemented
3#21445978C211yesmaximal, characteristic, complemented
4#21445820C211yesmaximal, characteristic
5C2×#16839489C211yesmaximal, characteristic
6#21443658C211yesmaximal, characteristic, complemented
7C61×#1296C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9C2×#14519739—21no—
10C2×C4⋊C4×D61—21no—
11C2×#14519711—21no—
12C2×#14519710—22no—
13C2×#14519699—21no—
14C2×C4×#92418C2211yescharacteristic
15#16839658—21no—
16#16839633—22no—
17#16839632C2211yescharacteristic
18#16839619—21no—
19C4×C4⋊Dic61C2211yescharacteristic
20#16839525—21no—
21#16839525—21no—
22#16839489C414yescomplemented
23C2×C4⋊C8×C61C2211yescharacteristic
24C2×C4⋊C8×C61C2211yescharacteristic
25C61×#384C2211yescharacteristic
26C22×D244D411yescharacteristic
27C22×C4×D61—21no—
28C22×C4×D61—22no—
29C22×#92418—21no—
30C4×D244—21no—
31C4×D244—21no—
32C4×D244C2×C412yes—
33C4×D244Q812yes—
34#14519739—42no—
35#14519739—42no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61D411yescharacteristic
40C2×C4⋊Dic61D411yescharacteristic
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758196—22no—
43C4⋊C4×D61—42no—
44C4⋊C4×D61—42no—
45C2×#11758198—21no—
46C2×#11758198—22no—
47C2×C4×Dic61—21no—
48#14519711—42no—
49#14519710—44no—
50#14519699—42no—
51C4×#92418—21no—
52C4×#92418C2×C412yescomplemented
53#14519684—22no—
54#14519684—21no—
55C22×C8×C61—21no—
56C22×C8×C61—21no—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61—21no—
59C2×C4⋊C4×C61D412yes—
60C2×C42×C61C2311yescharacteristic
61C4⋊C8×C61—21no—
62C4⋊C8×C61—21no—
63C4⋊C8×C61—21no—
64C4⋊C8×C61C2×C412yescomplemented
65C23×D61—21no—
66C22×C4×C61—22no—
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—41no—
74C2×D244—21no—
75C2×D244—21no—
76C2×D244—21no—
77C2×D244C4⋊C412yes—
78C2×D244D812yes—
79C2×D244Q1612yes—
80C2×C4×D61—22no—
81C2×C4×D61—22no—
82C2×C4×D61—41no—
83C2×C4×D61—41no—
84C2×C4×D61—41no—
85C2×C4×D61—41no—
86C2×C4×D61—44no—
87C2×C4×D61—44no—
88C2×C4×D61—82no—
89C2×C8×C61—22no—
90C2×C8×C61—41no—
91C2×C8×C61—41no—
92C2×C8×C61—41no—
93C2×C8×C61—41no—
94C2×C8×C61—41no—
95C4⋊C4×C61—42no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—41no—
99C4⋊C4×C61—41no—
100C42×C61C22×C411yescharacteristic
101C42×C61C2×Q811yescharacteristic
102C42×C61#4111yescharacteristic
103C42×C61#4111yescharacteristic
104#11758198—22no—
105#11758198—22no—
106#11758198—44no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110C4⋊Dic61—21no—
111C4⋊Dic61—42no—
112C4⋊Dic61—41no—
113C4⋊Dic61—41no—
114C4⋊Dic61—41no—
115C4⋊Dic61—41no—
116C4⋊Dic61C4⋊C412yes—
117C4⋊Dic61D812yes—
118#11758196—44no—
119C4×Dic61—41no—
120C4×Dic61—41no—
121C2×#92418—22no—
122C2×#92418—22no—
123C2×#92418—22no—
124C23×C61#12011yescharacteristic
125C22×D61—22no—
126C22×D61—22no—
127C22×D61—22no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61C2×C4⋊C411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61C22⋊Q811yescharacteristic
141C2×C4×C61C22⋊Q811yescharacteristic
142C2×C4×C61#12311yescharacteristic
143C2×C4×C61C2×D811yescharacteristic
144C2×C4×C61C2×D811yescharacteristic
145C2×C4×C61C2×Q1611yescharacteristic
146C2×C4×C61#13511yescharacteristic
147C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C61#13711yescharacteristic
150C2×C4×C61#13711yescharacteristic
151C2×Dic61—22no—
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—81no—
161C2×Dic61—81no—
162C2×Dic61—81no—
163D244—22no—
164D244—22no—
165D244—22no—
166D244#10714yescomplemented
167C4×D61—42no—
168C4×D61—42no—
169C4×D61—42no—
170C4×D61—42no—
171C4×D61—88no—
172C8×C61—42no—
173C8×C61—81no—
174#92418—24no—
175#8882—611nomaximal, Sylow
176C22×C61#38511yescharacteristic
177C22×C61#44711yescharacteristic
178C22×C61#44711yescharacteristic
179C22×C61#45111yescharacteristic
180C22×C61#45211yescharacteristic
181C22×C61#45311yescharacteristic
182C22×C61#45611yescharacteristic
183C2×D61—24no—
184C2×D61—42no—
185C2×D61—42no—
186C2×D61—42no—
187C2×D61—42no—
188C2×D61—42no—
189C2×D61—42no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—82no—
193C4×C61C2×#10711yescharacteristic
194C4×C61#42811yescharacteristic
195C4×C61C4×D811yescharacteristic
196C4×C61#44011yescharacteristic
197C4×C61#47411yescharacteristic
198C4×C61#47811yescharacteristic
199C4×C61#48011yescharacteristic
200C4×C61#48411yescharacteristic
201Dic61—42no—
202Dic61—42no—
203Dic61—81no—
204Dic61—161no—
205D4×C4⋊C4—611no—
206#1947—611no—
207#1845—611no—
208#1802—611no—
209C2×D4⋊C8—611no—
210#1296D6111yescharacteristic, complemented
211#1279—611no—
212C2×C61#161911yescharacteristic
213C2×C61#163211yescharacteristic
214C2×C61#164711yescharacteristic
215C2×C61#164911yescharacteristic
216C2×C61#185211yescharacteristic
217C2×C61#185911yescharacteristic
218C2×C61#186211yescharacteristic
219D61—44no—
220C2×C4×D4—611no—
221C2×C4×D4—1221no—
222C22×C4⋊C4—1221no—
223C2×#107—1221no—
224C2×#107—1222no—
225C2×C4⋊C8C2×D6111yescharacteristic
226C2×C4⋊C8C2×D6111yescharacteristic
227C2×#102—1221no—
228C2×C4×C8—611no—
229#389—1221no—
230#385—1222no—
231#384—611no—
232#384C2×D6111yescharacteristic
233#380—1221no—
234C4×C4⋊C4—611no—
235#340—1221no—
236#340—1221no—
237D4⋊C8—614no—
238C61#888211yescharacteristic, complemented, Sylow
239C22×D4—611no—
240C23×C4—1222no—
241C23×C4—1221no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—1221no—
245C4×D4—612no—
246C4×D4—612no—
247C4×D4—1221no—
248C4×D4—1221no—
249C4×D4—2442no—
250C4×D4—2442no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—611no—
255C2×C4⋊C4—611no—
256C2×C4⋊C4—1222no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—2442no—
261C2×C4⋊C4—2442no—
262C2×C4⋊C4D24412yes—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1222no—
265C2×C42—1221no—
266C2×C42C22×D6111yescharacteristic
267#107—2444no—
268#107—2442no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8C4×D6112yescomplemented
273#102—2442no—
274C4×C8—612no—
275C4×C8—1221no—
276#95—1222no—
277#95—1221no—
278C24—1221no—
279C2×D4—612no—
280C2×D4—612no—
281C2×D4—612no—
282C2×D4—1221no—
283C2×D4—1221no—
284C2×D4—1221no—
285C22×C4—22no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—1222no—
289C22×C4—1222no—
290C22×C4—1221no—
291C22×C4—4882no—
292C22×C4—2444no—
293C22×C4—2444no—
294C22×C4—2441no—
295C22×C4—2441no—
296C22×C4—2441no—
297C22×C4—2441no—
298C22×C4—2441no—
299C22×C4C2×D24411yescharacteristic
300C22×C4D4×D6111yescharacteristic
301C22×C4D4×D6111yescharacteristic
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—1222no—
309C2×C8—1222no—
310C2×C8—1222no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—612no—
317C4⋊C4—612no—
318C4⋊C4—1221no—
319C4⋊C4—1221no—
320C4⋊C4—1221no—
321C4⋊C4—1221no—
322C4⋊C4—2442no—
323C4⋊C4—2444no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C22⋊C4—1222no—
329C22⋊C4—1222no—
330C22⋊C4—2444no—
331C42—2441no—
332C42—2441no—
333C42C2×C4×D6111yescharacteristic
334C42#1175822211yescharacteristic
335C42#1175822411yescharacteristic
336C42Q8×D6111yescharacteristic
337C23—1222no—
338C23—1222no—
339C23—1222no—
340C23—2441no—
341C23—2441no—
342C23—2441no—
343C23—2441no—
344C23#1451974711yescharacteristic
345D4—614no—
346D4—1222no—
347D4—1222no—
348D4—1222no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—1222no—
355C2×C4—1222no—
356C2×C4—4888no—
357C2×C4—4881no—
358C2×C4—4881no—
359C2×C4—4881no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4—2442no—
368C2×C4—2442no—
369C2×C4—2442no—
370C2×C4—2442no—
371C2×C4C4⋊C4×D6111yescharacteristic
372C2×C4#1451973711yescharacteristic
373C2×C4C4×D24411yescharacteristic
374C2×C4#1451975511yescharacteristic
375C2×C4#1451975711yescharacteristic
376C2×C4#1451975911yescharacteristic
377C2×C4D8×D6111yescharacteristic
378C2×C4D8×D6111yescharacteristic
379C2×C4Q16×D6111yescharacteristic
380C2×C4#1451979811yescharacteristic
381C2×C4#1451980411yescharacteristic
382C2×C4#1451980411yescharacteristic
383C2×C4#1451980611yescharacteristic
384C2×C4#1451980911yescharacteristic
385C8—42no—
386C8—81no—
387C8—1224no—
388C22—1224no—
389C22—2442no—
390C22—2442no—
391C22—2442no—
392C22—2442no—
393C22—2442no—
394C22—2442no—
395C22#1683963611yescharacteristic
396C22#1683989711yescharacteristic
397C22#1683989711yescharacteristic
398C22#1683991911yescharacteristic
399C22#1683992711yescharacteristic
400C22#1683993211yescharacteristic
401C22#1683994711yescharacteristic
402C4—21no—
403C4—21no—
404C4—82no—
405C4—9761no—
406C4—4881no—
407C4—2442no—
408C4—2442no—
409C4#107×D6111yescharacteristic
410C4#1683980111yescharacteristic
411C4#1683984211yescharacteristic
412C4#1683986511yescharacteristic
413C4#1684004111yescharacteristic
414C4#1684006111yescharacteristic
415C4#1684007411yescharacteristic
416C4#1684009611yescharacteristic
417C2—2444no—
418C2#2144487411yescharacteristic
419C2#2144495411yescharacteristic
420C2#2144505511yescharacteristic
421C2#2144506911yescharacteristic
422C2#2144600811yescharacteristic
423C2#2144604811yescharacteristic
424C2#2144606711yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1296C2
socleC23×C61#120
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1296D61
hypocenterC61#8882
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #21446578C2semidirect
3. #21445978C2semidirect
6. #21443658C2semidirect
7. C61×#1296C2semidirect
22. #16839489C4semidirect
52. C4×#92418C2×C4semidirect
64. C4⋊C8×C61C2×C4semidirect
166. D244#107semidirect
210. #1296D61semidirect
238. C61#8882semidirect
272. C4⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214487192
#214465782
#214459782
#214458202
C2×#168394892
#214436582
C61×#12962
#888261