Finite Group Databasebeta

← #194942

Subgroups — #194942

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2640C211yesmaximal, characteristic
2#34377C211yesmaximal, characteristic
3#21840C211yesmaximal, characteristic, complemented
4#21833C211yesmaximal, characteristic, complemented
5C2×#1469C211yesmaximal, characteristic
6#9028C214yesmaximal, complemented
7#9025C214yesmaximal, complemented
8#8022C211yesmaximal, characteristic, complemented
9#8013C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#124C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2640—24no—
15#2640C418yescomplemented
16C2×#397C2214yes—
17C2×#393—22no—
18#2379—22no—
19#2367—22no—
20#2342—22no—
21Q8×C22⋊C4—21no—
22D4×C22⋊C4—21no—
23#2337—22no—
24#2326—22no—
25#2315C2212yes—
26#2312C2212yes—
27C2×#391—21no—
28C2×#390—21no—
29#2285C2211yescharacteristic
30#2283C2211yescharacteristic
31C2×#376C2211yescharacteristic
32#1993—22no—
33#1990—22no—
34#1988—22no—
35#1987—22no—
36#1823C2212yes—
37#1823C2214yes—
38#1823C2212yes—
39#1469—24no—
40C2×#323C2214yes—
41#1294C2212yes—
42#1294C2214yes—
43#1294C2212yes—
44C23×Q8—21no—
45C23×Q8D411yescharacteristic
46C23×D4D411yescharacteristic
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yescomplemented
55C2×#124—21no—
56C2×#124—21no—
57C2×#124—21no—
58C2×#124—21no—
59C2×#124—22no—
60C2×#124—22no—
61C2×#124C2×C414yes—
62C2×#124D414yes—
63C2×C4×Q8—22no—
64C2×C4×D4—22no—
65C2×#117—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—24no—
70C22×C22⋊C4D412yes—
71C22×C22⋊C4C2312yes—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#432—22no—
75#432—22no—
76#431—22no—
77#431—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2312yes—
82C2×#97—21no—
83C2×C4×C8—21no—
84#397—28no—
85#397C2×C4116yescomplemented
86#393—216no—
87#391—22no—
88#391—24no—
89#391—28no—
90#390—22no—
91#390—28no—
92#388—24no—
93#388—24no—
94#388—24no—
95#388—24no—
96#386—24no—
97#385—24no—
98#385—24no—
99#385—24no—
100#384D412yes—
101#383—21no—
102#382—21no—
103#382D412yes—
104#381—21no—
105#380—22no—
106#380—21no—
107#380C2311yescharacteristic
108#380C2312yes—
109#380C2311yescharacteristic
110#379—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4D412yes—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—21no—
117#376—21no—
118#376—22no—
119#376C2×C414yescomplemented
120C2×#95—21no—
121C2×#95—24no—
122C2×#95C2312yes—
123#340—24no—
124#340—24no—
125#340—28no—
126#323—28no—
127C25C2×D412yes—
128C22×Q8—24no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—44no—
133C22×Q8—41no—
134C22×Q8—41no—
135C22×Q8—42no—
136C22×Q8—82no—
137C22×Q8—82no—
138C22×Q8C22⋊C414yes—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—44no—
143C22×D4C22⋊C414yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—24no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161#124—24no—
162#124—24no—
163#124—24no—
164#124—24no—
165#124—24no—
166#124—24no—
167#124—24no—
168#124—24no—
169C4×Q8—84no—
170C4×D4—84no—
171#117—82no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C22×C414yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D414yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—216no—
202C2×C22⋊C4—84no—
203C2×C22⋊C4—84no—
204C2×C22⋊C4—84no—
205C2×C22⋊C4C22⋊C418yes—
206C2×C22⋊C4C22×C414yes—
207C2×C22⋊C4C2×D414yes—
208C2×C42—24no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—22no—
212C2×C42—44no—
213C2×C42C22×C412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C4⋊C8—42no—
222C4⋊C8—42no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227#97—42no—
228C4×C8—42no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—216no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95C22×C414yes—
242C24—22no—
243C24—22no—
244C24—24no—
245C24—24no—
246C24—44no—
247C24—44no—
248C24C2×C22⋊C414yes—
249C24C22×D411yescharacteristic
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—48no—
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—82no—
257C2×Q8—88no—
258C2×Q8—84no—
259C2×Q8—88no—
260C2×Q8—82no—
261C2×Q8—82no—
262C2×Q8—82no—
263C2×Q8—82no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—48no—
267C2×D4—44no—
268C2×D4—84no—
269C2×D4—82no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C414yes—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12312yes—
301C22×C4#12312yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—82no—
323C4⋊C4—84no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—28no—
329C22⋊C4—28no—
330C22⋊C4—42no—
331C22⋊C4—44no—
332C22⋊C4—42no—
333C22⋊C4—816no—
334C22⋊C4—84no—
335C22⋊C4—84no—
336C22⋊C4—84no—
337C42—48no—
338C42—44no—
339C42—82no—
340C42—84no—
341C42C4×D412yes—
342C42C4×D412yes—
343C42C4×D412yes—
344C42C4×D412yes—
345C42#12112yes—
346C42#12112yes—
347C42#12112yes—
348C42#12112yes—
349C23—22no—
350C23—22no—
351C23—24no—
352C23—28no—
353C23—44no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23—82no—
361C23—82no—
362C23—88no—
363C23—88no—
364C23C22×C22⋊C411yescharacteristic
365C23C2×#12011yescharacteristic
366C23C2×#12311yescharacteristic
367C23#57612yes—
368C23#57712yes—
369Q8—48no—
370Q8—84no—
371Q8—88no—
372Q8—88no—
373D4—48no—
374D4—84no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—48no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—82no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—168no—
414C2×C4#38612yes—
415C2×C4#38612yes—
416C2×C4#38612yes—
417C2×C4#38612yes—
418C2×C4#45514yes—
419C2×C4C2×C4×D411yescharacteristic
420C2×C4C2×C4×D411yescharacteristic
421C2×C4C2×#12111yescharacteristic
422C2×C4C2×#12111yescharacteristic
423C8—82no—
424C8—82no—
425C22—24no—
426C22—24no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—44no—
432C22—44no—
433C22—44no—
434C22—88no—
435C22—84no—
436C22—84no—
437C22—164no—
438C22C2×#38611yescharacteristic
439C22#289712yes—
440C22#300811yescharacteristic
441C22#301011yescharacteristic
442C22C2×#45512yes—
443C22#302812yes—
444C22#309612yes—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—44no—
453C4—84no—
454C4—84no—
455C4—82no—
456C4—82no—
457C4—84no—
458C4#188514yes—
459C2—22no—
460C2—22no—
461C2—84no—
462C2#1843612yes—
463C2#1843812yes—
464C2#2091411yescharacteristic
465C2C2×#188511yescharacteristic
466C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21840C2semidirect
4. #21833C2semidirect
6. #9028C2semidirect
7. #9025C2semidirect
8. #8022C2semidirect
9. #8013C2semidirect
15. #2640C4semidirect
54. C2×C4⋊Q8C2×C4semidirect
85. #397C2×C4semidirect
119. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26402
#343772
#218402
#218332
C2×#14692
#90282
#90252
#80222
#80132