Finite Group Databasebeta

← #192806

Subgroups — #192806

Order 512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34368C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#21961C211yesmaximal, characteristic, complemented
4#21869C211yesmaximal, characteristic
5C2×#1480C211yesmaximal, characteristic
6#8897C212yesmaximal, complemented
7#8890C212yesmaximal, complemented
8#8889C212yesmaximal, complemented
9#8882C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic, complemented
11#7900C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2379—22no—
17#2369—21no—
18#2366—21no—
19#2354—21no—
20#2354C2212yes—
21D4×C4⋊C4—21no—
22D4×C4⋊C4C2212yes—
23#2324—22no—
24#2324—22no—
25#2311—22no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29#1993—21no—
30#1961—21no—
31#1949—22no—
32#1947—21no—
33#1947—21no—
34#1947C2212yes—
35#1947C2212yes—
36#1875—22no—
37#1854—22no—
38#1845C2211yescharacteristic
39#1844C2211yescharacteristic
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1808—22no—
43#1802C2212yes—
44#1802C2212yes—
45#1480C418yescomplemented
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×#41—21no—
53C23×C8—21no—
54#532—22no—
55#532—22no—
56#532C2×C414yes—
57#532Q814yes—
58C2×C4⋊Q8D411yescharacteristic
59C2×#127D411yescharacteristic
60C2×#124—21no—
61C2×#121—21no—
62C2×#121D412yes—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4C2312yes—
67C2×#117—21no—
68C2×#117—22no—
69C2×#117—21no—
70C2×#117D412yes—
71C22×C4⋊C4—22no—
72C22×C42C2311yescharacteristic
73#432—22no—
74#432—22no—
75#431—22no—
76#431C2×C414yescomplemented
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#106—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8C2×C414yescomplemented
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87C2×#102—22no—
88C2×#102—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×#97—21no—
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#391—24no—
95#390—24no—
96#390—22no—
97#390—22no—
98#389—22no—
99#389—22no—
100#388—24no—
101#388—24no—
102#386—24no—
103#386—24no—
104#385—24no—
105#385—24no—
106#384D411yescharacteristic
107#384D411yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#382—21no—
113#382—21no—
114#382D412yes—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—22no—
119#380—22no—
120#380—21no—
121#380—21no—
122#380D412yes—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2312yes—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137D4⋊C8C2×C418yescomplemented
138D4⋊C8C2×C418yescomplemented
139C2×#41—24no—
140C2×#41—24no—
141C2×#41—81no—
142C2×#41—81no—
143C2×#41—81no—
144C2×#41—81no—
145C22×Q8—21no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C2×D412yes—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—24no—
159C22×C8—22no—
160C22×C8—81no—
161C22×C8—81no—
162C4⋊Q8—22no—
163C4⋊Q8C4⋊C414yes—
164#127—22no—
165#127C4⋊C414yes—
166#124—24no—
167#124—24no—
168#121—24no—
169#121—24no—
170#121—24no—
171#121C4⋊C418yes—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—82no—
177C4×D4—82no—
178C4×D4—84no—
179C4×D4C22×C414yes—
180C4×D4C2×Q814yes—
181#117—82no—
182#117—82no—
183#117—82no—
184#117—82no—
185#117—82no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—41no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—41no—
221C2×C42C22×C412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×Q812yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42C2411yescharacteristic
229#107—82no—
230#107—82no—
231#107—82no—
232#106—82no—
233C4⋊C8—24no—
234C4⋊C8—24no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8C22×C414yes—
239#102—82no—
240#102—82no—
241C22⋊C8—24no—
242C22⋊C8—42no—
243#97—42no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8C22×C412yes—
247C4×C8C22×C412yes—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—41no—
255#95—41no—
256C24—22no—
257C24#12011yescharacteristic
258#41—84no—
259#41—84no—
260#41—84no—
261#41—84no—
262C2×Q8—24no—
263C2×Q8—81no—
264C2×Q8—81no—
265C2×D4—28no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—81no—
276C2×D4—81no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4C2×C4⋊C414yes—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—84no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4C2×C4⋊C412yes—
305C22×C4C4×D412yes—
306C22×C4C4×D412yes—
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4C22⋊Q812yes—
311C22×C4C22⋊Q812yes—
312C22×C4#12312yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—84no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—84no—
356C4⋊C4C2×C4⋊C414yes—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—28no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C22⋊C4—84no—
363C22⋊C4—84no—
364C22⋊C4—84no—
365C22⋊C4—82no—
366C42—44no—
367C42—42no—
368C42—42no—
369C42—82no—
370C42—84no—
371C42C2×C4⋊C412yes—
372C42C4×D412yes—
373C42C4×D412yes—
374C42C22⋊Q812yes—
375C42C22⋊Q812yes—
376C42#12312yes—
377C42C23×C411yescharacteristic
378C42C22×Q811yescharacteristic
379C42C2×#4111yescharacteristic
380C42C2×#4111yescharacteristic
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—24no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23#38512yes—
395C23C2×#12011yescharacteristic
396Q8—84no—
397D4—24no—
398D4—24no—
399D4—48no—
400D4—42no—
401D4—42no—
402D4—84no—
403D4—88no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—168no—
446C2×C4#38512yes—
447C2×C4#38512yes—
448C2×C4#38512yes—
449C2×C4C22×C4⋊C411yescharacteristic
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×C22⋊Q811yescharacteristic
453C2×C4C2×C22⋊Q811yescharacteristic
454C2×C4C2×#12311yescharacteristic
455C2×C4#57611yescharacteristic
456C2×C4#57611yescharacteristic
457C2×C4#57612yes—
458C2×C4#57711yescharacteristic
459C2×C4#57711yescharacteristic
460C2×C4#57812yes—
461C8—42no—
462C8—42no—
463C8—44no—
464C8—82no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—44no—
473C22—44no—
474C22—44no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22C2×#38511yescharacteristic
480C22#300811yescharacteristic
481C22#301111yescharacteristic
482C22#301211yescharacteristic
483C22#301211yescharacteristic
484C22#301911yescharacteristic
485C22#302011yescharacteristic
486C4—22no—
487C4—21no—
488C4—21no—
489C4—22no—
490C4—22no—
491C4—44no—
492C4—82no—
493C4—82no—
494C4—82no—
495C4—82no—
496C4—161no—
497C4—161no—
498C4#291312yes—
499C4#294811yescharacteristic
500C4#294911yescharacteristic
501C4#308111yescharacteristic
502C4#308111yescharacteristic
503C4#309812yes—
504C2—42no—
505C2—84no—
506C2#1893411yescharacteristic
507C2#1894111yescharacteristic
508C2#1894611yescharacteristic
509C2#1895311yescharacteristic
510C2#2084711yescharacteristic
511C2#2086411yescharacteristic
512C2#2087911yescharacteristic
subgroupG/H
center, socleC23C2×#120
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
2. #21984C2semidirect
3. #21961C2semidirect
6. #8897C2semidirect
7. #8890C2semidirect
8. #8889C2semidirect
9. #8882C2semidirect
10. #8044C2semidirect
11. #7900C2semidirect
45. #1480C4semidirect
76. #431C2×C4semidirect
84. C2×C4⋊C8C2×C4semidirect
137. D4⋊C8C2×C4semidirect
138. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343682
#219842
#219612
#218692
C2×#14802
#88972
#88902
#88892
#88822
#80442
#79002