Finite Group Databasebeta

← #224528

Subgroups — #224528

Order 512.

906 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2513C211yesmaximal, characteristic
2#22627C212yesmaximal, complemented
3#9653C212yesmaximal
4#9652C212yesmaximal, complemented
5#9651C212yesmaximal, complemented
6#8561C212yesmaximal, complemented
7#8560C211yesmaximal, characteristic, complemented
8#8155C212yesmaximal, complemented
9#8153C211yesmaximal, characteristic, complemented
10#2513—22no—
11#2513—22no—
12#2513—22no—
13#2513—22no—
14C2×#397C412yes—
15C2×#397C412yes—
16C2×#390C411yescharacteristic
17C2×#390C411yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#389C411yescharacteristic
20C2×#388C2212yes—
21C2×#388C2212yes—
22C2×#382C411yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1940—22no—
26#1917—22no—
27#1916—22no—
28#1797C2212yes—
29#1797C2212yes—
30#1797C2212yes—
31#1771—22no—
32#1749C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2212yes—
36C2×#324C2211yescharacteristic
37C2×#324C2211yescharacteristic
38C2×#323C2212yes—
39C2×#323C2212yes—
40C2×#323C2212yes—
41C2×#323C2212yes—
42#1280C2212yes—
43#1280C2211yescharacteristic
44#1280C2211yescharacteristic
45C23×D4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C22⋊C4C2×C412yes—
53C22×C22⋊C4C2×C412yes—
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2312yes—
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2312yes—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C22⋊C8C2312yes—
66C2×C22⋊C8C2312yes—
67C2×C22⋊C8C2312yes—
68C2×C22⋊C8C2312yes—
69C2×#97—22no—
70C2×#97—22no—
71C2×C4×C8—22no—
72#397—22no—
73#397—22no—
74#397—22no—
75#397—22no—
76#397—22no—
77#397—22no—
78#397—22no—
79#397—22no—
80#390—21no—
81#390—21no—
82#390—22no—
83#390—22no—
84#390—21no—
85#390—21no—
86#390—22no—
87#390—22no—
88#389—22no—
89#389—21no—
90#389—21no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#382—22no—
100#382—21no—
101#382—21no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2311yescharacteristic
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#324—24no—
121#324—41no—
122#324—41no—
123#323—24no—
124#323—24no—
125#323—42no—
126#323—42no—
127C25C22×C412yes—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—41no—
131C22×D4—41no—
132C22×D4—42no—
133C22×D4—42no—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C412yes—
140C23×C4C22×C412yes—
141C23×C4C22×C412yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—42no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—42no—
200C2×C42—82no—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C4⋊C8—44no—
208C4⋊C8—82no—
209C4⋊C8—82no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8#4114yes—
218#97—44no—
219#97—82no—
220C4×C8—82no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24C23×C411yescharacteristic
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—84no—
252C2×D4—84no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4#13112yes—
305C22×C4C22×D412yes—
306C22×C4C2×#4112yes—
307C22×C4H(5,𝔽2)11yescharacteristic
308C22×C42−1+411yescharacteristic
309C2×C8—24no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C4⋊C4—24no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C42—24no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23C22×C22⋊C411yescharacteristic
386C23C4×#4111yescharacteristic
387C23#52011yescharacteristic
388C23C2×#13111yescharacteristic
389C23#56811yescharacteristic
390C23#56811yescharacteristic
391C23#56811yescharacteristic
392D4—84no—
393D4—84no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4#41012yes—
454C2×C4#41312yes—
455C2×C4#54812yes—
456C2×C4#55012yes—
457C8—44no—
458C8—82no—
459C8—82no—
460C8—82no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—162no—
481C22—162no—
482C22—162no—
483C22—162no—
484C22#234111yescharacteristic
485C22#287911yescharacteristic
486C22#288011yescharacteristic
487C22C2×#41011yescharacteristic
488C22#288311yescharacteristic
489C22C2×#41311yescharacteristic
490C22#288811yescharacteristic
491C22#296711yescharacteristic
492C22#296811yescharacteristic
493C22#299211yescharacteristic
494C22#299511yescharacteristic
495C4—22no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4—82no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—162no—
514C2#1739811yescharacteristic
515C2#1740811yescharacteristic
516C2#1741811yescharacteristic
517C2#1742911yescharacteristic
518C2#2253311yescharacteristic
519C2#2253511yescharacteristic
520C2#2253911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22627C2semidirect
4. #9652C2semidirect
5. #9651C2semidirect
6. #8561C2semidirect
7. #8560C2semidirect
8. #8155C2semidirect
9. #8153C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25132
#226272
#96532
#96522
#96512
#85612
#85602
#81552
#81532