Finite Group Databasebeta

← #219536

Subgroups — #219536

Order 512.

590 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21695C211yesmaximal, characteristic
2#17879C211yesmaximal, characteristic, complemented
3#17878C211yesmaximal, characteristic, complemented
4#9672C211yesmaximal, characteristic, complemented
5#9672C211yesmaximal, characteristic, complemented
6#9105C211yesmaximal, characteristic
7#9104C211yesmaximal, characteristic, complemented
8#9101C211yesmaximal, characteristic
9#9100C211yesmaximal, characteristic, complemented
10#9094C211yesmaximal, characteristic, complemented
11#9091C211yesmaximal, characteristic, complemented
12#9090C211yesmaximal, characteristic, complemented
13#9087C211yesmaximal, characteristic, complemented
14#8781C211yesmaximal, characteristic, complemented
15#8140C211yesmaximal, characteristic, complemented
16C2×#492C411yescharacteristic
17#2983C2211yescharacteristic
18#2982C2211yescharacteristic
19#2975C2211yescharacteristic
20#2972C2211yescharacteristic
21C2×#440C2211yescharacteristic
22C2×#440C2211yescharacteristic
23C2×C4×SD16C2211yescharacteristic
24C2×C4×SD16C2211yescharacteristic
25#2328C2211yescharacteristic
26#2327C2211yescharacteristic
27#1973C411yescharacteristic
28#1973C411yescharacteristic
29#1968C411yescharacteristic
30#1968C411yescharacteristic
31#1961C411yescharacteristic
32#1960C411yescharacteristic
33#1846C411yescharacteristic
34#1726C2211yescharacteristic
35#1588—22no—
36#1585—22no—
37#1583—22no—
38#1582—22no—
39#1564C2211yescharacteristic
40#1564C2211yescharacteristic
41#1479C2211yescharacteristic
42#1479C2211yescharacteristic
43#1479C2211yescharacteristic
44#1479C2211yescharacteristic
45#1478C2211yescharacteristic
46#1478C2211yescharacteristic
47#1478C2211yescharacteristic
48#1478C2211yescharacteristic
49#1461C2211yescharacteristic
50#1461C2211yescharacteristic
51#1460C2211yescharacteristic
52#1460C2211yescharacteristic
53#1452C2211yescharacteristic
54#1452C2211yescharacteristic
55#1281C2211yescharacteristic
56#1281C2211yescharacteristic
57#1275C2211yescharacteristic
58#1275C2211yescharacteristic
59#1274C2211yescharacteristic
60#1274C2211yescharacteristic
61#1274C2211yescharacteristic
62#1274C2211yescharacteristic
63C22×SD16C2×C411yescharacteristic
64C22×SD16C2×C411yescharacteristic
65C22×SD16C2×C411yescharacteristic
66C22×SD16C2×C411yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×#127C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73#492—24no—
74#446—22no—
75#446—41no—
76C8×Q8—22no—
77C8×Q8—41no—
78#440—41no—
79#440—41no—
80#440—41no—
81#440—41no—
82C4×SD16—21no—
83C4×SD16—21no—
84C4×SD16—21no—
85C4×SD16—21no—
86C4×SD16—21no—
87C4×SD16—21no—
88C4×SD16—21no—
89C4×SD16—21no—
90#436—22no—
91#436—41no—
92C8×D4—22no—
93C8×D4—41no—
94#432—41no—
95#431—41no—
96#431C2311yescharacteristic
97#431C2311yescharacteristic
98#431C2311yescharacteristic
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#431C2311yescharacteristic
103#431C2311yescharacteristic
104C2×#107C2×C411yescharacteristic
105C2×#107C2×C411yescharacteristic
106C2×#106C2×C411yescharacteristic
107C2×#106C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116#407—41no—
117C2×C22⋊C8—41no—
118C2×#97C2×C411yescharacteristic
119C2×#97C2×C411yescharacteristic
120C2×#97C2311yescharacteristic
121C2×C4×C8C2×C411yescharacteristic
122C2×C4×C8C2×C411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391C2×C411yescharacteristic
125#391C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#387—41no—
129#387—41no—
130#386—41no—
131#386—41no—
132#382C2×C411yescharacteristic
133#382C2×C411yescharacteristic
134#376C2311yescharacteristic
135#335—21no—
136#335—21no—
137#334—21no—
138#334—21no—
139Q8⋊C8—22no—
140Q8⋊C8—22no—
141Q8⋊C8—41no—
142Q8⋊C8—41no—
143D4⋊C8—22no—
144D4⋊C8—22no—
145D4⋊C8—41no—
146D4⋊C8—41no—
147#324—41no—
148#324—41no—
149#323—41no—
150#323—41no—
151#322—21no—
152#322—21no—
153#322D412yescomplemented
154#322D412yescomplemented
155C22×Q8C22×C411yescharacteristic
156C22×Q8C22×C411yescharacteristic
157C22×D4—41no—
158C22×D4C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C2×SD16—22no—
163C2×SD16—22no—
164C2×SD16—21no—
165C2×SD16—21no—
166C2×SD16—21no—
167C2×SD16—21no—
168C2×SD16—22no—
169C2×SD16—41no—
170C2×SD16—41no—
171C2×SD16—41no—
172C2×SD16—41no—
173C2×SD16—41no—
174C2×SD16—41no—
175C2×SD16M1614yescomplemented
176C2×M16—42no—
177C22×C8—42no—
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184#127—22no—
185#127—22no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C2411yescharacteristic
218#107—41no—
219#107—41no—
220#106—21no—
221#106—21no—
222#106—21no—
223#106—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234#103—21no—
235#103—21no—
236#103—21no—
237#103—21no—
238#103—41no—
239#103—41no—
240#102—21no—
241#102—21no—
242#102—21no—
243#102—21no—
244#102—41no—
245#102—41no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—42no—
255C22⋊C8—42no—
256#97—22no—
257#97—21no—
258#97—21no—
259#97—41no—
260#97—42no—
261C4×C8—22no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—41no—
269C4×C8—42no—
270C4×C8C2×D412yes—
271C4×C8C2×D412yes—
272C4×C8C2×D411yescharacteristic
273C4×C8C2×D411yescharacteristic
274C4×C8#4111yescharacteristic
275C4×C8#4111yescharacteristic
276#95—21no—
277#95—21no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—41no—
282C24—21no—
283C24—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8C2×M1612yes—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—44no—
308C2×D4—161no—
309C2×D4—161no—
310C2×D4C2×M1612yes—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—161no—
324C22×C4—161no—
325C22×C4C23×C411yescharacteristic
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328SD16—44no—
329SD16—42no—
330SD16—42no—
331SD16—42no—
332SD16—42no—
333SD16—44no—
334M16—162no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—162no—
353C2×C8C4×D411yescharacteristic
354C2×C8C4×D411yescharacteristic
355C2×C8C4×D411yescharacteristic
356C2×C8C4×D411yescharacteristic
357C2×C8C2×M1612yes—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—44no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—161no—
376C22⋊C4—161no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42#13112yes—
385C42C22×D411yescharacteristic
386C42C2×#4111yescharacteristic
387C42H(5,𝔽2)11yescharacteristic
388C42H(5,𝔽2)11yescharacteristic
389C23—22no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23#51811yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—41no—
403Q8—41no—
404Q8—41no—
405Q8—41no—
406Q8—41no—
407Q8—41no—
408D4—22no—
409D4—22no—
410D4—41no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—41no—
416D4—164no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—161no—
442C2×C4—161no—
443C2×C4C2×C4×D411yescharacteristic
444C2×C4#51811yescharacteristic
445C2×C4#53511yescharacteristic
446C2×C4#53511yescharacteristic
447C2×C4#54011yescharacteristic
448C2×C4#54011yescharacteristic
449C2×C4C22×M1611yescharacteristic
450C2×C4C2×#13111yescharacteristic
451C2×C4#56811yescharacteristic
452C2×C4#56811yescharacteristic
453C8—21no—
454C8—21no—
455C8—21no—
456C8—21no—
457C8—22no—
458C8—42no—
459C8—42no—
460C8—41no—
461C8—81no—
462C8—81no—
463C8—81no—
464C8—81no—
465C8—81no—
466C8—81no—
467C22—21no—
468C22—21no—
469C22—42no—
470C22—82no—
471C22—82no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—161no—
478C22—161no—
479C22#232811yescharacteristic
480C22#297411yescharacteristic
481C22#298011yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—42no—
487C4—42no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4D4×M1611yescharacteristic
495C4#293511yescharacteristic
496C4#297311yescharacteristic
497C4#298111yescharacteristic
498C2—21no—
499C2—21no—
500C2—82no—
501C2—161no—
502C2#2226611yescharacteristic
503C2#2228411yescharacteristic
504C2#2228411yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17879C2semidirect
3. #17878C2semidirect
4. #9672C2semidirect
5. #9672C2semidirect
7. #9104C2semidirect
9. #9100C2semidirect
10. #9094C2semidirect
11. #9091C2semidirect
12. #9090C2semidirect
13. #9087C2semidirect
14. #8781C2semidirect
15. #8140C2semidirect
153. #322D4semidirect
154. #322D4semidirect
175. C2×SD16M16semidirect

Maximal subgroups

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

subgroupindex
#216952
#178792
#178782
#96722
#91052
#91042
#91012
#91002
#90942
#90912
#90902
#90872
#87812
#81402