Finite Group Databasebeta

← #223198

Subgroups — #223198

Order 512.

903 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#22650C211yesmaximal, characteristic
3D4×C4⋊C8C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9135C212yesmaximal, complemented
7#9127C212yesmaximal, complemented
8#8727C212yesmaximal, complemented
9#8587C211yesmaximal, characteristic, complemented
10C4×#333C211yesmaximal, characteristic, complemented
11#8045C212yesmaximal, complemented
12C2×#436—21no—
13C2×C8×D4—21no—
14C2×#423C2212yes—
15C22×C4⋊C8C2212yes—
16#2868C2211yescharacteristic
17#2857C2212yes—
18#2571—21no—
19#2563—21no—
20#2538C412yes—
21#2497—22no—
22#2492C414yes—
23#2439—22no—
24#2369—21no—
25#2355C412yes—
26#2354—21no—
27C4×C4⋊Q8C2211yescharacteristic
28#2285C2212yes—
29C42×D4C2211yescharacteristic
30#1939—21no—
31#1939—21no—
32#1851—22no—
33#1850—22no—
34#1842C2211yescharacteristic
35#1842C2211yescharacteristic
36#1797C2212yes—
37#1797C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C2×#333C2211yescharacteristic
41C2×#333C2211yescharacteristic
42C2×#333C2212yes—
43#1296C2211yescharacteristic
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2211yescharacteristic
48#1296C2212yes—
49#1296C2212yes—
50C22×M16—22no—
51C23×C8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C412yes—
55C2×C4⋊Q8C2311yescharacteristic
56C2×C22⋊Q8C2×C414yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×#117—22no—
62C22×C42C2312yes—
63#436—44no—
64C8×D4—44no—
65#423—24no—
66#423—42no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—42no—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#400—24no—
86#396—24no—
87#393—24no—
88#391—24no—
89#391C2×C412yes—
90#391C2×C414yes—
91#390—22no—
92#389—22no—
93#389—21no—
94#389—21no—
95#389—22no—
96#389—22no—
97#388—24no—
98#387—24no—
99#384—21no—
100#384—21no—
101#384C2×C412yes—
102#384C2311yescharacteristic
103#384C2312yes—
104#382—22no—
105#382—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380C2×C414yes—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2×C414yes—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4D412yes—
120C4×C22⋊C4D412yes—
121C43C2311yescharacteristic
122#336—22no—
123#336—22no—
124#336—21no—
125#336—21no—
126#333—22no—
127#333—22no—
128#333—42no—
129C22×Q8—22no—
130C22×Q8C22×C412yes—
131C22×D4C22⋊C411yescharacteristic
132C23×C4C22×C412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C2×M16—44no—
136C2×M16—44no—
137C2×M16—82no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—44no—
145C22×C8—44no—
146C22×C8—82no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—81no—
154C4⋊Q8—81no—
155C4⋊Q8—82no—
156C22⋊Q8—84no—
157C4×Q8—44no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—42no—
161C4×D4SD1614yes—
162C4×D4Q1614yes—
163#117—82no—
164#117—82no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C414yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4C22⋊C412yes—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2411yescharacteristic
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—44no—
213C4⋊C8—44no—
214C4⋊C8—42no—
215C4⋊C8—42no—
216C4⋊C8—81no—
217C4⋊C8—81no—
218C4⋊C8#4112yes—
219C4⋊C8#4112yes—
220C22⋊C8—44no—
221C22⋊C8—44no—
222C4×C8—42no—
223C4×C8—42no—
224#95—24no—
225#95—24no—
226#95—22no—
227#95—22no—
228#95—24no—
229#95—24no—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233C24C2×C22⋊C412yes—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—44no—
239C2×Q8—82no—
240C2×Q8—82no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4#10314yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
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—24no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×SD1614yes—
269C22×C4C2×Q1614yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273M16—88no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—88no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—44no—
311C4⋊C4—84no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4#10314yes—
315C22⋊C4—24no—
316C22⋊C4—42no—
317C22⋊C4—84no—
318C22⋊C4—84no—
319C22⋊C4#10318yes—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—22no—
331C42—41no—
332C42—44no—
333C42—84no—
334C42C2×SD1612yes—
335C42C2×Q1612yes—
336C42AGL(1,ℤ/8)12yes—
337C42#13712yes—
338C42C2×#4111yescharacteristic
339C42C2×#4111yescharacteristic
340C42H(5,𝔽2)11yescharacteristic
341C422−1+411yescharacteristic
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23C2×#10314yes—
349C23C22×C22⋊C411yescharacteristic
350Q8—82no—
351Q8—82no—
352Q8—164no—
353D4—28no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4#41312yes—
391C2×C4C2×#10312yes—
392C2×C4C2×#10314yes—
393C2×C4C2×#10312yes—
394C2×C4#41712yes—
395C2×C4#41712yes—
396C2×C4C4×#4111yescharacteristic
397C2×C4#52011yescharacteristic
398C2×C4#54611yescharacteristic
399C2×C4#54611yescharacteristic
400C2×C4#54711yescharacteristic
401C2×C4#54711yescharacteristic
402C2×C4C22×SD1611yescharacteristic
403C2×C4C22×Q1611yescharacteristic
404C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
405C2×C4C2×#13711yescharacteristic
406C2×C4#57611yescharacteristic
407C2×C4#57711yescharacteristic
408C2×C4#57711yescharacteristic
409C2×C4#57811yescharacteristic
410C8—42no—
411C8—42no—
412C8—81no—
413C8—81no—
414C8—82no—
415C8—82no—
416C22—24no—
417C22—24no—
418C22—24no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22#233911yescharacteristic
424C22C2×#41311yescharacteristic
425C22#288811yescharacteristic
426C22C22×#10311yescharacteristic
427C22C2×#41711yescharacteristic
428C22#289711yescharacteristic
429C22#289811yescharacteristic
430C4—22no—
431C4—22no—
432C4—21no—
433C4—21no—
434C4—24no—
435C4—41no—
436C4—42no—
437C4—42no—
438C4—84no—
439C4—82no—
440C4—82no—
441C4#155014yes—
442C4#329511yescharacteristic
443C4#329911yescharacteristic
444C4#332411yescharacteristic
445C4#332811yescharacteristic
446C4#333511yescharacteristic
447C4#333811yescharacteristic
448C4#335311yescharacteristic
449C4#335611yescharacteristic
450C2—44no—
451C2C2×#155011yescharacteristic
452C2#1772211yescharacteristic
453C2#1776211yescharacteristic
454C2#1778111yescharacteristic
455C2#2248311yescharacteristic
456C2#2248711yescharacteristic
457C2#2250411yescharacteristic
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. #9896C2semidirect
6. #9135C2semidirect
7. #9127C2semidirect
8. #8727C2semidirect
9. #8587C2semidirect
10. C4×#333C2semidirect
11. #8045C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#226502
D4×C4⋊C82
#98982
#98962
#91352
#91272
#87272
#85872
C4×#3332
#80452