Finite Group Databasebeta

← #223161

Subgroups — #223161

Order 512.

963 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic
3D4×C4⋊C8C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9898C211yesmaximal, characteristic, complemented
6#9131C212yesmaximal, complemented
7#9130C212yesmaximal, complemented
8#8726C212yesmaximal, complemented
9#8586C211yesmaximal, characteristic, complemented
10C4×#332C211yesmaximal, characteristic, complemented
11#8039C212yesmaximal, complemented
12C2×#556—21no—
13C2×C8×D4—21no—
14C2×#431C2212yes—
15C22×C4⋊C8C2212yes—
16#2868C2211yescharacteristic
17#2857C2212yes—
18#2568—21no—
19#2547—21no—
20#2547C412yes—
21#2499—22no—
22#2496—22no—
23#2496C414yes—
24#2365—21no—
25#2365C412yes—
26C4×C4⋊Q8C2211yescharacteristic
27#2285C2212yes—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1939—21no—
31#1923—22no—
32#1850—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1797C2212yes—
36#1797C2212yes—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C2×#332C2211yescharacteristic
41C2×#332C2211yescharacteristic
42C2×#332C2212yes—
43#1296C2211yescharacteristic
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2211yescharacteristic
48#1296C2212yes—
49#1296C2212yes—
50C23×C8—22no—
51#556—44no—
52#556—44no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8C2311yescharacteristic
57C2×#126—24no—
58C2×#124—22no—
59C2×C22⋊Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—21no—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×#117—22no—
66C22×C42C2312yes—
67C8×D4—44no—
68#431—24no—
69#431—42no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—42no—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89#400—24no—
90#397—24no—
91#397—24no—
92#391—22no—
93#391—22no—
94#391—21no—
95#391—22no—
96#389—21no—
97#389—22no—
98#389C2×C412yes—
99#389C2×C414yes—
100#387—24no—
101#387—24no—
102#384—21no—
103#384—21no—
104#384C2×C412yes—
105#384C2311yescharacteristic
106#384C2312yes—
107#382—22no—
108#380—22no—
109#380—22no—
110#380C2×C414yes—
111#380C2×C414yes—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C414yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C22⋊C4C2×C412yes—
120C4×C22⋊C4D412yes—
121C4×C22⋊C4D412yes—
122C43C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—21no—
126#336—21no—
127#332—22no—
128#332—22no—
129#332—42no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×D4C22⋊C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—44no—
143C22×C8—44no—
144C22×C8—82no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—44no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154#126—416no—
155#124—48no—
156C22⋊Q8—44no—
157C22⋊Q8—44no—
158C4×Q8—48no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—42no—
164C4×D4D814yes—
165C4×D4SD1614yes—
166#117—44no—
167#117—44no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C414yes—
183C2×C4⋊C4C22×C412yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—24no—
194C2×C42—22no—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C412yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2411yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—44no—
216C4⋊C8—42no—
217C4⋊C8—42no—
218C4⋊C8—81no—
219C4⋊C8—81no—
220C4⋊C8#4112yes—
221C4⋊C8#4112yes—
222C22⋊C8—44no—
223C22⋊C8—44no—
224C22⋊C8—82no—
225C4×C8—44no—
226C4×C8—42no—
227C4×C8—42no—
228#95—24no—
229#95—22no—
230#95—24no—
231#95—22no—
232#95—24no—
233#95—24no—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237C24C2×C22⋊C412yes—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4#10214yes—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×D814yes—
275C22×C4C2×SD1614yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—88no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—48no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4#10214yes—
320C22⋊C4—24no—
321C22⋊C4—42no—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C22⋊C4#10218yes—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—22no—
336C42—41no—
337C42—48no—
338C42—44no—
339C42C2×D812yes—
340C42C2×SD1612yes—
341C42#13512yes—
342C42#13512yes—
343C42C2×#4111yescharacteristic
344C42C2×#4111yescharacteristic
345C42H(5,𝔽2)11yescharacteristic
346C422−1+411yescharacteristic
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23C2×#10214yes—
354C23C22×C22⋊C411yescharacteristic
355Q8—88no—
356Q8—81no—
357Q8—81no—
358Q8—81no—
359Q8—81no—
360D4—28no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4#41312yes—
398C2×C4C2×#10214yes—
399C2×C4C2×#10212yes—
400C2×C4C2×#10212yes—
401C2×C4#41612yes—
402C2×C4#41612yes—
403C2×C4C4×#4111yescharacteristic
404C2×C4#52011yescharacteristic
405C2×C4#54611yescharacteristic
406C2×C4#54611yescharacteristic
407C2×C4#54711yescharacteristic
408C2×C4#54711yescharacteristic
409C2×C4C22×D811yescharacteristic
410C2×C4C22×SD1611yescharacteristic
411C2×C4C2×#13511yescharacteristic
412C2×C4C2×#13511yescharacteristic
413C2×C4#57711yescharacteristic
414C2×C4#57711yescharacteristic
415C2×C4#57811yescharacteristic
416C2×C4#57811yescharacteristic
417C8—42no—
418C8—42no—
419C8—81no—
420C8—81no—
421C8—82no—
422C8—82no—
423C22—24no—
424C22—24no—
425C22—24no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22#233911yescharacteristic
431C22C2×#41311yescharacteristic
432C22#288911yescharacteristic
433C22C22×#10211yescharacteristic
434C22C2×#41611yescharacteristic
435C22#289811yescharacteristic
436C22#289811yescharacteristic
437C4—22no—
438C4—22no—
439C4—21no—
440C4—21no—
441C4—24no—
442C4—41no—
443C4—42no—
444C4—42no—
445C4—84no—
446C4—82no—
447C4—82no—
448C4#154014yes—
449C4#329411yescharacteristic
450C4#329811yescharacteristic
451C4#330211yescharacteristic
452C4#330411yescharacteristic
453C4#333411yescharacteristic
454C4#333511yescharacteristic
455C4#334011yescharacteristic
456C4#334211yescharacteristic
457C2—44no—
458C2C2×#154011yescharacteristic
459C2#1772111yescharacteristic
460C2#1772511yescharacteristic
461C2#1777811yescharacteristic
462C2#2248411yescharacteristic
463C2#2248511yescharacteristic
464C2#2249011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9898C2semidirect
5. #9898C2semidirect
6. #9131C2semidirect
7. #9130C2semidirect
8. #8726C2semidirect
9. #8586C2semidirect
10. C4×#332C2semidirect
11. #8039C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#227952
D4×C4⋊C82
#98982
#91312
#91302
#87262
#85862
C4×#3322
#80392