Finite Group Databasebeta

← #195728

Subgroups — #195728

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic
2#34393C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic, complemented
4#21811C211yesmaximal, characteristic, complemented
5C2×#1553C211yesmaximal, characteristic
6#9059C214yesmaximal, complemented
7#9057C214yesmaximal, complemented
8#7900C211yesmaximal, characteristic, complemented
9#7894C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3227—24no—
13#3227C418yescomplemented
14C2×#498C2212yes—
15C2×#486C2212yes—
16C2×#481—22no—
17C2×#476—22no—
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2377—22no—
22#2366C2212yes—
23#2365C2212yes—
24#2357—21no—
25#2349—21no—
26#2341—21no—
27#2339—21no—
28#2326—22no—
29#2324—22no—
30#2318—22no—
31#2285C2211yescharacteristic
32#2283C2211yescharacteristic
33#1990—22no—
34#1988C2214yes—
35#1981C2214yes—
36#1960—21no—
37#1960—21no—
38#1823—21no—
39#1823—21no—
40#1787—22no—
41#1553—24no—
42C2×#332C2212yes—
43C2×#331C2212yes—
44#1279C2214yes—
45#1279C2214yes—
46C22×#41—21no—
47C23×Q8—21no—
48#532—22no—
49#532—22no—
50#532C2×C414yes—
51#532D414yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yescomplemented
59C2×C4⋊Q8C2312yes—
60C2×C4⋊Q8C2311yescharacteristic
61C2×#127C2311yescharacteristic
62C2×#124—21no—
63C2×#121—21no—
64C2×#121D412yes—
65C2×C4×Q8—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4D412yes—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#498—24no—
75#498C2×C418yescomplemented
76#486—24no—
77#486C2×C418yes—
78#481—216no—
79#476—216no—
80#432—22no—
81#432—22no—
82#431—22no—
83#431—22no—
84#431—22no—
85#431C2×C414yescomplemented
86C2×#106—24no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×#103—24no—
91C2×#103—24no—
92C2×#102—24no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×#97—21no—
97C2×C4×C8—21no—
98C2×C4×C8C2312yes—
99#391—22no—
100#390—24no—
101#390—22no—
102#390—22no—
103#389—22no—
104#388—24no—
105#388—24no—
106#387—24no—
107#386—24no—
108#386—24no—
109#385—24no—
110#385—24no—
111#384D412yes—
112#383—21no—
113#382—21no—
114#382D411yescharacteristic
115#382D411yescharacteristic
116#381—22no—
117#381—21no—
118#380—22no—
119#380—21no—
120#380D412yes—
121#380D411yescharacteristic
122#380D411yescharacteristic
123C4×C4⋊C4—22no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2312yes—
126C4×C4⋊C4C2312yes—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—21no—
129#376—21no—
130C2×#95—22no—
131#340—24no—
132#340—24no—
133#332—24no—
134#331—24no—
135C2×#41—24no—
136C2×#41—24no—
137C2×#41—84no—
138C22×Q8—21no—
139C22×Q8—24no—
140C22×Q8—22no—
141C22×Q8—41no—
142C22×Q8—42no—
143C22×Q8—82no—
144C22×Q8—82no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C2×D412yes—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—41no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161C4⋊Q8—22no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8C22×C414yes—
170C4⋊Q8C22×C412yes—
171C4⋊Q8C2×D412yes—
172#127—21no—
173#127—21no—
174#127C22×C412yes—
175#127C2×D412yes—
176#124—24no—
177#124—24no—
178#121—24no—
179#121—24no—
180#121—24no—
181#121C22⋊C418yes—
182C4×Q8—84no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—84no—
187C4×D4—84no—
188C4×D4C22⋊C418yes—
189#117—82no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42C22×C412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226#106—216no—
227C4⋊C8—24no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232#103—84no—
233#103—84no—
234#102—216no—
235C22⋊C8—24no—
236C22⋊C8—42no—
237C22⋊C8—42no—
238#97—42no—
239C4×C8—22no—
240C4×C8—42no—
241C4×C8C22×C414yes—
242#95—24no—
243#95—24no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—22no—
252#95—22no—
253C24—22no—
254C24#12011yescharacteristic
255#41—88no—
256#41—88no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—88no—
266C2×Q8—82no—
267C2×Q8—82no—
268C2×Q8—88no—
269C2×D4—28no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—44no—
276C2×D4—42no—
277C2×D4—44no—
278C2×D4—42no—
279C2×D4—84no—
280C2×D4C2×C22⋊C414yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—28no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—84no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12112yes—
306C22×C4#12112yes—
307C22×C4#12312yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C2×C8—24no—
312C2×C8—44no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—44no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—84no—
334C4⋊C4C2×C22⋊C414yes—
335C22⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—82no—
338C22⋊C4—81no—
339C22⋊C4—81no—
340C22⋊C4—84no—
341C22⋊C4—84no—
342C22⋊C4—84no—
343C42—44no—
344C42—44no—
345C42—84no—
346C42C2×C22⋊C412yes—
347C42C4×D412yes—
348C42C4×D412yes—
349C42#12112yes—
350C42#12112yes—
351C42#12312yes—
352C42C23×C411yescharacteristic
353C42C22×D411yescharacteristic
354C42C2×#4111yescharacteristic
355C42C2×#4111yescharacteristic
356C23—21no—
357C23—22no—
358C23—21no—
359C23—22no—
360C23—24no—
361C23—42no—
362C23—44no—
363C23—42no—
364C23—44no—
365C23#38612yes—
366C23C2×#12011yescharacteristic
367Q8—44no—
368Q8—44no—
369Q8—48no—
370Q8—88no—
371Q8—84no—
372D4—44no—
373D4—44no—
374D4—44no—
375D4—44no—
376D4—44no—
377D4—82no—
378D4—88no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—88no—
406C2×C4—88no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—168no—
410C2×C4#38612yes—
411C2×C4#38612yes—
412C2×C4#38612yes—
413C2×C4C22×C22⋊C411yescharacteristic
414C2×C4C2×C4×D411yescharacteristic
415C2×C4C2×C4×D411yescharacteristic
416C2×C4C2×#12111yescharacteristic
417C2×C4C2×#12111yescharacteristic
418C2×C4C2×#12311yescharacteristic
419C2×C4#57612yes—
420C2×C4#57712yes—
421C2×C4#57714yes—
422C8—44no—
423C8—82no—
424C8—82no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—42no—
429C22—42no—
430C22—44no—
431C22—44no—
432C22—44no—
433C22—84no—
434C22—84no—
435C22C2×#38611yescharacteristic
436C22#300811yescharacteristic
437C22#300912yes—
438C22#301011yescharacteristic
439C22#302212yes—
440C4—22no—
441C4—22no—
442C4—21no—
443C4—21no—
444C4—22no—
445C4—44no—
446C4—82no—
447C4—82no—
448C4—82no—
449C4—82no—
450C4—84no—
451C4#294612yes—
452C4#294912yes—
453C4#304212yes—
454C4#304512yes—
455C2—42no—
456C2—84no—
457C2#1896712yes—
458C2#1897112yes—
459C2#2093211yescharacteristic
460C2#2093311yescharacteristic
461C2#2095511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21833C2semidirect
4. #21811C2semidirect
6. #9059C2semidirect
7. #9057C2semidirect
8. #7900C2semidirect
9. #7894C2semidirect
13. #3227C4semidirect
58. C2×C4⋊Q8C2×C4semidirect
75. #498C2×C4semidirect
85. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
#343932
#218332
#218112
C2×#15532
#90592
#90572
#79002
#78942