Finite Group Databasebeta

← #196437

Subgroups — #196437

Order 512.

1247 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2982C211yesmaximal, characteristic, complemented
2#35111C211yesmaximal, characteristic
3C2×#1521C211yesmaximal, characteristic
4C2×#1500C211yesmaximal, characteristic
5#13961C211yesmaximal, characteristic, complemented
6#9114C214yesmaximal
7#9112C214yesmaximal, complemented
8#8025C211yesmaximal, characteristic, complemented
9#8013C211yesmaximal, characteristic, complemented
10C2×#529C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2982—24no—
13#2982—24no—
14C2×C8×Q8—22no—
15C2×C8×Q8—22no—
16C2×#431C2214yes—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2855C2212yes—
20#2854C2212yes—
21#2623—21no—
22#2614—21no—
23#2604—22no—
24#2599—22no—
25#2592C412yescomplemented
26#2584C412yescomplemented
27#2581C412yescomplemented
28#2580C412yescomplemented
29#2322—22no—
30#2283C2211yescharacteristic
31C2×#383C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1521—24no—
34#1500—24no—
35C2×#328C2212yes—
36C2×#328C2212yes—
37C2×#327C2212yes—
38C2×#327C2212yes—
39#1446—21no—
40#1445—21no—
41#1436—22no—
42#1434C2214yes—
43#1294C2214yes—
44#1294C2214yes—
45C23×Q8D411yescharacteristic
46#529—22no—
47#529—22no—
48#529—22no—
49#529—22no—
50C2×#126—21no—
51C2×#125C2×C411yescharacteristic
52C2×#124C2×C411yescharacteristic
53C2×#123—22no—
54C2×#123—21no—
55C2×C22⋊Q8C2311yescharacteristic
56C2×#121C2311yescharacteristic
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8D412yes—
65C2×C4×Q8D412yes—
66C2×C4×D4—21no—
67C2×C4×D4C2×C411yescharacteristic
68C2×#117—21no—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C8×Q8—216no—
75C8×Q8—216no—
76#431—22no—
77#431—22no—
78#431—28no—
79#431—28no—
80#431D414yes—
81#431D414yes—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2312yes—
87C2×C4×C8—24no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90#400—22no—
91#399—22no—
92#398—22no—
93#397—22no—
94#395—22no—
95#395—22no—
96#393—22no—
97#393—22no—
98#391C2×C414yes—
99#390C2×C414yes—
100#389—24no—
101#389C2×C414yes—
102#389C2×C414yes—
103#388—24no—
104#388—24no—
105#387—24no—
106#387—24no—
107#386—22no—
108#385—22no—
109#385—24no—
110#383—21no—
111#383—22no—
112#383—22no—
113#382C2×C412yes—
114#381—22no—
115#381—21no—
116#380C2312yes—
117C4×C4⋊C4C2×C412yes—
118C4×C22⋊C4—21no—
119#376—21no—
120#376—22no—
121#376—22no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95C2312yes—
128C2×#95C2312yes—
129#328—24no—
130#328—24no—
131#327—24no—
132#327—24no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—28no—
136C22×Q8C22⋊C412yes—
137C22×Q8C22⋊C412yes—
138C22×Q8C22×C411yescharacteristic
139C22×D4—21no—
140C22×D4C22×C411yescharacteristic
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151#126—42no—
152#126—42no—
153#125—22no—
154#125—22no—
155#124—22no—
156#124—22no—
157#123—42no—
158#123—42no—
159#123—48no—
160C22⋊Q8—21no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164#121—21no—
165#121—21no—
166#121—21no—
167#121—21no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—24no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—42no—
181C4×D4—42no—
182#117—42no—
183#117—42no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4C22⋊C414yes—
195C2×C4⋊C4C22⋊C414yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C414yes—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—24no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—44no—
216C2×C42—42no—
217C2×C42C22×C414yes—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D414yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4112yes—
226C2×C42#4112yes—
227C4⋊C8—216no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8C2×D414yes—
233C22⋊C8C2×D414yes—
234C4×C8—216no—
235C4×C8—24no—
236C4×C8—24no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—48no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—48no—
254C24—21no—
255C24C22×D411yescharacteristic
256C2×Q8—216no—
257C2×Q8—216no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—28no—
261C2×Q8—28no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—42no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—44no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—82no—
296C22×C4C2×C22⋊C414yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C414yes—
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4#12112yes—
305C22×C4#12112yes—
306C22×C4#13518yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—44no—
316C2×C8—44no—
317C4⋊C4—28no—
318C4⋊C4—28no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—44no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—48no—
332C4⋊C4—84no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—44no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—48no—
342C22⋊C4—84no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—24no—
347C42—48no—
348C42—42no—
349C42—82no—
350C42C4×D412yes—
351C42C4×D412yes—
352C42C4×D412yes—
353C42C4×D412yes—
354C42#12012yes—
355C42#12012yes—
356C42#12312yes—
357C42#12312yes—
358C23—22no—
359C23—22no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—41no—
364C23C22×C22⋊C411yescharacteristic
365C23C2×#12111yescharacteristic
366C23C2×#12111yescharacteristic
367C23C2×#13514yes—
368Q8—216no—
369Q8—216no—
370Q8—81no—
371Q8—81no—
372D4—81no—
373D4—81no—
374D4—82no—
375D4—82no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—28no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—44no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—84no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4#38612yes—
420C2×C4#38612yes—
421C2×C4#38612yes—
422C2×C4#38612yes—
423C2×C4#41618yes—
424C2×C4C2×C4×D411yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4C2×#12011yescharacteristic
427C2×C4C2×#12311yescharacteristic
428C8—44no—
429C8—44no—
430C22—21no—
431C22—21no—
432C22—24no—
433C22—24no—
434C22—21no—
435C22—21no—
436C22—81no—
437C22—84no—
438C22—81no—
439C22—81no—
440C22C2×#38611yescharacteristic
441C22#288412yes—
442C22C2×#41612yes—
443C22#290112yes—
444C22#305012yes—
445C22#305012yes—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—28no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—44no—
456C4—84no—
457C4—81no—
458C4—84no—
459C4—81no—
460C4—81no—
461C4—162no—
462C2—22no—
463C2—22no—
464C2—161no—
465C2#1761614yes—
466C2#2090311yescharacteristic
467C2#2096711yescharacteristic
468C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2982C2semidirect
5. #13961C2semidirect
7. #9112C2semidirect
8. #8025C2semidirect
9. #8013C2semidirect
25. #2592C4semidirect
26. #2584C4semidirect
27. #2581C4semidirect
28. #2580C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29822
#351112
C2×#15212
C2×#15002
#139612
#91142
#91122
#80252
#80132