Finite Group Databasebeta

← #193305

Subgroups — #193305

Order 512.

1056 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#21988C211yesmaximal, characteristic, complemented
3#21961C211yesmaximal, characteristic, complemented
4#21805C211yesmaximal, characteristic
5C2×#1551C211yesmaximal, characteristic
6#8929C212yesmaximal, complemented
7#8927C212yesmaximal, complemented
8#8921C212yesmaximal, complemented
9#8920C212yesmaximal, complemented
10#7900C211yesmaximal, characteristic, complemented
11#7893C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2379—22no—
17#2369C2212yes—
18#2366C2212yes—
19#2354—21no—
20#2354—21no—
21D4×C4⋊C4—21no—
22D4×C4⋊C4—21no—
23#2324—22no—
24#2324—22no—
25#2311—22no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29#1983—22no—
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1948C2212yes—
35#1947—21no—
36#1947—21no—
37#1947C2212yes—
38#1947C2212yes—
39#1947C2212yes—
40#1854—22no—
41#1847—21no—
42#1836—21no—
43#1808—22no—
44#1788—22no—
45#1551C418yescomplemented
46C2×#333C2212yes—
47C2×#331C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×#41D412yes—
53C23×C8—21no—
54#532—22no—
55#532—22no—
56#532C2×C414yes—
57#532Q814yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×#127C2311yescharacteristic
60C2×#124D412yes—
61C2×#121—21no—
62C2×#121—21no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×#117—22no—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117—21no—
71C22×C4⋊C4—22no—
72C22×C42C2311yescharacteristic
73#432—22no—
74#432C2×C414yescomplemented
75#431—22no—
76#431—22no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#106—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2×C414yescomplemented
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87C2×#103—22no—
88C2×#103—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×#97—21no—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#391—24no—
97#390—24no—
98#390—22no—
99#390—22no—
100#389—22no—
101#389—22no—
102#388—24no—
103#388—24no—
104#386—24no—
105#386—24no—
106#385—24no—
107#385—24no—
108#384D411yescharacteristic
109#384D411yescharacteristic
110#384D411yescharacteristic
111#384D411yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#382—21no—
115#382—21no—
116#382—21no—
117#380—22no—
118#380—22no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380D412yes—
124#380D412yes—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2312yes—
132C4×C4⋊C4C2312yes—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#333C2×C418yescomplemented
138#331C2×C418yes—
139C2×#41—24no—
140C2×#41—82no—
141C2×#41—82no—
142C2×#41C4⋊C418yes—
143C22×Q8C2×D412yes—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C2×D412yes—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—24no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—81no—
159C22×C8—81no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8C22×C412yes—
163C4⋊Q8C2×Q812yes—
164#127—21no—
165#127—21no—
166#127C22×C412yes—
167#127C2×Q812yes—
168#124—24no—
169#124C4⋊C418yes—
170#121—24no—
171#121—24no—
172#121—24no—
173#121—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—82no—
179C4×D4—82no—
180C4×D4—82no—
181C4×D4—82no—
182#117—82no—
183#117—81no—
184#117—81no—
185#117—82no—
186#117—84no—
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—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—84no—
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—22no—
235C4⋊C8—22no—
236C4⋊C8—42no—
237C4⋊C8C22×C414yes—
238C4⋊C8C22×C414yes—
239#103—82no—
240#103—82no—
241#102—82no—
242#102—82no—
243C22⋊C8—24no—
244C22⋊C8—42no—
245#97—24no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—41no—
257#95—41no—
258C24—22no—
259C24#12011yescharacteristic
260#41—88no—
261#41—88no—
262C2×Q8—22no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8C2×C4⋊C414yes—
266C2×D4—28no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—82no—
281C2×D4—82no—
282C2×D4C2×C4⋊C414yes—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—28no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—44no—
298C22×C4—44no—
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—24no—
317C2×C8—24no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—84no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—84no—
357C22⋊C4—28no—
358C22⋊C4—28no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—84no—
362C22⋊C4—84no—
363C22⋊C4—84no—
364C22⋊C4—82no—
365C42—42no—
366C42—42no—
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—44no—
397Q8—82no—
398D4—44no—
399D4—42no—
400D4—44no—
401D4—42no—
402D4—42no—
403D4—42no—
404D4—44no—
405D4—82no—
406D4—88no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—416no—
425C2×C4—42no—
426C2×C4—42no—
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#57612yes—
457C2×C4#57711yescharacteristic
458C2×C4#57711yescharacteristic
459C2×C4#57811yescharacteristic
460C2×C4#57812yes—
461C8—44no—
462C8—44no—
463C8—81no—
464C8—81no—
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#301311yescharacteristic
484C22#301911yescharacteristic
485C22#301911yescharacteristic
486C4—21no—
487C4—22no—
488C4—21no—
489C4—22no—
490C4—22no—
491C4—44no—
492C4—82no—
493C4—82no—
494C4—82no—
495C4—82no—
496C4—161no—
497C4—161no—
498C4#294611yescharacteristic
499C4#294711yescharacteristic
500C4#295112yes—
501C4#308311yescharacteristic
502C4#308312yes—
503C4#308411yescharacteristic
504C2—42no—
505C2—84no—
506C2#1893411yescharacteristic
507C2#1893611yescharacteristic
508C2#1893911yescharacteristic
509C2#1894111yescharacteristic
510C2#2085611yescharacteristic
511C2#2087811yescharacteristic
512C2#2088111yescharacteristic
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. #21988C2semidirect
3. #21961C2semidirect
6. #8929C2semidirect
7. #8927C2semidirect
8. #8921C2semidirect
9. #8920C2semidirect
10. #7900C2semidirect
11. #7893C2semidirect
45. #1551C4semidirect
74. #432C2×C4semidirect
84. C2×C4⋊C8C2×C4semidirect
137. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343682
#219882
#219612
#218052
C2×#15512
#89292
#89272
#89212
#89202
#79002
#78932