Finite Group Databasebeta

← #196432

Subgroups — #196432

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic, complemented
2#35249C211yesmaximal, characteristic
3C2×#1517C211yesmaximal, characteristic
4C2×#1514C211yesmaximal, characteristic
5#14346C211yesmaximal, characteristic, complemented
6#9118C214yesmaximal, complemented
7#9114C214yesmaximal, complemented
8#8025C211yesmaximal, characteristic, complemented
9#8017C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2984—24no—
13#2984—24no—
14C2×#446—22no—
15C2×#446—21no—
16C2×C8×Q8—21no—
17C2×#432C2212yes—
18C2×#432C2211yescharacteristic
19C2×#431C2212yes—
20C2×#431C2211yescharacteristic
21#2854C2214yes—
22#2632—21no—
23#2632—21no—
24#2580C412yescomplemented
25#2580C412yescomplemented
26#2567C412yescomplemented
27#2548C412yescomplemented
28#2501—21no—
29#2436—21no—
30#2406—22no—
31#2326—22no—
32#2283C2211yescharacteristic
33C2×#383C2211yescharacteristic
34C2×#382C2211yescharacteristic
35#1517—24no—
36#1514—24no—
37C2×#327C2214yes—
38C2×#327C2212yes—
39C2×#327C2212yes—
40#1446—21no—
41#1445—21no—
42#1441—22no—
43#1434C2214yes—
44#1294C2214yes—
45#1294C2212yes—
46#1294C2212yes—
47C22×#41—21no—
48C23×Q8D411yescharacteristic
49#532—22no—
50#532—22no—
51#532—22no—
52#532—22no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#124—22no—
56C2×#124—21no—
57C2×#123—24no—
58C2×#121—21no—
59C2×#121C2312yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8D412yes—
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2×C412yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4D411yescharacteristic
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77#446—216no—
78#446—28no—
79C8×Q8—28no—
80#432—22no—
81#432—24no—
82#432—24no—
83#432D414yes—
84#431—22no—
85#431—24no—
86#431—24no—
87#431D414yes—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—21no—
92C2×C22⋊C8C2311yescharacteristic
93C2×C22⋊C8C2312yes—
94C2×C22⋊C8C2311yescharacteristic
95C2×#97—22no—
96C2×#97—21no—
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99#399—22no—
100#399—22no—
101#395—22no—
102#395—22no—
103#394—22no—
104#394—22no—
105#393—22no—
106#393—22no—
107#391—22no—
108#390—22no—
109#390C2×C414yes—
110#390C2×C414yes—
111#389C2×C414yes—
112#389C2×C414yes—
113#388—24no—
114#388—24no—
115#386—24no—
116#385—24no—
117#385—24no—
118#384—22no—
119#383—21no—
120#383—22no—
121#383—22no—
122#382—22no—
123#382—22no—
124#382C2×C412yes—
125#381—21no—
126#380C2312yes—
127C4×C4⋊C4C2×C412yes—
128C4×C22⋊C4—21no—
129#376—21no—
130C2×#95—24no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—24no—
135C2×#95C2312yes—
136C2×#95C2312yes—
137#327—28no—
138#327—24no—
139#327—24no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—82no—
143C2×#41—82no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—44no—
148C22×Q8C22⋊C411yescharacteristic
149C22×Q8C22⋊C412yes—
150C22×Q8C22⋊C411yescharacteristic
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4C22×C412yes—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168#127—22no—
169#127—22no—
170#124—48no—
171#124—42no—
172#124—42no—
173#123—84no—
174#123—84no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—22no—
179#121—42no—
180#121—42no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8—24no—
189C4×Q8—22no—
190C4×Q8—24no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—42no—
196C4×D4—42no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C414yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42C22×C412yes—
228C2×C42C22×C412yes—
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4112yes—
238C2×C42#4112yes—
239C4⋊C8—28no—
240C4⋊C8—28no—
241C4⋊C8—24no—
242C4⋊C8—24no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—22no—
246C22⋊C8C2×D412yes—
247C22⋊C8C2×D414yes—
248C22⋊C8C2×D412yes—
249#97—28no—
250#97—24no—
251C4×C8—28no—
252C4×C8—24no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—24no—
257#95—24no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—48no—
264#95—42no—
265#95—42no—
266#95—42no—
267#95—84no—
268C24—22no—
269C24C22×D411yescharacteristic
270#41—84no—
271#41—84no—
272#41—88no—
273C2×Q8—22no—
274C2×Q8—24no—
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—48no—
281C2×Q8—48no—
282C2×Q8—42no—
283C2×Q8—82no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—44no—
287C2×D4—44no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—82no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—84no—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C412yes—
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4#12112yes—
331C22×C4#12112yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—22no—
341C2×C8—42no—
342C2×C8—44no—
343C2×C8—42no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C22⋊C4—48no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—44no—
364C22⋊C4—48no—
365C22⋊C4—84no—
366C22⋊C4—84no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—44no—
376C42C4×D412yes—
377C42C4×D412yes—
378C42C4×D412yes—
379C42C4×D412yes—
380C42#12012yes—
381C42#12012yes—
382C42#12312yes—
383C42#12312yes—
384C23—22no—
385C23—22no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—42no—
390C23C22×C22⋊C411yescharacteristic
391C23C2×#12111yescharacteristic
392C23C2×#12111yescharacteristic
393C23#57612yes—
394C23#57712yes—
395Q8—28no—
396Q8—48no—
397Q8—44no—
398Q8—84no—
399D4—82no—
400D4—82no—
401D4—84no—
402D4—84no—
403D4—82no—
404D4—82no—
405D4—84no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—88no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—84no—
444C2×C4—88no—
445C2×C4—84no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4#38612yes—
451C2×C4#38612yes—
452C2×C4#38612yes—
453C2×C4#38612yes—
454C2×C4#42114yes—
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×C4×D411yescharacteristic
457C2×C4C2×#12011yescharacteristic
458C2×C4C2×#12311yescharacteristic
459C8—42no—
460C8—44no—
461C8—42no—
462C22—21no—
463C22—21no—
464C22—24no—
465C22—24no—
466C22—21no—
467C22—21no—
468C22—82no—
469C22—88no—
470C22—82no—
471C22—82no—
472C22C2×#38611yescharacteristic
473C22#288412yes—
474C22#289712yes—
475C22C2×#42112yes—
476C22#305711yescharacteristic
477C22#305811yescharacteristic
478C22#306212yes—
479C4—22no—
480C4—22no—
481C4—24no—
482C4—22no—
483C4—22no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—84no—
492C4—82no—
493C4—162no—
494C4—162no—
495C2—21no—
496C2—22no—
497C2—21no—
498C2—162no—
499C2#1760214yes—
500C2#2090211yescharacteristic
501C2#2097511yescharacteristic
502C2#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
1. C2×#2984C2semidirect
5. #14346C2semidirect
6. #9118C2semidirect
7. #9114C2semidirect
8. #8025C2semidirect
9. #8017C2semidirect
24. #2580C4semidirect
25. #2580C4semidirect
26. #2567C4semidirect
27. #2548C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
#352492
C2×#15172
C2×#15142
#143462
#91182
#91142
#80252
#80172