Finite Group Databasebeta

← #204412

Subgroups — #204412

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2#21821C211yesmaximal, characteristic
3#21818C211yesmaximal, characteristic
4C2×#1572C211yesmaximal, characteristic
5C2×#1485C211yesmaximal, characteristic
6C2×#1491C211yesmaximal, characteristic
7C2×#1453C211yesmaximal, characteristic
8#9664C212yesmaximal, complemented
9#9662C212yesmaximal, complemented
10#9658C212yesmaximal, complemented
11#9656C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3104—24no—
15#3104—24no—
16C2×#484C412yes—
17C2×#477C412yes—
18C2×#462C412yes—
19C2×#451C412yes—
20C2×C4×Q16C2212yes—
21C2×C4×Q16C2212yes—
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27#1985—22no—
28#1984—22no—
29#1984C2212yes—
30#1984C2212yes—
31#1984C2212yes—
32#1984C2212yes—
33#1818—22no—
34#1817—22no—
35#1572—24no—
36C2×#335C2212yes—
37C2×#334C2212yes—
38#1491—24no—
39#1485—24no—
40C2×Q8⋊C8C2212yes—
41C2×Q8⋊C8C2212yes—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44#1453—24no—
45C2×#322C2211yescharacteristic
46C2×#322C2211yescharacteristic
47C2×#322C2211yescharacteristic
48C2×C82C2211yescharacteristic
49C23×Q8C2×C411yescharacteristic
50C22×Q16C2×C412yes—
51C22×Q16C2×C412yes—
52C22×M16—21no—
53#533—22no—
54#533—22no—
55#533—22no—
56#533—22no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2×C412yes—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8C2×C412yes—
70C2×C4×Q8C2312yes—
71C2×C4×Q8C2312yes—
72C2×#117—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#484—28no—
78#477—28no—
79#462—28no—
80#451—28no—
81C4×Q16—24no—
82C4×Q16—24no—
83C4×Q16—24no—
84C4×Q16—24no—
85#432—22no—
86#432—22no—
87#431—22no—
88#431—22no—
89#431—22no—
90#431—22no—
91#431D414yes—
92#431D414yes—
93C2×#107C2×C412yes—
94C2×#107C2×C412yes—
95C2×C4⋊C8C2×C412yes—
96C2×C4⋊C8D412yes—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2312yes—
99C2×#103C2×C412yes—
100C2×#103C2×C412yes—
101C2×#103C2×C412yes—
102C2×#103C2×C412yes—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8C2×C412yes—
106C2×C22⋊C8D412yes—
107C4×M16—22no—
108C4×M16D414yes—
109C2×#97—21no—
110C2×#97—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#336—24no—
119#336—24no—
120#336—24no—
121#336—24no—
122#335—24no—
123#334—24no—
124Q8⋊C8—24no—
125Q8⋊C8—24no—
126Q8⋊C8—24no—
127Q8⋊C8—24no—
128#322—22no—
129#322—22no—
130#322—22no—
131C82—22no—
132C22×Q8—24no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—42no—
138C22×Q8C22×C412yes—
139C22×Q8C22×C412yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4#4111yescharacteristic
145C2×Q16—24no—
146C2×Q16—24no—
147C2×Q16—24no—
148C2×Q16—24no—
149C2×Q16—42no—
150C2×Q16—42no—
151C2×M16—24no—
152C2×M16—44no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C22×C412yes—
158C22×C8C22×C412yes—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165#125—42no—
166#125—42no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C4×Q8—24no—
172C4×Q8—24no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187#117—42no—
188#117—42no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C22⋊C4—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42#4112yes—
215C2×C42#4111yescharacteristic
216C2×C42C2411yescharacteristic
217#107—24no—
218#107—24no—
219C4⋊C8—24no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8C22⋊C418yes—
225#103—24no—
226#103—24no—
227#103—24no—
228#103—24no—
229C22⋊C8—24no—
230C22⋊C8—42no—
231C22⋊C8—42no—
232C22⋊C8C22⋊C418yes—
233#97—24no—
234#97—42no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—42no—
242C4×C8C2×D412yes—
243C4×C8C2×D412yes—
244C4×C8C2×D412yes—
245C4×C8C2×D412yes—
246C4×C8C2×D412yes—
247C4×C8C2×D412yes—
248C24C4×D411yescharacteristic
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—48no—
262C2×Q8—48no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—82no—
267C2×Q8#13114yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—82no—
281C22×C4C2×C22⋊C412yes—
282C22×C4C4×D412yes—
283C22×C4C4×D411yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4#12412yes—
289C22×C4#12712yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4111yescharacteristic
293Q16—48no—
294Q16—48no—
295M16—48no—
296C2×C8—24no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8C2×C22⋊C414yes—
313C2×C8C2×C22⋊C414yes—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C22⋊C4—44no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—44no—
334C42C2×C22⋊C412yes—
335C42C4×D412yes—
336C42#12112yes—
337C42#12112yes—
338C42#12412yes—
339C42#12712yes—
340C42C22×D411yescharacteristic
341C42C22×D411yescharacteristic
342C42C22×D411yescharacteristic
343C42C2×#4111yescharacteristic
344C23—21no—
345C23—21no—
346C23—22no—
347C23—22no—
348C23#39012yes—
349C23C2×C4×D411yescharacteristic
350Q8—24no—
351Q8—24no—
352Q8—48no—
353Q8—42no—
354Q8—42no—
355Q8—42no—
356Q8—42no—
357Q8—42no—
358Q8—42no—
359Q8—84no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—48no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—84no—
383C2×C4#39012yes—
384C2×C4#39012yes—
385C2×C4#39012yes—
386C2×C4C22×C22⋊C411yescharacteristic
387C2×C4C2×C4×D411yescharacteristic
388C2×C4C2×#12111yescharacteristic
389C2×C4C2×#12111yescharacteristic
390C2×C4C2×#12411yescharacteristic
391C2×C4C2×#12711yescharacteristic
392C2×C4C2×#13112yes—
393C2×C4#56812yes—
394C2×C4#57814yes—
395C8—24no—
396C8—24no—
397C8—24no—
398C8—42no—
399C8—42no—
400C8—42no—
401C8—82no—
402C8—82no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—42no—
407C22—42no—
408C22C2×#39011yescharacteristic
409C22#292911yescharacteristic
410C22#293011yescharacteristic
411C22#295012yes—
412C22#295612yes—
413C4—22no—
414C4—22no—
415C4—21no—
416C4—21no—
417C4—22no—
418C4—44no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4#288012yes—
423C4#293312yes—
424C4#305912yes—
425C4#315612yes—
426C2—42no—
427C2#1792512yes—
428C2#1795312yes—
429C2#2119111yescharacteristic
430C2#2123211yescharacteristic
431C2#2124111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #9664C2semidirect
9. #9662C2semidirect
10. #9658C2semidirect
11. #9656C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
#218212
#218182
C2×#15722
C2×#14852
C2×#14912
C2×#14532
#96642
#96622
#96582
#96562