Finite Group Databasebeta

← #196780

Subgroups — #196780

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2991C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#21849C211yesmaximal, characteristic
4C2×#1549C211yesmaximal, characteristic, complemented
5C2×#1546C211yesmaximal, characteristic, complemented
6#9139C214yesmaximal, complemented
7#9132C214yesmaximal, complemented
8#8041C211yesmaximal, characteristic, complemented
9#8038C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#2991—24no—
13#2991—24no—
14C2×C8×Q8—21no—
15C2×#436—22no—
16C2×#436—21no—
17C2×#431C2211yescharacteristic
18C2×#431C2212yes—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#423C2212yes—
22C2×#423C2211yescharacteristic
23#2857C2214yes—
24#2628C414yes—
25#2568—21no—
26#2567—21no—
27#2563—21no—
28#2542—21no—
29#2487—22no—
30#2409—22no—
31#2317C412yes—
32#2311C412yes—
33#2285C2211yescharacteristic
34#1984—21no—
35#1954—21no—
36#1870—22no—
37#1797C2214yes—
38#1549—24no—
39#1546—24no—
40C2×#332C2214yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1296C2214yes—
46C22×#41C2×C411yescharacteristic
47C23×Q8—21no—
48C22×M16—22no—
49C22×M16—21no—
50C22×M16—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C412yes—
58C2×C4⋊Q8C2312yes—
59C2×C22⋊Q8—24no—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41—22no—
63C4×#41—22no—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D412yes—
68C2×C4×D4D412yes—
69C2×C4×D4D411yescharacteristic
70C2×#117C2×C412yes—
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C412yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C412yes—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C8×Q8—28no—
78#436—216no—
79#436—28no—
80#431—22no—
81#431—24no—
82#431—24no—
83#431—22no—
84#431—22no—
85#431D414yes—
86#423—22no—
87#423—24no—
88#423—24no—
89#423—22no—
90#423—22no—
91#423D414yes—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2311yescharacteristic
100C2×C22⋊C8—22no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—24no—
104C2×C4×C8—22no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107#400—24no—
108#399—24no—
109#395—24no—
110#393—24no—
111#391—22no—
112#391—24no—
113#390—22no—
114#389—24no—
115#389C2×C418yes—
116#386—24no—
117#385—24no—
118#384—22no—
119#384C2312yes—
120#382—22no—
121#382C2×C412yes—
122#380—24no—
123#380—21no—
124#380—21no—
125#380C2×C418yes—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128#336—24no—
129#332—28no—
130#332—24no—
131#332—24no—
132C2×#41—24no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—42no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—82no—
141C22×Q8—82no—
142C22×Q8C22⋊C411yescharacteristic
143C22×D4C22⋊C411yescharacteristic
144C22×D4C22⋊C412yes—
145C23×C4—22no—
146C23×C4C22×C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×M16—28no—
153C2×M16—24no—
154C2×M16—42no—
155C2×M16—82no—
156C2×M16—84no—
157C2×M16—82no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C4⋊Q8—24no—
164C4⋊Q8—24no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C22⋊Q8—84no—
174C22⋊Q8—84no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187#117—22no—
188#117—22no—
189#117—24no—
190#117—24no—
191#117—44no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C22×C414yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22⋊C412yes—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42C22×C412yes—
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42C2411yescharacteristic
229C4⋊C8—28no—
230C4⋊C8—24no—
231C4⋊C8—22no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—42no—
239C4⋊C8C2×D412yes—
240C4⋊C8C2×D414yes—
241C4⋊C8C2×D412yes—
242C22⋊C8—28no—
243C22⋊C8—24no—
244C22⋊C8—42no—
245C22⋊C8—84no—
246C4×C8—28no—
247C4×C8—24no—
248C4×C8—42no—
249#95—22no—
250#95—22no—
251#95—28no—
252#95—28no—
253C24C2×C22⋊C412yes—
254C24C2×C22⋊C411yescharacteristic
255#41—48no—
256#41—48no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—84no—
265C2×Q8—88no—
266C2×Q8—84no—
267C2×Q8—84no—
268C2×Q8#10114yes—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—42no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—84no—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C4×D412yes—
304C22×C4C4×D412yes—
305C22×C4#12012yes—
306C22×C4#12112yes—
307C22×C4#12112yes—
308C22×C4#12312yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312M16—84no—
313M16—88no—
314M16—84no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—22no—
318C2×C8—44no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—44no—
325C2×C8—42no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—88no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—44no—
346C22⋊C4—88no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—44no—
354C42C4×D412yes—
355C42C4×D412yes—
356C42#12012yes—
357C42#12112yes—
358C42#12112yes—
359C42#12312yes—
360C42C22×D411yescharacteristic
361C42C22×D411yescharacteristic
362C42C2×#4111yescharacteristic
363C42C2×#4111yescharacteristic
364C23—21no—
365C23—21no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—24no—
371C23—24no—
372C23—22no—
373C23#38612yes—
374C23C22×C22⋊C411yescharacteristic
375Q8—28no—
376Q8—88no—
377Q8—84no—
378Q8—82no—
379Q8—82no—
380D4—48no—
381D4—44no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—48no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—88no—
414C2×C4#38612yes—
415C2×C4#38612yes—
416C2×C4#38612yes—
417C2×C4C2×#10114yes—
418C2×C4C2×#10112yes—
419C2×C4#41312yes—
420C2×C4C2×C4×D411yescharacteristic
421C2×C4C2×C4×D411yescharacteristic
422C2×C4C2×#12011yescharacteristic
423C2×C4C2×#12111yescharacteristic
424C2×C4C2×#12111yescharacteristic
425C2×C4C2×#12311yescharacteristic
426C2×C4#57612yes—
427C2×C4#57712yes—
428C2×C4#57712yes—
429C2×C4#57812yes—
430C8—44no—
431C8—42no—
432C8—42no—
433C8—82no—
434C8—82no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—42no—
439C22—42no—
440C22—44no—
441C22—44no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22C2×#38611yescharacteristic
446C22C22×#10111yescharacteristic
447C22C2×#41311yescharacteristic
448C22#289712yes—
449C22#289812yes—
450C4—22no—
451C4—22no—
452C4—21no—
453C4—21no—
454C4—24no—
455C4—42no—
456C4—42no—
457C4—84no—
458C4—84no—
459C4#190114yes—
460C4#300811yescharacteristic
461C4#301011yescharacteristic
462C4#305711yescharacteristic
463C4#305911yescharacteristic
464C4#306312yes—
465C4#309612yes—
466C2—42no—
467C2—82no—
468C2#1777712yes—
469C2#1778112yes—
470C2#2091411yescharacteristic
471C2#2097411yescharacteristic
472C2C2×#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×#1549C2semidirect
5. C2×#1546C2semidirect
6. #9139C2semidirect
7. #9132C2semidirect
8. #8041C2semidirect
9. #8038C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29912
#352792
#218492
C2×#15492
C2×#15462
#91392
#91322
#80412
#80382