Finite Group Databasebeta

← #30600245

Subgroups — #30600245

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×#92420C211yesmaximal, characteristic, complemented
2#21461114C211yesmaximal, characteristic, complemented
3C2×D4⋊#92420C211yesmaximal, characteristic, complemented
4#21461059C211yesmaximal, characteristic, complemented
5#21460665C211yesmaximal, characteristic, complemented
6#21460652C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×Dic61C211yesmaximal, characteristic
8Q8×#92420—21no—
9D4×#92420—21no—
10D4×#92420—22no—
11#16840778—21no—
12#16840778C2211yescharacteristic
13#16840778C2211yescharacteristic
14C2×C4⋊#92420C2211yescharacteristic
15C2×C4×#92420—21no—
16C2×C4×#92420C2211yescharacteristic
17Q8⋊#92420—21no—
18D4⋊#92420—21no—
19D4⋊#92420—21no—
20D4⋊#92420—21no—
21#16840605—22no—
22#41×Dic61C2211yescharacteristic
23C2×D4×Dic61C2211yescharacteristic
24#16839874—21no—
25#16839863—21no—
26SD16×Dic61—21no—
27D8×Dic61—21no—
28#16839804C411yescharacteristic
29#16839767C411yescharacteristic
30#16839763C411yescharacteristic
31M16×Dic61C2211yescharacteristic
32C2×AGL(1,ℤ/8)×C61C411yescharacteristic
33C22×#92420—22no—
34C22×#92420—21no—
35C22×#92420—21no—
36C4⋊#92420—22no—
37C4⋊#92420—21no—
38C4⋊#92420—21no—
39C4⋊#92420—21no—
40C4⋊#92420—21no—
41C22⋊#92420—22no—
42C22⋊#92420—21no—
43C22⋊#92420D412yes—
44C4×#92420—22no—
45C4×#92420—21no—
46C4×#92420—21no—
47C4×#92420—21no—
48C4×#92420D411yescharacteristic
49C4×#92420D411yescharacteristic
50C23×Dic61—21no—
51C2×#11758194C2×C411yescharacteristic
52Q8×Dic61D411yescharacteristic
53Q8×Dic61D411yescharacteristic
54D4×Dic61—21no—
55D4×Dic61—21no—
56D4×Dic61—21no—
57D4×Dic61—21no—
58D4×Dic61—21no—
59D4×Dic61—21no—
60D4×Dic61D411yescharacteristic
61D4×Dic61D411yescharacteristic
62#14519736—21no—
63#14519736C2×C411yescharacteristic
64C2×C4⋊Dic61C2×C411yescharacteristic
65C2×C22⋊Dic61—21no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61C2311yescharacteristic
68#14519711—21no—
69#14519709—21no—
70#14519701—21no—
71#14519698—21no—
72#14519698—21no—
73#14519698—21no—
74#14519687—21no—
75C8×Dic61—21no—
76C2×#41×C61C2×C411yescharacteristic
77C22×D4×C61C2×C411yescharacteristic
78AGL(1,ℤ/8)×C61—22no—
79AGL(1,ℤ/8)×C61C814yescomplemented
80C2×SD16×C61—21no—
81C2×D8×C61—21no—
82C2×M16×C61C2×C411yescharacteristic
83C24×C61—21no—
84#41×C61—21no—
85#41×C61—21no—
86#41×C61C2×C812yes—
87#41×C61M1612yes—
88C2×Q8×C61C22⋊C411yescharacteristic
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61C22⋊C411yescharacteristic
96C2×D4×C61C2×C812yescomplemented
97C22×C4×C61—21no—
98C22×C4×C61C22×C411yescharacteristic
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—41no—
103C22×Dic61—41no—
104C22×Dic61—81no—
105C22×Dic61—81no—
106C22×Dic61C2×D411yescharacteristic
107C22×Dic61#4111yescharacteristic
108C2×#92420—21no—
109C2×#92420—22no—
110C2×#92420—21no—
111C2×#92420—42no—
112C2×#92420—44no—
113C2×#92420—41no—
114C2×#92420—41no—
115C2×#92420—81no—
116SD16×C61—24no—
117D8×C61—24no—
118M16×C61—21no—
119M16×C61C2×C812yescomplemented
120C2×C8×C61—21no—
121C22⋊Dic61—21no—
122C22⋊Dic61—21no—
123C22⋊Dic61—21no—
124C22⋊Dic61—41no—
125C22⋊Dic61—41no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61C22⋊C411yescharacteristic
131C4⋊Dic61C22⋊C411yescharacteristic
132C4×Dic61—21no—
133C4×Dic61—21no—
134C4×Dic61—21no—
135C4×Dic61C2×D411yescharacteristic
136C4×Dic61C2×D411yescharacteristic
137C4×Dic61C2×D411yescharacteristic
138C4×Dic61#4111yescharacteristic
139#11758194—21no—
140#11758194—21no—
141C2×#92418—21no—
142C23×C61—21no—
143C23×C61—22no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—81no—
147C23×C61—81no—
148C23×C61C4×D411yescharacteristic
149Q8×C61—21no—
150Q8×C61C22⋊C812yes—
151D4×C61—22no—
152D4×C61—22no—
153D4×C61—22no—
154D4×C61—21no—
155D4×C61—22no—
156D4×C61—41no—
157D4×C61—41no—
158D4×C61C22⋊C812yes—
159C2×C4×C61—21no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61C2×C22⋊C411yescharacteristic
163C2×C4×C61C4×D411yescharacteristic
164C2×C4×C61C22×C811yescharacteristic
165C2×C4×C61C2×M1611yescharacteristic
166C2×C4×C61#13111yescharacteristic
167C2×C4×C61#13111yescharacteristic
168C2×Dic61—21no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—41no—
173C2×Dic61—42no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—82no—
178C2×Dic61—81no—
179C2×Dic61—81no—
180C2×Dic61—81no—
181C2×Dic61#12011yescharacteristic
182C2×Dic61#12111yescharacteristic
183C2×Dic61#12111yescharacteristic
184C2×Dic61#12311yescharacteristic
185#92420—22no—
186#92420—42no—
187#92420—41no—
188#92420—81no—
189#92420—81no—
190C8×C61—22no—
191#92418—41no—
192#9095—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—22no—
196C22×C61—41no—
197C22×C61—41no—
198C22×C61—42no—
199C22×C61—82no—
200C22×C61—82no—
201C22×C61—81no—
202C22×C61—81no—
203C22×C61#38611yescharacteristic
204C22×C61C8×D411yescharacteristic
205C22×C61#43511yescharacteristic
206C4×C61—22no—
207C4×C61C2×C22⋊C811yescharacteristic
208C4×C61#40811yescharacteristic
209C4×C61C8×D411yescharacteristic
210C4×C61#43511yescharacteristic
211Dic61—21no—
212Dic61—21no—
213Dic61—21no—
214Dic61—41no—
215Dic61—41no—
216Dic61—81no—
217Dic61#47111yescharacteristic
218Dic61#47211yescharacteristic
219C8×#41—611no—
220C4×AGL(1,ℤ/8)—611no—
221#1477—611no—
222C2×D4⋊C8—611no—
223#1460—611no—
224#1283—611no—
225#1278—611no—
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—42no—
229C2×C61—82no—
230C2×C61#186911yescharacteristic
231C2×C61#190611yescharacteristic
232C2×C61#190711yescharacteristic
233C2×AGL(1,ℤ/8)#692311yescharacteristic
234C4×#41—611no—
235C2×C4×D4—611no—
236C8×Q8—1221no—
237#442—1221no—
238#440—1221no—
239C4×SD16—1221no—
240C4×D8—1221no—
241C8×D4—1222no—
242C8×D4—1221no—
243#431—611no—
244#431—611no—
245#431—1221no—
246#428—611no—
247C2×C4⋊C8—611no—
248#418—611no—
249#417—611no—
250C4×M16—611no—
251C2×C4×C8—611no—
252C2×C4×C8—1221no—
253Q8⋊C8—1221no—
254D4⋊C8—1221no—
255D4⋊C8—1221no—
256D4⋊C8—1221no—
257#323—1222no—
258C61#909511yescharacteristic, complemented, Sylow
259C2×#41C2×#692311yescharacteristic
260C22×D4C2×#692311yescharacteristic
261C23×C4—1221no—
262AGL(1,ℤ/8)—22no—
263AGL(1,ℤ/8)#9242014yescomplemented
264C2×SD16—21no—
265C2×D8—21no—
266C2×M16—611no—
267C2×M16C2×#692311yescharacteristic
268C22×C8—1221no—
269C22×C8—1222no—
270C22×C8—1221no—
271C4×Q8—611no—
272C4×Q8—611no—
273C4×D4—611no—
274C4×D4—611no—
275C4×D4—1221no—
276C4×D4—1221no—
277C4×D4—1221no—
278C4×D4—1221no—
279C4×D4—1221no—
280C4×D4—1221no—
281#117—611no—
282#117—1221no—
283C2×C4⋊C4—611no—
284C2×C22⋊C4—1221no—
285C2×C42—611no—
286C2×C42—1221no—
287#107—1221no—
288#106—1221no—
289C4⋊C8—1221no—
290C4⋊C8—1221no—
291C4⋊C8—1221no—
292C4⋊C8—1221no—
293C4⋊C8—1222no—
294#103—1221no—
295#102—1221no—
296#102—1221no—
297#102—1221no—
298C22⋊C8—612no—
299C22⋊C8—1222no—
300C22⋊C8—1221no—
301#97—1221no—
302C4×C8—611no—
303C4×C8—611no—
304C4×C8—1221no—
305C4×C8—1221no—
306C4×C8—1221no—
307C4×C8—1221no—
308C4×C8—1222no—
309C24—21no—
310#41—21no—
311#41—21no—
312#41C2×#9242012yes—
313#41#1175821712yes—
314C2×Q8C22⋊#692311yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4C2×#9242012yescomplemented
322C2×D4C22⋊#692311yescharacteristic
323C22×C4—21no—
324C22×C4—611no—
325C22×C4—611no—
326C22×C4—1221no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—4881no—
330C22×C4—4881no—
331C22×C4—2441no—
332C22×C4—2441no—
333C22×C4C22×#692311yescharacteristic
334SD16—24no—
335D8—24no—
336M16—21no—
337M16—1221no—
338M16—1221no—
339M16C2×#9242012yescomplemented
340C2×C8—21no—
341C2×C8—1221no—
342C2×C8—1221no—
343C2×C8—1222no—
344C2×C8—1221no—
345C2×C8—4881no—
346C2×C8—2442no—
347C2×C8—2444no—
348C2×C8—2441no—
349C2×C8—2441no—
350C4⋊C4—611no—
351C4⋊C4—611no—
352C4⋊C4—1221no—
353C4⋊C4—1221no—
354C4⋊C4—1221no—
355C4⋊C4—1221no—
356C22⋊C4—1221no—
357C22⋊C4—1221no—
358C22⋊C4—1221no—
359C22⋊C4—2441no—
360C22⋊C4—2441no—
361C42—611no—
362C42—611no—
363C42—611no—
364C42—611no—
365C42—1221no—
366C42—1221no—
367C42—1221no—
368C23—21no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—81no—
373C23—81no—
374C23D4×#692311yescharacteristic
375Q8—21no—
376Q8C22⋊#9242012yes—
377D4—22no—
378D4—22no—
379D4—22no—
380D4—21no—
381D4—22no—
382D4—41no—
383D4—41no—
384D4C22⋊#9242012yes—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—611no—
389C2×C4—611no—
390C2×C4—611no—
391C2×C4—611no—
392C2×C4—1221no—
393C2×C4—1221no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—4881no—
397C2×C4—4881no—
398C2×C4—4882no—
399C2×C4—4881no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2442no—
404C2×C4—2441no—
405C2×C4C2×C22⋊#692311yescharacteristic
406C2×C4D4×#692311yescharacteristic
407C2×C4C22×#9242011yescharacteristic
408C2×C4C2×#1175821711yescharacteristic
409C2×C4#1451986411yescharacteristic
410C2×C4#1451986411yescharacteristic
411C8—22no—
412C8—1222no—
413C8—4881no—
414C8—4881no—
415C8—2441no—
416C8—2442no—
417C8—2441no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—42no—
424C22—82no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22#1684071011yescharacteristic
429C22D4×#9242011yescharacteristic
430C22#1684079311yescharacteristic
431C4—22no—
432C4—611no—
433C4—611no—
434C4—1221no—
435C4—1221no—
436C4—1221no—
437C4—4881no—
438C4—2441no—
439C4—2441no—
440C4C2×C22⋊#9242011yescharacteristic
441C4#1684073511yescharacteristic
442C4D4×#9242011yescharacteristic
443C4#1684079311yescharacteristic
444C2—21no—
445C2—21no—
446C2—42no—
447C2—82no—
448C2#2146179211yescharacteristic
449C2#2146185611yescharacteristic
450C2#2146185811yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
Fitting, prime radicalC2×AGL(1,ℤ/8)×C61C4
socleC22×C61#386
cosocleC2×C4×Dic61C23
hypercenterC2×AGL(1,ℤ/8)#6923
hypocenterC61#9095
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×#92420C2semidirect
2. #21461114C2semidirect
3. C2×D4⋊#92420C2semidirect
4. #21461059C2semidirect
5. #21460665C2semidirect
6. #21460652C2semidirect
79. AGL(1,ℤ/8)×C61C8semidirect
96. C2×D4×C61C2×C8semidirect
119. M16×C61C2×C8semidirect
258. C61#9095semidirect
263. AGL(1,ℤ/8)#92420semidirect
321. C2×D4C2×#92420semidirect
339. M16C2×#92420semidirect

Maximal subgroups

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

subgroupindex
#41×#924202
#214611142
C2×D4⋊#924202
#214610592
#214606652
#214606522
AGL(1,ℤ/8)×Dic612
#909561