Finite Group Databasebeta

← #49134685

Subgroups — #49134685

Order 36224.

1012 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827093C211yesmaximal, characteristic, complemented
2#31827088C212yesmaximal, complemented
3#31827084C212yesmaximal, complemented
4#31827076C212yesmaximal, complemented
5#31827075C212yesmaximal, complemented
6#31827059C214yesmaximal, complemented
7#31827043C211yesmaximal, characteristic
8#31827039C211yesmaximal, characteristic, complemented
9#31827020C212yesmaximal, complemented
10#31827017C212yesmaximal, complemented
11#31827001C212yesmaximal, complemented
12#31826994C212yesmaximal, complemented
13#31826992C214yesmaximal, complemented
14#41×Dic283C211yesmaximal, characteristic, complemented
15#31826933C211yesmaximal, characteristic, complemented
16C2×Q8×Dic283C211yesmaximal, characteristic, complemented
17#536×C283C211yesmaximal, characteristic, complemented
18C2×#17985392C2211yescharacteristic
19#22583610C2212yes—
20#22583609C2212yes—
21#22583608C2212yes—
22#22583605C2214yes—
23#22583605C2214yes—
24#22583603C2214yes—
25#22583603C2214yes—
26#22583602C2212yes—
27#22583601C2212yes—
28#22583601C2214yes—
29#22583600C2214yes—
30#22583600C2214yes—
31#22583599C2212yes—
32#22583596C2214yes—
33#22583594C2214yes—
34#22583593C2214yes—
35#22583592C2214yes—
36#22583590C2214yes—
37#22583589C2212yes—
38#22583587C2214yes—
39#22583587C2214yes—
40#22583586C2212yes—
41#22583586C2212yes—
42#22583586C2212yes—
43#22583584C2212yes—
44#22583582C2214yes—
45#22583581C2212yes—
46Q8×Dic283C2212yes—
47Q8×Dic283C2212yes—
48Q8×Dic283C2211yescharacteristic
49Q8×Dic283C2212yes—
50Q8×Dic283C2211yescharacteristic
51Q8×Dic283C2212yes—
52#22583575C2214yes—
53#22583575C2211yescharacteristic
54#22583575C2211yescharacteristic
55D4×Dic283C2212yes—
56D4×Dic283C2211yescharacteristic
57D4×Dic283C2212yes—
58D4×Dic283C2211yescharacteristic
59#22583572C2211yescharacteristic
60#22583572C2211yescharacteristic
61(C22⋊D283)⋊C4C2212yes—
62(C22⋊D283)⋊C4C2214yes—
63(C22⋊D283)⋊C4C2212yes—
64#22583569C2211yescharacteristic
65#22583569C2212yes—
66#22583568C2214yes—
67#22583567C2212yes—
68#22583566C2214yes—
69C2×C4⋊Dic283C2211yescharacteristic
70C2×C4⋊Dic283C2212yes—
71C2×C4×Dic283C2212yes—
72C2×C4×Dic283C2212yes—
73C2×C4×Dic283C2211yescharacteristic
74C42×D283C2212yes—
75C2×#41×C283C2211yescharacteristic
76C22×Q8×C283C2211yescharacteristic
77C4⋊Q8×C283C2212yes—
78#124×C283C2212yes—
79#123×C283C2214yes—
80C22⋊Q8×C283C2212yes—
81C22⋊Q8×C283C2212yes—
82#117×C283C2211yescharacteristic
83#17985392—24no—
84C2×Dic566C2311yescharacteristic
85C2×Dic566C2314yes—
86C2×D1132C2311yescharacteristic
87C2×C22⋊D283C2312yes—
88C2×C22⋊D283C2314yes—
89C22×Dic283C2311yescharacteristic
90C22×Dic283C2312yes—
91C22×Dic283C2311yescharacteristic
92C22×Dic283C2312yes—
93C22×Dic283C2312yes—
94C2×C4×D283C2312yes—
95C2×C4×D283C2312yes—
96C2×C4×D283C2312yes—
97C4⋊Dic283C2312yes—
98C4⋊Dic283C2312yes—
99C4⋊Dic283C2314yes—
100C4⋊Dic283C2314yes—
101C4⋊Dic283C2314yes—
102C4⋊Dic283C2311yescharacteristic
103C4⋊Dic283C2311yescharacteristic
104C4⋊Dic283C2312yes—
105C4⋊Dic283C2312yes—
106Dic283⋊C4C2314yes—
107Dic283⋊C4C2314yes—
108Dic283⋊C4C2312yes—
109Dic283⋊C4C2312yes—
110Dic283⋊C4C2314yes—
111Dic283⋊C4C2314yes—
112Dic283⋊C4C2314yes—
113C22⋊Dic283C2314yes—
114C22⋊Dic283C2312yes—
115C22⋊Dic283C2312yes—
116C22⋊Dic283C2312yes—
117(C2×D283)⋊C4C2312yes—
118(C2×D283)⋊C4C2312yes—
119(C2×D283)⋊C4C2314yes—
120(C2×D283)⋊C4C2314yes—
121(C2×D283)⋊C4C2314yes—
122C4×Dic283C2311yescharacteristic
123C4×Dic283C2312yes—
124C4×Dic283C2312yes—
125C4×Dic283C2311yescharacteristic
126C4×Dic283C2314yes—
127C4×Dic283C2311yescharacteristic
128C4×Dic283C2312yes—
129C4×Dic283C2311yescharacteristic
130C4×Dic283C2312yes—
131C4×Dic283C2312yes—
132C4×Dic283C2314yes—
133#41×C283—24no—
134C2×Q8×C283—24no—
135C2×Q8×C283C2312yes—
136C2×Q8×C283C2311yescharacteristic
137C2×Q8×C283C2312yes—
138C2×D4×C283C2311yescharacteristic
139C2×D4×C283C2312yes—
140C22×C4×C283C2312yes—
141C22×C4×C283C2312yes—
142C22×C4×C283C2311yescharacteristic
143C4⋊C4×C283C2314yes—
144C4⋊C4×C283C2311yescharacteristic
145C4⋊C4×C283C2311yescharacteristic
146C4⋊C4×C283C2314yes—
147C22⋊C4×C283C2314yes—
148C22⋊C4×C283C2312yes—
149C22⋊C4×C283C2314yes—
150C42×C283C2312yes—
151C22×D283C2412yes—
152Dic566—22no—
153Dic566—44no—
154D1132—22no—
155C22⋊D283—24no—
156C22⋊D283—44no—
157C2×Dic283—22no—
158C2×Dic283—24no—
159C2×Dic283—41no—
160C2×Dic283—42no—
161C2×Dic283—42no—
162C2×Dic283C2411yescharacteristic
163C2×Dic283C2411yescharacteristic
164C2×Dic283C2411yescharacteristic
165C2×Dic283C2412yes—
166C2×Dic283C2412yes—
167C2×Dic283C2414yes—
168C2×Dic283C2412yes—
169C2×Dic283C2411yescharacteristic
170C4×D283—24no—
171C4×D283—24no—
172C4×D283—42no—
173C23×C283C2412yes—
174C23×C283C2411yescharacteristic
175Q8×C283—24no—
176Q8×C283—24no—
177Q8×C283—22no—
178D4×C283—24no—
179D4×C283—22no—
180C2×C4×C283—24no—
181C2×C4×C283—24no—
182C2×C4×C283#4114yes—
183C2×C4×C283C2412yes—
184C2×C4×C283C2412yes—
185C2×C4×C283C2414yes—
186C2×C4×C283C2412yes—
187C2×C4×C283C2411yescharacteristic
188C2×C4×C283C2411yescharacteristic
189C22×C283—21no—
190C22×C283—21no—
191C22×C283—22no—
192C22×C283—22no—
193C22×C283—22no—
194C22×C283C2×#4112yes—
195C22×C283C2511yescharacteristic
196C2×D283—22no—
197C2×D283—22no—
198C2×D283—22no—
199C4×C283—22no—
200C4×C283—24no—
201C4×C283—22no—
202C4×C283—22no—
203C4×C283C2×#4112yes—
204C4×C283C2×#4112yes—
205Dic283—22no—
206Dic283—21no—
207Dic283—22no—
208Dic283—21no—
209Dic283—21no—
210Dic283—24no—
211Dic283—22no—
212Dic2832−1+412yes—
213C2×C283—22no—
214C2×C283—42no—
215C2×C283C22×#4111yescharacteristic
216C2×C283C2×2−1+411yescharacteristic
217C2×C283#58511yescharacteristic
218D283—42no—
219C283#346111yescharacteristic, complemented, Sylow
220#3461—2831nomaximal, Sylow
221#544—2831no—
222#542—2832no—
223#542—2832no—
224#541—2832no—
225#541—2832no—
226#539—2834no—
227#537—2831no—
228#536—2831no—
229#536D28311yescharacteristic, complemented
230#533—2832no—
231#533—2832no—
232#529—2832no—
233#529—2834no—
234#529—2832no—
235C4×#41—2831no—
236C4×#41—2831no—
237C2×C4×Q8—2831no—
238C2×#41—2831no—
239C2×#41C2×D28311yescharacteristic
240C22×Q8C2×D28311yescharacteristic
241C4⋊Q8—2832no—
242C4⋊Q8—2832no—
243C4⋊Q8—2832no—
244C4⋊Q8C2×D28312yes—
245#126—2834no—
246#126—2834no—
247#126—2834no—
248#126—2834no—
249#125—2834no—
250#125—2832no—
251#125—2834no—
252#125—2834no—
253#125—2832no—
254#124—2832no—
255#124—2834no—
256#124C2×D28312yes—
257#123—2834no—
258#123—2834no—
259#123—2834no—
260#123C2×D28314yes—
261C22⋊Q8—2832no—
262C22⋊Q8—2832no—
263C22⋊Q8—2832no—
264C22⋊Q8—2832no—
265C22⋊Q8—2834no—
266C22⋊Q8—2834no—
267C22⋊Q8—2834no—
268C22⋊Q8C2×D28312yes—
269C22⋊Q8C2×D28312yes—
270#121—2834no—
271#121—2832no—
272#121—2832no—
273C4×Q8—2832no—
274C4×Q8—2832no—
275C4×Q8—2832no—
276C4×Q8—2832no—
277C4×Q8—2831no—
278C4×Q8—2831no—
279C4×Q8—2831no—
280C4×Q8—2831no—
281C4×Q8—2834no—
282C4×D4—2831no—
283C4×D4—2832no—
284C4×D4—2832no—
285C4×D4—2831no—
286C4×D4—2831no—
287C4×D4—2831no—
288C4×D4—2832no—
289C4×D4—2832no—
290C4×D4—2834no—
291#117—2831no—
292#117—2832no—
293#117—2834no—
294#117—2834no—
295#117—2832no—
296#117C2×D28311yescharacteristic
297C2×C4⋊C4—2831no—
298C2×C4⋊C4—2832no—
299C2×C42—2831no—
300C2×C42—2832no—
301C2×C42—2832no—
302C2×C42—2832no—
303#41—24no—
304#41—5664no—
305C2×Q8—24no—
306C2×Q8—2831no—
307C2×Q8—2834no—
308C2×Q8C22×D28312yes—
309C2×Q8C22×D28311yescharacteristic
310C2×Q8C22×D28312yes—
311C2×D4—2832no—
312C2×D4—2834no—
313C2×D4—2831no—
314C2×D4C22×D28312yes—
315C2×D4C22×D28311yescharacteristic
316C22×C4—2831no—
317C22×C4—2832no—
318C22×C4—2831no—
319C22×C4—2832no—
320C22×C4—2832no—
321C22×C4—2832no—
322C22×C4—2832no—
323C22×C4—2832no—
324C22×C4C22×D28312yes—
325C22×C4C22×D28312yes—
326C22×C4C22×D28311yescharacteristic
327C4⋊C4—2832no—
328C4⋊C4—2832no—
329C4⋊C4—2834no—
330C4⋊C4—2834no—
331C4⋊C4—2834no—
332C4⋊C4—2834no—
333C4⋊C4—2834no—
334C4⋊C4—2832no—
335C4⋊C4—2832no—
336C4⋊C4—2834no—
337C4⋊C4—2834no—
338C4⋊C4—2832no—
339C4⋊C4—2831no—
340C4⋊C4—2831no—
341C4⋊C4—2832no—
342C4⋊C4—2834no—
343C4⋊C4C22×D28314yes—
344C4⋊C4C22×D28311yescharacteristic
345C4⋊C4C22×D28311yescharacteristic
346C4⋊C4C22×D28314yes—
347C22⋊C4—2832no—
348C22⋊C4—2832no—
349C22⋊C4—2832no—
350C22⋊C4—2834no—
351C22⋊C4—2832no—
352C22⋊C4—2832no—
353C22⋊C4—2834no—
354C22⋊C4—2834no—
355C22⋊C4—2834no—
356C22⋊C4C22×D28314yes—
357C22⋊C4C22×D28312yes—
358C22⋊C4C22×D28314yes—
359C42—2831no—
360C42—2831no—
361C42—2832no—
362C42—2834no—
363C42—2832no—
364C42—2831no—
365C42—2831no—
366C42—2832no—
367C42—2832no—
368C42—2832no—
369C42—2834no—
370C42C22×D28312yes—
371C23—2832no—
372C23C23×D28312yes—
373C23C23×D28311yescharacteristic
374Q8—24no—
375Q8—24no—
376Q8—22no—
377Q8—5662no—
378Q8—11324no—
379D4—24no—
380D4—22no—
381D4—5664no—
382D4—5662no—
383D4—11324no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—2831no—
387C2×C4—2831no—
388C2×C4—2831no—
389C2×C4—2831no—
390C2×C4—2832no—
391C2×C4—2832no—
392C2×C4—2832no—
393C2×C4—2834no—
394C2×C4—5664no—
395C2×C4—5662no—
396C2×C4—5664no—
397C2×C4—5664no—
398C2×C4—11322no—
399C2×C4—11321no—
400C2×C4—11322no—
401C2×C4—11322no—
402C2×C4#1798539114yes—
403C2×C4C23×D28314yes—
404C2×C4C23×D28312yes—
405C2×C4C23×D28312yes—
406C2×C4C23×D28312yes—
407C2×C4C23×D28311yescharacteristic
408C2×C4C23×D28311yescharacteristic
409C22—21no—
410C22—21no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—5662no—
415C22—5662no—
416C22—5662no—
417C22C2×#1798539112yes—
418C22C24×D28311yescharacteristic
419C4—22no—
420C4—24no—
421C4—22no—
422C4—22no—
423C4—2832no—
424C4—5662no—
425C4—5661no—
426C4—5662no—
427C4—5661no—
428C4—5661no—
429C4—5664no—
430C4—5662no—
431C4C2×#1798539112yes—
432C4C2×#1798539112yes—
433C2—22no—
434C2—42no—
435C2—11322no—
436C2C22×#1798539111yescharacteristic
437C22−1+4×D28311yescharacteristic
438C2#3182730211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#536×C283C2
prime radical#31827043C2
hypercenter#536D283
hypocenterC283#3461
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #31827093C2semidirect
2. #31827088C2semidirect
3. #31827084C2semidirect
4. #31827076C2semidirect
5. #31827075C2semidirect
6. #31827059C2semidirect
8. #31827039C2semidirect
9. #31827020C2semidirect
10. #31827017C2semidirect
11. #31827001C2semidirect
12. #31826994C2semidirect
13. #31826992C2semidirect
14. #41×Dic283C2semidirect
15. #31826933C2semidirect
16. C2×Q8×Dic283C2semidirect
17. #536×C283C2semidirect
219. C283#3461semidirect
229. #536D283semidirect

Maximal subgroups

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

subgroupindex
#318270932
#318270882
#318270842
#318270762
#318270752
#318270592
#318270432
#318270392
#318270202
#318270172
#318270012
#318269942
#318269922
#41×Dic2832
#318269332
C2×Q8×Dic2832
#536×C2832
#3461283