Finite Group Databasebeta

← #196594

Subgroups — #196594

Order 512.

1182 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2971C211yesmaximal, characteristic
2#35333C211yesmaximal, characteristic
3#21380C211yesmaximal, characteristic
4C2×#1528C211yesmaximal, characteristic, complemented
5C2×#1528C211yesmaximal, characteristic, complemented
6#9125C214yesmaximal, complemented
7#9122C214yesmaximal, complemented
8#8031C211yesmaximal, characteristic, complemented
9#8028C211yesmaximal, characteristic, complemented
10C22×#125C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#2971—24no—
13#2971—24no—
14C2×C8×Q8—21no—
15C2×#436—21no—
16C2×#436—22no—
17C2×#432C2212yes—
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×C4×M16C2212yes—
21C2×C4×M16C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2851C2214yes—
24#2629C414yes—
25#2579—22no—
26#2579—22no—
27#2566—21no—
28#2566—21no—
29#2565—21no—
30#2565—21no—
31#2317C411yescharacteristic
32#2317C411yescharacteristic
33#2311C411yescharacteristic
34#2311C411yescharacteristic
35#2285C2211yescharacteristic
36#1984—21no—
37#1954—21no—
38#1870—22no—
39#1749C2214yes—
40#1528—24no—
41#1528—24no—
42C2×#330C2212yes—
43C2×#330C2214yes—
44C2×#330C2212yes—
45#1295C2212yes—
46#1295C2212yes—
47#1295C2214yes—
48C22×#41C2×C411yescharacteristic
49C22×M16—22no—
50C22×M16—21no—
51C22×M16—21no—
52C2×#125—21no—
53C2×#125—21no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125C2×C412yes—
59C2×#125C2312yes—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41—22no—
63C4×#41—22no—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D412yes—
69C2×C4×D4D412yes—
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C412yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C8×Q8—28no—
81#436—216no—
82#436—28no—
83#432—22no—
84#432—22no—
85#432—22no—
86#432—24no—
87#432—24no—
88#432D414yes—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—24no—
96C4×M16—22no—
97C4×M16—24no—
98C4×M16—24no—
99C4×M16—22no—
100C4×M16—22no—
101C4×M16D414yes—
102C2×#97C2312yes—
103C2×#97C2312yes—
104C2×#97C2311yescharacteristic
105C2×#97C2312yes—
106C2×#97C2311yescharacteristic
107C2×C4×C8—22no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110#398—24no—
111#398—24no—
112#396—24no—
113#396—24no—
114#391—21no—
115#391—21no—
116#390—21no—
117#390—21no—
118#389C2×C414yes—
119#389C2×C414yes—
120#387—24no—
121#387—24no—
122#386—22no—
123#386—22no—
124#385—24no—
125#385—24no—
126#385—22no—
127#385—22no—
128#384—21no—
129#384—21no—
130#384C2×C412yes—
131#382—21no—
132#382—21no—
133#382C2312yes—
134#380—21no—
135#380—21no—
136#380C2×C414yes—
137#380C2×C414yes—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140#336—24no—
141#330—28no—
142#330—24no—
143#330—24no—
144C2×#41—24no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—42no—
148C22×Q8C22⋊C411yescharacteristic
149C22×D4C22⋊C411yescharacteristic
150C22×D4C22⋊C412yes—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C2×M16—28no—
160C2×M16—24no—
161C2×M16—44no—
162C2×M16—48no—
163C2×M16—44no—
164C2×M16—42no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—22no—
170#125—24no—
171#125—24no—
172#125—22no—
173#125—22no—
174#125—22no—
175#125—22no—
176#125—42no—
177#125—42no—
178#125—42no—
179#125—42no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192#117—22no—
193#117—22no—
194#117—24no—
195#117—24no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C22⋊C412yes—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42C22×C412yes—
238C2×C42C22×C411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4112yes—
246C2×C42#4112yes—
247C2×C42C2411yescharacteristic
248C4⋊C8—28no—
249C4⋊C8—24no—
250C4⋊C8—42no—
251C22⋊C8—28no—
252C22⋊C8—24no—
253C22⋊C8—48no—
254C22⋊C8—42no—
255#97—21no—
256#97—21no—
257#97—22no—
258#97—22no—
259#97—22no—
260#97—22no—
261#97—22no—
262#97C2×D412yes—
263#97C2×D414yes—
264#97C2×D412yes—
265C4×C8—28no—
266C4×C8—24no—
267C4×C8—42no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—22no—
273#95—22no—
274C24C2×C22⋊C412yes—
275C24C2×C22⋊C411yescharacteristic
276#41—48no—
277#41—48no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8#10114yes—
281C2×D4—24no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—42no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C4×D412yes—
316C22×C4C4×D412yes—
317C22×C4#12012yes—
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4#12312yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324M16—48no—
325M16—416no—
326M16—48no—
327C2×C8—24no—
328C2×C8—22no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—22no—
332C2×C8—42no—
333C2×C8—44no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—44no—
337C2×C8—42no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—84no—
366C22⋊C4—84no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—42no—
374C42—42no—
375C42C4×D412yes—
376C42C4×D412yes—
377C42#12012yes—
378C42#12112yes—
379C42#12112yes—
380C42#12312yes—
381C42C22×D411yescharacteristic
382C42C22×D411yescharacteristic
383C42C2×#4111yescharacteristic
384C42C2×#4111yescharacteristic
385C23—21no—
386C23—21no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—24no—
392C23—24no—
393C23—22no—
394C23#38612yes—
395C23C22×C22⋊C411yescharacteristic
396Q8—28no—
397D4—48no—
398D4—44no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4#38612yes—
436C2×C4#38612yes—
437C2×C4#38612yes—
438C2×C4C2×#10114yes—
439C2×C4C2×#10112yes—
440C2×C4#41312yes—
441C2×C4C2×C4×D411yescharacteristic
442C2×C4C2×C4×D411yescharacteristic
443C2×C4C2×#12011yescharacteristic
444C2×C4C2×#12111yescharacteristic
445C2×C4C2×#12111yescharacteristic
446C2×C4C2×#12311yescharacteristic
447C8—24no—
448C8—44no—
449C8—44no—
450C8—42no—
451C8—82no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—42no—
456C22—42no—
457C22—44no—
458C22—44no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22C2×#38611yescharacteristic
463C22C22×#10111yescharacteristic
464C22C2×#41311yescharacteristic
465C22#290112yes—
466C22#290112yes—
467C4—21no—
468C4—21no—
469C4—22no—
470C4—21no—
471C4—21no—
472C4—22no—
473C4—21no—
474C4—21no—
475C4—24no—
476C4—42no—
477C4—42no—
478C4—82no—
479C4—82no—
480C4—82no—
481C4—82no—
482C4#190114yes—
483C2—42no—
484C2—82no—
485C2#1768712yes—
486C2#1769212yes—
487C2#2090311yescharacteristic
488C2#2098011yescharacteristic
489C2C2×#190111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1528C2semidirect
5. C2×#1528C2semidirect
6. #9125C2semidirect
7. #9122C2semidirect
8. #8031C2semidirect
9. #8028C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29712
#353332
#213802
C2×#15282
#91252
#91222
#80312
#80282