Finite Group Databasebeta

← #197916

Subgroups — #197916

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3235C211yesmaximal, characteristic
2C2×#2981C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic, complemented
4#21859C211yesmaximal, characteristic
5C2×#1554C211yesmaximal, characteristic, complemented
6#9227C212yesmaximal, complemented
7#9223C212yesmaximal, complemented
8#9219C212yesmaximal, complemented
9#9215C212yesmaximal, complemented
10#8040C211yesmaximal, characteristic, complemented
11#7878C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3235—24no—
15#3235C418yescomplemented
16C2×#501—21no—
17C2×#490—21no—
18C2×#482—22no—
19C2×#476C2212yes—
20C2×#474C2212yes—
21#2981—24no—
22#2981—24no—
23C2×#436C2212yes—
24C2×#435—22no—
25C2×C8×D4C2212yes—
26C2×#432C2211yescharacteristic
27C2×#432C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#408—22no—
31C2×#384C2211yescharacteristic
32#1990—21no—
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1957—21no—
36#1956C2211yescharacteristic
37#1956C2211yescharacteristic
38#1874—22no—
39#1872—22no—
40#1803—22no—
41#1802C2212yes—
42#1802C2212yes—
43#1554—24no—
44C2×#333C2212yes—
45C2×#332C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49#1277C2212yes—
50C22×#41—21no—
51C22×M16—22no—
52C23×C8—22no—
53#533—22no—
54#533—22no—
55#533—22no—
56#533C2×C414yes—
57#532—22no—
58#532—22no—
59#532C2×C414yes—
60#532D414yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×#127D411yescharacteristic
65C2×#125—21no—
66C2×#124—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×#121—21no—
70C2×#121D412yes—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2312yes—
75C2×#117—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C42C2311yescharacteristic
80#501—28no—
81#490—28no—
82#482—216no—
83#476—24no—
84#476C2×C418yescomplemented
85#474—24no—
86#474C2×C418yescomplemented
87#436—24no—
88#436—24no—
89#435—216no—
90C8×D4—24no—
91C8×D4—24no—
92#432—22no—
93#432—22no—
94#432C2×C414yes—
95#432D414yes—
96#431—22no—
97#431—22no—
98#431—22no—
99#431—22no—
100C2×#107—22no—
101C2×#106—22no—
102C2×C4⋊C8D412yes—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2312yes—
106C2×#103—22no—
107C2×#103—22no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—22no—
112#408—216no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8D412yes—
119C2×#97—21no—
120C2×#97—21no—
121C2×C4×C8—21no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#384—21no—
125#384—21no—
126#384—21no—
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#333—24no—
134#332—24no—
135C2×#41—24no—
136C2×#41—24no—
137C2×#41—82no—
138C2×#41—82no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C2×D412yes—
146C23×C4—22no—
147C23×C4—41no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4#4111yescharacteristic
151C2×M16—28no—
152C2×M16—82no—
153C2×M16—82no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—28no—
157C22×C8—22no—
158C22×C8—82no—
159C22×C8—82no—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C4⋊Q8—22no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8C22⋊C414yes—
170C4⋊Q8C22×C412yes—
171C4⋊Q8C22×C412yes—
172#127—22no—
173#127C22⋊C414yes—
174#125—24no—
175#125—42no—
176#124—24no—
177#124—24no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182#121—24no—
183#121—24no—
184#121—24no—
185#121C22⋊C418yes—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4C22×C414yes—
195C4×D4C2×D414yes—
196#117—24no—
197#117—42no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C2×D412yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—41no—
221C2×C42—41no—
222C2×C42C22×C412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42#4111yescharacteristic
228C2×C42#4112yes—
229C2×C42C2411yescharacteristic
230#107—28no—
231#106—28no—
232C4⋊C8—24no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8C22×C414yes—
238C4⋊C8C2×D414yes—
239#103—82no—
240#103—82no—
241#102—28no—
242#102—28no—
243#102—82no—
244#102—82no—
245C22⋊C8—28no—
246C22⋊C8—28no—
247C22⋊C8—24no—
248C22⋊C8—24no—
249C22⋊C8—24no—
250C22⋊C8—42no—
251#97—24no—
252#97—24no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—42no—
258C24—22no—
259C24#12111yescharacteristic
260#41—88no—
261#41—88no—
262C2×Q8—24no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—81no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×Q8—81no—
273C2×D4—28no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—81no—
282C2×D4—81no—
283C2×D4—81no—
284C2×D4—81no—
285C2×D4—82no—
286C2×D4—82no—
287C2×D4C2×C22⋊C414yes—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4#12012yes—
312C22×C4#12112yes—
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12312yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320M16—88no—
321C2×C8—24no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—88no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4C2×C22⋊C414yes—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—28no—
362C22⋊C4—44no—
363C42—44no—
364C42C2×C22⋊C412yes—
365C42C4×D412yes—
366C42C4×D412yes—
367C42#12012yes—
368C42#12112yes—
369C42#12312yes—
370C42C23×C411yescharacteristic
371C42C22×D411yescharacteristic
372C42C22×D411yescharacteristic
373C42C2×#4111yescharacteristic
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—24no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23#38612yes—
388C23C2×#12111yescharacteristic
389Q8—44no—
390Q8—84no—
391Q8—82no—
392Q8—82no—
393Q8—82no—
394D4—24no—
395D4—24no—
396D4—42no—
397D4—42no—
398D4—84no—
399D4—84no—
400D4—88no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—164no—
435C2×C4#38612yes—
436C2×C4#38612yes—
437C2×C4#38612yes—
438C2×C4C22×C22⋊C411yescharacteristic
439C2×C4C2×C4×D411yescharacteristic
440C2×C4C2×C4×D411yescharacteristic
441C2×C4C2×#12011yescharacteristic
442C2×C4C2×#12111yescharacteristic
443C2×C4C2×#12311yescharacteristic
444C2×C4#57612yes—
445C2×C4#57712yes—
446C2×C4#57712yes—
447C2×C4#57812yes—
448C8—44no—
449C8—42no—
450C8—42no—
451C8—44no—
452C8—82no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22C2×#38611yescharacteristic
468C22#304111yescharacteristic
469C22#304211yescharacteristic
470C22#305711yescharacteristic
471C22#305911yescharacteristic
472C22#306811yescharacteristic
473C22#306911yescharacteristic
474C4—21no—
475C4—21no—
476C4—22no—
477C4—22no—
478C4—22no—
479C4—44no—
480C4—82no—
481C4—82no—
482C4—161no—
483C4—161no—
484C4#289812yes—
485C4#294612yes—
486C4#301212yes—
487C4#306312yes—
488C2—42no—
489C2—84no—
490C2#1950411yescharacteristic
491C2#1950711yescharacteristic
492C2#1951211yescharacteristic
493C2#1951511yescharacteristic
494C2#2092911yescharacteristic
495C2#2097411yescharacteristic
496C2#2100411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21860C2semidirect
5. C2×#1554C2semidirect
6. #9227C2semidirect
7. #9223C2semidirect
8. #9219C2semidirect
9. #9215C2semidirect
10. #8040C2semidirect
11. #7878C2semidirect
15. #3235C4semidirect
84. #476C2×C4semidirect
86. #474C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32352
C2×#29812
#218602
#218592
C2×#15542
#92272
#92232
#92192
#92152
#80402
#78782