Finite Group Databasebeta

← #196852

Subgroups — #196852

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2C2×#2356C211yesmaximal, characteristic, complemented
3#21818C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic
5C2×#1502C211yesmaximal, characteristic
6#9157C214yesmaximal, complemented
7#9149C214yesmaximal, complemented
8#8026C211yesmaximal, characteristic, complemented
9#8014C211yesmaximal, characteristic, complemented
10C2×#536—21no—
11C2×#532C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2854C2214yes—
16#2618—22no—
17#2548C412yescomplemented
18#2546C412yescomplemented
19#2545—22no—
20#2538C414yescomplemented
21#2536—22no—
22#2356—24no—
23#2356—24no—
24#2311—21no—
25C2×#391—21no—
26C2×#387—22no—
27#2285C2211yescharacteristic
28C2×#383C2212yes—
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2212yes—
33#1984—22no—
34#1984—22no—
35#1817C2214yes—
36#1517—24no—
37#1502—24no—
38C2×#327C2212yes—
39C2×#327C2212yes—
40C2×#327C2214yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C22×#41—21no—
45C23×Q8D411yescharacteristic
46#536—44no—
47#536—44no—
48#532—22no—
49#532—22no—
50#532—22no—
51#532—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#124—22no—
56C2×#124—21no—
57C2×#123—24no—
58C2×C22⋊Q8—24no—
59C2×#121—21no—
60C2×#121C2312yes—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8C2×C412yes—
67C2×C4×Q8D412yes—
68C2×C4×Q8D412yes—
69C2×C4×D4—21no—
70C2×C4×D4C2×C412yes—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C42—22no—
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#431—22no—
81#431—22no—
82#431—22no—
83#431—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C22⋊C8C2312yes—
87C2×C22⋊C8C2312yes—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90#398—24no—
91#396—28no—
92#394—24no—
93#391—28no—
94#391—22no—
95#390—22no—
96#390C2×C418yes—
97#389C2×C418yes—
98#388—28no—
99#387—216no—
100#386—24no—
101#384C2×C412yes—
102#383—24no—
103#383—24no—
104#382—22no—
105#382—21no—
106#382D414yes—
107#380—21no—
108#380—24no—
109#380C2312yes—
110C4×C4⋊C4—28no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4D414yes—
115#376—24no—
116#376—24no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95C2311yescharacteristic
122C2×#95C2312yes—
123C2×#95C2311yescharacteristic
124#336—28no—
125#327—24no—
126#327—24no—
127#327—28no—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—42no—
132C2×#41—42no—
133C2×#41—42no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8C22⋊C412yes—
142C22×Q8C22⋊C412yes—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C22×C412yes—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—48no—
161#127—22no—
162#127—22no—
163#124—48no—
164#124—42no—
165#124—42no—
166#123—48no—
167#123—48no—
168C22⋊Q8—416no—
169#121—22no—
170#121—22no—
171#121—22no—
172#121—22no—
173#121—42no—
174#121—42no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—42no—
186C4×D4—42no—
187#117—42no—
188#117—42no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C414yes—
205C2×C4⋊C4C22⋊C414yes—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C414yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—28no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—28no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—48no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42C22×C414yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D414yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4112yes—
232C2×C42#4112yes—
233C4⋊C8—42no—
234C4⋊C8—42no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C4×C8—42no—
240C4×C8—42no—
241#95—28no—
242#95—28no—
243#95—28no—
244#95—22no—
245#95—28no—
246#95—24no—
247#95—21no—
248#95—22no—
249#95—21no—
250#95—22no—
251#95C2×D412yes—
252#95C2×D414yes—
253#95C2×D412yes—
254C24—22no—
255C24C22×D411yescharacteristic
256#41—84no—
257#41—84no—
258#41—84no—
259#41—84no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—44no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—82no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4C2×C22⋊C414yes—
319C22×C4C2×C22⋊C414yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4#12012yes—
325C22×C4#12012yes—
326C22×C4#12312yes—
327C22×C4#12312yes—
328C22×C4#13512yes—
329C22×C4#13512yes—
330C22×C4AGL(1,ℤ/8)12yes—
331C22×C4AGL(1,ℤ/8)12yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C2×C8—44no—
339C2×C8—44no—
340C2×C8—42no—
341C2×C8—42no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—28no—
345C4⋊C4—28no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—44no—
349C4⋊C4—48no—
350C4⋊C4—416no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—48no—
354C4⋊C4—82no—
355C22⋊C4—416no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—44no—
361C22⋊C4—88no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—24no—
366C42—416no—
367C42—44no—
368C42—48no—
369C42C4×D412yes—
370C42C4×D412yes—
371C42C4×D412yes—
372C42C4×D412yes—
373C42#12112yes—
374C42#12112yes—
375C42#12112yes—
376C42#12112yes—
377C23—22no—
378C23—22no—
379C23—44no—
380C23—42no—
381C23—42no—
382C23—44no—
383C23—42no—
384C23C22×C22⋊C411yescharacteristic
385C23C2×#12011yescharacteristic
386C23C2×#12311yescharacteristic
387C23C2×#13511yescharacteristic
388C23C2×#13511yescharacteristic
389C23C2×AGL(1,ℤ/8)11yescharacteristic
390C23C2×AGL(1,ℤ/8)11yescharacteristic
391Q8—48no—
392Q8—48no—
393Q8—84no—
394D4—82no—
395D4—82no—
396D4—84no—
397D4—82no—
398D4—82no—
399D4—82no—
400D4—82no—
401D4—84no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—28no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—84no—
442C2×C4—82no—
443C2×C4—84no—
444C2×C4—88no—
445C2×C4#38612yes—
446C2×C4#38612yes—
447C2×C4#38612yes—
448C2×C4#38612yes—
449C2×C4#41612yes—
450C2×C4#41612yes—
451C2×C4#41812yes—
452C2×C4#41812yes—
453C2×C4#44914yes—
454C2×C4#44914yes—
455C2×C4#48214yes—
456C2×C4#48214yes—
457C2×C4C2×C4×D411yescharacteristic
458C2×C4C2×C4×D411yescharacteristic
459C2×C4C2×#12111yescharacteristic
460C2×C4C2×#12111yescharacteristic
461C8—82no—
462C8—82no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—82no—
472C22—84no—
473C22—84no—
474C22—82no—
475C22—82no—
476C22C2×#38611yescharacteristic
477C22#288412yes—
478C22C2×#41611yescharacteristic
479C22C2×#41811yescharacteristic
480C22#290112yes—
481C22C2×#44911yescharacteristic
482C22C2×#44911yescharacteristic
483C22C2×#48211yescharacteristic
484C22C2×#48211yescharacteristic
485C4—22no—
486C4—22no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—44no—
492C4—44no—
493C4—44no—
494C4—82no—
495C4—84no—
496C4—82no—
497C4—84no—
498C4#187514yes—
499C4#187514yes—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—162no—
505C2#1761612yes—
506C2#1763112yes—
507C2C2×#187511yescharacteristic
508C2#2098011yescharacteristic
509C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2356C2semidirect
3. #21818C2semidirect
6. #9157C2semidirect
7. #9149C2semidirect
8. #8026C2semidirect
9. #8014C2semidirect
17. #2548C4semidirect
18. #2546C4semidirect
20. #2538C4semidirect

Maximal subgroups

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

subgroupindex
#351782
C2×#23562
#218182
C2×#15172
C2×#15022
#91572
#91492
#80262
#80142