Finite Group Databasebeta

← #197905

Subgroups — #197905

Order 512.

1128 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3216C211yesmaximal, characteristic
2C2×#2986C211yesmaximal, characteristic, complemented
3#21885C211yesmaximal, characteristic, complemented
4#21863C211yesmaximal, characteristic
5C2×#1540C211yesmaximal, characteristic
6#9230C212yesmaximal, complemented
7#9226C212yesmaximal, complemented
8#9222C212yesmaximal, complemented
9#9218C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic, complemented
11#7879C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3216—24no—
15#3216C418yescomplemented
16C2×#497—21no—
17C2×#489—21no—
18C2×#462C2212yes—
19C2×#460C2212yes—
20C2×#452—22no—
21#2986—24no—
22#2986—24no—
23C2×#446—21no—
24C2×#446C2212yes—
25C2×C8×Q8—21no—
26C2×C8×Q8C2212yes—
27C2×#433—22no—
28C2×#432C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32#2285C2211yescharacteristic
33#1990C2212yes—
34#1987—21no—
35#1981—21no—
36#1957C2212yes—
37#1892—22no—
38#1875—22no—
39#1804—21no—
40#1804—21no—
41#1803C2212yes—
42#1803C2212yes—
43#1540—24no—
44C2×#332C2211yescharacteristic
45C2×#332C2211yescharacteristic
46C2×#331C2211yescharacteristic
47C2×#331C2211yescharacteristic
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51#1277C2212yes—
52C22×#41—21no—
53C22×Q16—22no—
54C22×SD16—22no—
55C23×C8—21no—
56#533—22no—
57#533—22no—
58#533C2×C414yes—
59#533D414yes—
60#532—22no—
61#532—22no—
62#532—22no—
63#532C2×C414yes—
64C2×C4⋊Q8D411yescharacteristic
65C2×C4⋊Q8D411yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#125—21no—
69C2×#124—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8D412yes—
72C2×#121—21no—
73C2×#121—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8C2312yes—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×#117—21no—
79C2×#117—21no—
80C22×C42C2311yescharacteristic
81#497—28no—
82#489—28no—
83#462—24no—
84#462C2×C418yes—
85#460—24no—
86#460C2×C418yes—
87#452—216no—
88#446—28no—
89#446—24no—
90#446—24no—
91C8×Q8—28no—
92C8×Q8—24no—
93C8×Q8—24no—
94#433—216no—
95#432—22no—
96#432C2×C414yes—
97#431—22no—
98#431—22no—
99#431—22no—
100#431D414yes—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8D411yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×#103—22no—
114C2×#103—22no—
115C2×#103—22no—
116C2×#102—22no—
117C2×#102—22no—
118C2×#102—22no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8D412yes—
122C2×#97—22no—
123C2×#97—21no—
124C2×C4×C8—22no—
125C2×C4×C8—21no—
126C2×C4×C8C2312yes—
127#384C2312yes—
128#382—21no—
129#380—21no—
130#380—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#332—22no—
138#332—22no—
139#331—22no—
140#331—22no—
141C2×#41—24no—
142C2×#41—42no—
143C2×#41—82no—
144C2×#41—82no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8C2×D412yes—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—41no—
152C23×C4—41no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4#4111yescharacteristic
156C2×Q16—28no—
157C2×Q16—82no—
158C2×Q16—82no—
159C2×SD16—28no—
160C2×SD16—82no—
161C2×SD16—82no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—42no—
168C22×C8—82no—
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8C22⋊C414yes—
177C4⋊Q8C22⋊C414yes—
178C4⋊Q8C22×C412yes—
179#127—21no—
180#127—21no—
181#127—21no—
182#127C22×C412yes—
183#125—24no—
184#125—24no—
185#124—24no—
186#124—42no—
187C22⋊Q8—24no—
188C22⋊Q8—24no—
189C22⋊Q8—24no—
190C22⋊Q8C22⋊C418yes—
191#121—24no—
192#121—24no—
193#121—42no—
194#121—42no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8C22×C414yes—
200C4×Q8C2×D414yes—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—42no—
204C4×D4—42no—
205#117—24no—
206#117—24no—
207#117—81no—
208#117—81no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4C2×D412yes—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C42—22no—
225C2×C42—41no—
226C2×C42C22×C412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42#4111yescharacteristic
232C2×C42#4112yes—
233C2×C42C2411yescharacteristic
234C4⋊C8—28no—
235C4⋊C8—28no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—22no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—42no—
246C4⋊C8C22×C414yes—
247C4⋊C8C2×D412yes—
248C4⋊C8C2×D412yes—
249#103—28no—
250#103—82no—
251#103—82no—
252#102—28no—
253#102—82no—
254#102—82no—
255C22⋊C8—24no—
256C22⋊C8—24no—
257C22⋊C8—24no—
258#97—28no—
259#97—24no—
260C4×C8—28no—
261C4×C8—24no—
262C4×C8—22no—
263C4×C8—22no—
264#95—41no—
265#95—41no—
266C24—41no—
267C24#12111yescharacteristic
268#41—88no—
269#41—84no—
270#41—84no—
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—44no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8C2×C22⋊C414yes—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—44no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—82no—
297C2×D4—84no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—84no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C4×D412yes—
313C22×C4C4×D412yes—
314C22×C4#12012yes—
315C22×C4#12112yes—
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12312yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4111yescharacteristic
323Q16—88no—
324SD16—88no—
325C2×C8—24no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—84no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—28no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—82no—
361C4⋊C4C2×C22⋊C414yes—
362C22⋊C4—28no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—82no—
366C42—24no—
367C42—24no—
368C42—82no—
369C42C2×C22⋊C412yes—
370C42C4×D412yes—
371C42C4×D412yes—
372C42#12012yes—
373C42#12112yes—
374C42#12312yes—
375C42C23×C411yescharacteristic
376C42C22×D411yescharacteristic
377C42C22×D411yescharacteristic
378C42C2×#4111yescharacteristic
379C23—21no—
380C23—21no—
381C23—22no—
382C23—22no—
383C23—42no—
384C23—81no—
385C23—81no—
386C23—82no—
387C23—82no—
388C23#38612yes—
389C23C2×#12111yescharacteristic
390Q8—24no—
391Q8—24no—
392Q8—48no—
393Q8—42no—
394Q8—42no—
395Q8—44no—
396Q8—81no—
397Q8—81no—
398D4—44no—
399D4—42no—
400D4—42no—
401D4—82no—
402D4—82no—
403D4—88no—
404D4—81no—
405D4—81no—
406D4—81no—
407D4—81no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—88no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#38612yes—
434C2×C4#38612yes—
435C2×C4#38612yes—
436C2×C4C22×C22⋊C411yescharacteristic
437C2×C4C2×C4×D411yescharacteristic
438C2×C4C2×C4×D411yescharacteristic
439C2×C4C2×#12011yescharacteristic
440C2×C4C2×#12111yescharacteristic
441C2×C4C2×#12311yescharacteristic
442C2×C4#57612yes—
443C2×C4#57712yes—
444C2×C4#57711yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57811yescharacteristic
447C2×C4#57811yescharacteristic
448C8—44no—
449C8—42no—
450C8—42no—
451C8—42no—
452C8—42no—
453C8—81no—
454C8—81no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—42no—
459C22—42no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—162no—
464C22—162no—
465C22C2×#38611yescharacteristic
466C22#304311yescharacteristic
467C22#304411yescharacteristic
468C22#306212yes—
469C22#306712yes—
470C4—21no—
471C4—21no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—44no—
476C4—44no—
477C4—82no—
478C4—162no—
479C4#289712yes—
480C4#294711yescharacteristic
481C4#294711yescharacteristic
482C4#301311yescharacteristic
483C4#301311yescharacteristic
484C4#305711yescharacteristic
485C4#305811yescharacteristic
486C2—42no—
487C2—162no—
488C2#1956912yes—
489C2#1957512yes—
490C2#2093011yescharacteristic
491C2#2097511yescharacteristic
492C2#2100311yescharacteristic
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
2. C2×#2986C2semidirect
3. #21885C2semidirect
6. #9230C2semidirect
7. #9226C2semidirect
8. #9222C2semidirect
9. #9218C2semidirect
10. #8046C2semidirect
11. #7879C2semidirect
15. #3216C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32162
C2×#29862
#218852
#218632
C2×#15402
#92302
#92262
#92222
#92182
#80462
#78792