Finite Group Databasebeta

← #29575482

Subgroups — #29575482

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×#92418C211yesmaximal, characteristic, complemented
2#21444206C211yesmaximal, characteristic, complemented
3C2×D4⋊#92418C211yesmaximal, characteristic, complemented
4#21444140C211yesmaximal, characteristic, complemented
5#21443670C211yesmaximal, characteristic, complemented
6#21443657C211yesmaximal, characteristic, complemented
7C4×AGL(1,ℤ/8)×C61C211yesmaximal, characteristic
8Q8×#92418—21no—
9D4×#92418—22no—
10D4×#92418—21no—
11#16839815—21no—
12#16839815C2211yescharacteristic
13#16839815C2211yescharacteristic
14C2×C4⋊#92418C2211yescharacteristic
15C2×C4×#92418—21no—
16C2×C4×#92418C2211yescharacteristic
17Q8⋊#92418—21no—
18D4⋊#92418—21no—
19D4⋊#92418—21no—
20D4⋊#92418—21no—
21#16839486—22no—
22C2×AGL(1,ℤ/8)×C61C411yescharacteristic
23C4×#41×C61C2211yescharacteristic
24C2×C4×D4×C61C2211yescharacteristic
25C61×#442—21no—
26C61×#440—21no—
27C4×SD16×C61—21no—
28C4×D8×C61—21no—
29C61×#428C411yescharacteristic
30C61×#418C411yescharacteristic
31C61×#417C411yescharacteristic
32C4×M16×C61C2211yescharacteristic
33C22×#92418—22no—
34C22×#92418—21no—
35C22×#92418—21no—
36C4⋊#92418—22no—
37C4⋊#92418—21no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40C4⋊#92418—21no—
41C22⋊#92418—22no—
42C22⋊#92418—21no—
43C22⋊#92418D412yes—
44C4×#92418—22no—
45C4×#92418—21no—
46C4×#92418—21no—
47C4×#92418—21no—
48C4×#92418D411yescharacteristic
49C4×#92418D411yescharacteristic
50C2×#41×C61C2×C411yescharacteristic
51C22×D4×C61C2×C411yescharacteristic
52C23×C4×C61—21no—
53AGL(1,ℤ/8)×C61—22no—
54AGL(1,ℤ/8)×C61C814yescomplemented
55C2×SD16×C61—21no—
56C2×D8×C61—21no—
57C2×M16×C61C2×C411yescharacteristic
58C2×M16×C61C2×C411yescharacteristic
59C4×Q8×C61D411yescharacteristic
60C4×Q8×C61D411yescharacteristic
61C4×D4×C61—21no—
62C4×D4×C61—21no—
63C4×D4×C61—21no—
64C4×D4×C61—21no—
65C4×D4×C61—21no—
66C4×D4×C61—21no—
67C4×D4×C61D411yescharacteristic
68C4×D4×C61D411yescharacteristic
69#117×C61—21no—
70#117×C61C2×C411yescharacteristic
71C2×C4⋊C4×C61C2×C411yescharacteristic
72C2×C22⋊C4×C61—21no—
73C2×C42×C61—21no—
74C2×C42×C61C2311yescharacteristic
75#107×C61—21no—
76#106×C61—21no—
77#103×C61—21no—
78#102×C61—21no—
79#102×C61—21no—
80#102×C61—21no—
81#97×C61—21no—
82C4×C8×C61—21no—
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×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61—81no—
104C22×C4×C61—81no—
105C22×C4×C61C22×C411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61#4111yescharacteristic
108SD16×C61—24no—
109D8×C61—24no—
110M16×C61—21no—
111M16×C61—21no—
112M16×C61—21no—
113M16×C61C2×C812yescomplemented
114C2×C8×C61—21no—
115C2×C8×C61—21no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61C22⋊C411yescharacteristic
121C4⋊C4×C61C22⋊C411yescharacteristic
122C22⋊C4×C61—21no—
123C22⋊C4×C61—21no—
124C22⋊C4×C61—21no—
125C22⋊C4×C61—41no—
126C22⋊C4×C61—41no—
127C42×C61—21no—
128C42×C61—21no—
129C42×C61—21no—
130C42×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61C2×D411yescharacteristic
133C42×C61#4111yescharacteristic
134C2×#92418—21no—
135C2×#92418—22no—
136C2×#92418—21no—
137C2×#92418—42no—
138C2×#92418—44no—
139C2×#92418—41no—
140C2×#92418—41no—
141C2×#92418—81no—
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×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—41no—
167C2×C4×C61—42no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—82no—
172C2×C4×C61—81no—
173C2×C4×C61—81no—
174C2×C4×C61—81no—
175C2×C4×C61C2×C22⋊C411yescharacteristic
176C2×C4×C61C4×D411yescharacteristic
177C2×C4×C61#12011yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61#12311yescharacteristic
181C2×C4×C61C22×C811yescharacteristic
182C2×C4×C61C2×M1611yescharacteristic
183C2×C4×C61#13111yescharacteristic
184C2×C4×C61#13111yescharacteristic
185C8×C61—22no—
186C8×C61—41no—
187#92418—22no—
188#92418—42no—
189#92418—41no—
190#92418—81no—
191#92418—81no—
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×C61—21no—
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—41no—
211C4×C61—41no—
212C4×C61—81no—
213C4×C61C2×C22⋊C811yescharacteristic
214C4×C61#40811yescharacteristic
215C4×C61C8×D411yescharacteristic
216C4×C61#43511yescharacteristic
217C4×C61#47111yescharacteristic
218C4×C61#47211yescharacteristic
219C8×#41—611no—
220C4×AGL(1,ℤ/8)D6111yescharacteristic
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)Dic6111yescharacteristic
234C4×#41C2×D6111yescharacteristic
235C2×C4×D4C2×D6111yescharacteristic
236C8×Q8—1221no—
237#442—21no—
238#440—21no—
239C4×SD16—21no—
240C4×D8—21no—
241C8×D4—1222no—
242C8×D4—1221no—
243#431—611no—
244#431—611no—
245#431—1221no—
246#428Dic6111yescharacteristic
247C2×C4⋊C8—611no—
248#418Dic6111yescharacteristic
249#417Dic6111yescharacteristic
250C4×M16C2×D6111yescharacteristic
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×Dic6111yescharacteristic
260C22×D4C2×Dic6111yescharacteristic
261C23×C4—21no—
262AGL(1,ℤ/8)—22no—
263AGL(1,ℤ/8)#9241814yescomplemented
264C2×SD16—21no—
265C2×D8—21no—
266C2×M16C2×Dic6111yescharacteristic
267C2×M16C2×Dic6111yescharacteristic
268C22×C8—1222no—
269C22×C8—1221no—
270C22×C8—1221no—
271C4×Q8C22⋊D6111yescharacteristic
272C4×Q8C22⋊D6111yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4C22⋊D6111yescharacteristic
280C4×D4C22⋊D6111yescharacteristic
281#117—21no—
282#117C2×Dic6111yescharacteristic
283C2×C4⋊C4C2×Dic6111yescharacteristic
284C2×C22⋊C4—21no—
285C2×C42—21no—
286C2×C42C22×D6111yescharacteristic
287#107—21no—
288#106—21no—
289C4⋊C8—1222no—
290C4⋊C8—1221no—
291C4⋊C8—1221no—
292C4⋊C8—1221no—
293C4⋊C8—1221no—
294#103—21no—
295#102—21no—
296#102—21no—
297#102—21no—
298C22⋊C8—612no—
299C22⋊C8—1222no—
300C22⋊C8—1221no—
301#97—21no—
302C4×C8—21no—
303C4×C8—611no—
304C4×C8—611no—
305C4×C8—1222no—
306C4×C8—1221no—
307C4×C8—1221no—
308C4×C8—1221no—
309C24—21no—
310#41—21no—
311#41—21no—
312#41C2×#9241812yes—
313#41#1175819412yes—
314C2×Q8C22⋊Dic6111yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4C2×#9241812yescomplemented
322C2×D4C22⋊Dic6111yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4D4×D6111yescharacteristic
332C22×C4#1175822411yescharacteristic
333C22×C4C22×Dic6111yescharacteristic
334SD16—24no—
335D8—24no—
336M16—21no—
337M16—21no—
338M16—21no—
339M16C2×#9241812yescomplemented
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—1221no—
343C2×C8—1221no—
344C2×C8—1222no—
345C2×C8—4881no—
346C2×C8—2442no—
347C2×C8—2444no—
348C2×C8—2441no—
349C2×C8—2441no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4C22⋊Dic6111yescharacteristic
355C4⋊C4C22⋊Dic6111yescharacteristic
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42D4×D6111yescharacteristic
365C42#1175822411yescharacteristic
366C42C2×C22⋊D6111yescharacteristic
367C42C2×C22⋊D6111yescharacteristic
368C23—21no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—81no—
373C23—81no—
374C23D4×Dic6111yescharacteristic
375Q8—21no—
376Q8C22⋊#9241812yes—
377D4—22no—
378D4—22no—
379D4—22no—
380D4—21no—
381D4—22no—
382D4—41no—
383D4—41no—
384D4C22⋊#9241812yes—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4C2×C22⋊Dic6111yescharacteristic
402C2×C4D4×Dic6111yescharacteristic
403C2×C4#1451974511yescharacteristic
404C2×C4#1451974911yescharacteristic
405C2×C4#1451974911yescharacteristic
406C2×C4#1451975811yescharacteristic
407C2×C4C22×#9241811yescharacteristic
408C2×C4C2×#1175819411yescharacteristic
409C2×C4#1451978411yescharacteristic
410C2×C4#1451978411yescharacteristic
411C8—22no—
412C8—41no—
413C8—1222no—
414C8—4881no—
415C8—4881no—
416C8—2441no—
417C8—2442no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—42no—
424C22—82no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22#1683964211yescharacteristic
429C22D4×#9241811yescharacteristic
430C22#1683983411yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—22no—
435C4—41no—
436C4—41no—
437C4—81no—
438C4C2×C22⋊#9241811yescharacteristic
439C4#1683972911yescharacteristic
440C4D4×#9241811yescharacteristic
441C4#1683983411yescharacteristic
442C4#1684002311yescharacteristic
443C4#1684003011yescharacteristic
444C2—21no—
445C2—21no—
446C2—42no—
447C2—82no—
448C2#2144611211yescharacteristic
449C2#2144632211yescharacteristic
450C2#2144632911yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutatorC2×C4×C61C22×C8
FrattiniC2×C42C22×D61
FittingC4×AGL(1,ℤ/8)×C61C2
socleC22×C61#386
prime radicalC2×AGL(1,ℤ/8)×C61C4
cosocleC2×C42×C61C23
hypercenterC4×AGL(1,ℤ/8)D61
hypocenterC61#9095
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×#92418C2semidirect
2. #21444206C2semidirect
3. C2×D4⋊#92418C2semidirect
4. #21444140C2semidirect
5. #21443670C2semidirect
6. #21443657C2semidirect
54. AGL(1,ℤ/8)×C61C8semidirect
96. C2×D4×C61C2×C8semidirect
113. M16×C61C2×C8semidirect
258. C61#9095semidirect
263. AGL(1,ℤ/8)#92418semidirect
321. C2×D4C2×#92418semidirect
339. M16C2×#92418semidirect

Maximal subgroups

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

subgroupindex
#41×#924182
#214442062
C2×D4⋊#924182
#214441402
#214436702
#214436572
C4×AGL(1,ℤ/8)×C612
#909561