Finite Group Databasebeta

← #196567

Subgroups — #196567

Order 512.

1160 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2923C211yesmaximal, characteristic
2#35138C211yesmaximal, characteristic
3#21379C211yesmaximal, characteristic
4C2×#1528C211yesmaximal, characteristic, complemented
5C22×#330C211yesmaximal, characteristic, complemented
6#9126C214yesmaximal, complemented
7#9125C214yesmaximal, complemented
8#8036C211yesmaximal, characteristic, complemented
9#8029C211yesmaximal, characteristic, complemented
10C22×#125C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#2923—24no—
13#2923—24no—
14C2×#433—22no—
15C2×#432C2211yescharacteristic
16C2×#432C2212yes—
17C2×#432C2212yes—
18C2×#424—21no—
19C2×#408—21no—
20C2×C4×M16C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×#97C2211yescharacteristic
23#2851C2214yes—
24#2634—21no—
25#2633—21no—
26#2611—22no—
27#2584C412yes—
28#2562C412yes—
29#2487C412yes—
30#2486C412yes—
31#2484—21no—
32#2406—21no—
33#2322—22no—
34#2283C2211yescharacteristic
35#1985—21no—
36#1955—21no—
37#1869—22no—
38#1749C2214yes—
39#1528—24no—
40C2×#330—24no—
41C2×#330C2212yes—
42C2×#330C2214yes—
43C2×#330C2212yes—
44#1295C2214yes—
45#1295C2212yes—
46#1295C2212yes—
47C22×#41D411yescharacteristic
48C22×M16—21no—
49C23×C8—21no—
50C23×C8—21no—
51C2×#125—21no—
52C2×#125—21no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125C2×C412yes—
58C2×#125C2312yes—
59C2×#123—22no—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41—22no—
63C4×#41—22no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C412yes—
68C2×C4×D4D412yes—
69C2×C4×D4D411yescharacteristic
70C2×#117D412yes—
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#433—216no—
79#432—22no—
80#432—22no—
81#432—24no—
82#432—24no—
83#432—24no—
84#432—24no—
85#424—28no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89#408—28no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—24no—
94C4×M16—22no—
95C4×M16D414yes—
96C2×#97—21no—
97C2×#97—22no—
98C2×#97—22no—
99C2×#97—22no—
100C2×#97—21no—
101C2×#97D414yes—
102C2×#97C2312yes—
103C2×#97C2312yes—
104C2×#97C2311yescharacteristic
105C2×#97C2311yescharacteristic
106C2×#97C2312yes—
107C2×C4×C8—21no—
108#400—22no—
109#400—22no—
110#399—22no—
111#398—22no—
112#398—22no—
113#397—22no—
114#395—22no—
115#394—22no—
116#389—24no—
117#389C2×C414yes—
118#389C2×C414yes—
119#388—24no—
120#387—24no—
121#387—24no—
122#386—22no—
123#385—24no—
124#385—24no—
125#385—22no—
126#383—21no—
127#382C2312yes—
128#381—21no—
129#381—22no—
130#380C2×C414yes—
131#380C2×C414yes—
132#380C2×C412yes—
133C4×C4⋊C4C2×C412yes—
134C4×C22⋊C4—21no—
135#376—21no—
136#336—24no—
137#330—24no—
138#330—24no—
139#330—28no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—44no—
143C22×Q8C22⋊C411yescharacteristic
144C22×D4C22⋊C412yes—
145C22×D4C22⋊C411yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C22×C412yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C2×M16—24no—
155C2×M16—44no—
156C22×C8—24no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—44no—
163C22×C8—44no—
164C22×C8—42no—
165#125—24no—
166#125—24no—
167#125—22no—
168#125—22no—
169#125—22no—
170#125—22no—
171#125—42no—
172#125—42no—
173#125—42no—
174#125—42no—
175#123—48no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188#117—22no—
189#117—22no—
190#117—24no—
191#117—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C414yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42C22×C412yes—
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4112yes—
235C2×C42#4112yes—
236C2×C42C2411yescharacteristic
237C4⋊C8—28no—
238C4⋊C8—28no—
239C4⋊C8—42no—
240C22⋊C8—28no—
241C22⋊C8—28no—
242C22⋊C8—48no—
243C22⋊C8—42no—
244#97—24no—
245#97—21no—
246#97—21no—
247#97—22no—
248#97—22no—
249#97—22no—
250#97—22no—
251#97—22no—
252#97—42no—
253#97C2×D412yes—
254#97C2×D412yes—
255#97C2×D414yes—
256C4×C8—24no—
257#95—22no—
258#95—22no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—22no—
264#95—22no—
265C24C2×C22⋊C412yes—
266C24C2×C22⋊C411yescharacteristic
267#41—48no—
268#41—48no—
269C2×Q8—22no—
270C2×Q8—42no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—42no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—82no—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C4×D412yes—
303C22×C4C4×D412yes—
304C22×C4#12012yes—
305C22×C4#12112yes—
306C22×C4#12112yes—
307C22×C4#12312yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311M16—48no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—48no—
320C2×C8—44no—
321C2×C8—48no—
322C2×C8—42no—
323C2×C8—44no—
324C2×C8—42no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—44no—
334C4⋊C4—48no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—84no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—48no—
348C22⋊C4—44no—
349C22⋊C4—84no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—82no—
357C42C4×D412yes—
358C42C4×D412yes—
359C42#12012yes—
360C42#12112yes—
361C42#12112yes—
362C42#12312yes—
363C42C22×D411yescharacteristic
364C42C22×D411yescharacteristic
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C23—22no—
368C23—22no—
369C23—24no—
370C23—22no—
371C23—22no—
372C23—24no—
373C23—22no—
374C23#38612yes—
375C23#42114yes—
376C23C22×C22⋊C411yescharacteristic
377Q8—44no—
378D4—44no—
379D4—48no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4#38612yes—
416C2×C4#38612yes—
417C2×C4#38612yes—
418C2×C4C2×C4×D411yescharacteristic
419C2×C4C2×C4×D411yescharacteristic
420C2×C4C2×#12011yescharacteristic
421C2×C4C2×#12111yescharacteristic
422C2×C4C2×#12111yescharacteristic
423C2×C4C2×#12311yescharacteristic
424C8—44no—
425C8—42no—
426C8—42no—
427C8—44no—
428C8—82no—
429C22—28no—
430C22—22no—
431C22—22no—
432C22—44no—
433C22—44no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22#189714yes—
438C22C2×#38611yescharacteristic
439C22#288912yes—
440C22C2×#42112yes—
441C22#290112yes—
442C4—21no—
443C4—21no—
444C4—22no—
445C4—21no—
446C4—21no—
447C4—22no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—84no—
455C4—82no—
456C4—82no—
457C2—24no—
458C2—82no—
459C2#1768014yes—
460C2#2090311yescharacteristic
461C2C2×#189711yescharacteristic
462C2#2099111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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. C22×#330C2semidirect
6. #9126C2semidirect
7. #9125C2semidirect
8. #8036C2semidirect
9. #8029C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29232
#351382
#213792
C2×#15282
C22×#3302
#91262
#91252
#80362
#80292