Finite Group Databasebeta

← #213803

Subgroups — #213803

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35623C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic, complemented
3#17370C211yesmaximal, characteristic, complemented
4#9105C212yesmaximal, complemented
5#9104C212yesmaximal, complemented
6#8964C212yesmaximal, complemented
7#8961C212yesmaximal, complemented
8#8733C212yesmaximal, complemented
9#8325C212yesmaximal, complemented
10C2×#542C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2983C2212yes—
13#2975C2212yes—
14#2568C411yescharacteristic
15#2563C411yescharacteristic
16#2545C411yescharacteristic
17#2544C411yescharacteristic
18#2504C411yescharacteristic
19#2503C411yescharacteristic
20#2481—22no—
21#2477—22no—
22#2459—22no—
23#2457—22no—
24#2432C411yescharacteristic
25#2431C411yescharacteristic
26#2328C2211yescharacteristic
27#2327C2211yescharacteristic
28C4×#124C2211yescharacteristic
29C4×#124C2211yescharacteristic
30#2298C2211yescharacteristic
31#1479C2214yes—
32#1478C2214yes—
33#1466—22no—
34#1465—22no—
35#1464—22no—
36#1463—22no—
37#1461C2212yes—
38#1460C2212yes—
39#1458C2212yes—
40#1458C2212yes—
41#1276C2214yes—
42#1274C2214yes—
43C22×#41C2×C411yescharacteristic
44#542—44no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#127C2×C411yescharacteristic
48C2×#126—21no—
49C2×#124—21no—
50C2×#124C2×C411yescharacteristic
51C2×#124C2×C411yescharacteristic
52C2×#124C2×C411yescharacteristic
53C2×#123—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2×C411yescharacteristic
56C2×#121C2×C411yescharacteristic
57#520—44no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D412yes—
63C2×C4×D4D412yes—
64C2×C4×D4C2311yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×#117D412yes—
67C2×#117C2311yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4D412yes—
70#446—44no—
71#436—44no—
72#432—24no—
73#431C2314yes—
74#431C2314yes—
75#407—24no—
76#400—22no—
77#400—22no—
78#397—22no—
79#396—22no—
80#394—22no—
81#394—22no—
82#393—22no—
83#392—22no—
84#391—22no—
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#390C2×C411yescharacteristic
90#390C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#390C2×C411yescharacteristic
93#388—22no—
94#388—21no—
95#388—22no—
96#388—21no—
97#388—22no—
98#387—22no—
99#387—21no—
100#387—21no—
101#387—22no—
102#386—22no—
103#386—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#385—22no—
108#382C2×C411yescharacteristic
109#382C2×C411yescharacteristic
110#382C2×C411yescharacteristic
111#382C2×C411yescharacteristic
112#381—22no—
113#380—22no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4C2×C411yescharacteristic
119C4×C22⋊C4C2×C411yescharacteristic
120#376C2311yescharacteristic
121C43C2311yescharacteristic
122Q8⋊C8—44no—
123D4⋊C8—44no—
124#324—22no—
125#324—22no—
126#324—24no—
127#323—22no—
128#323—22no—
129#323—24no—
130C2×#41—42no—
131C2×#41—42no—
132C2×#41—42no—
133C22×Q8—21no—
134C22×Q8C22×C411yescharacteristic
135C22×Q8C22×C411yescharacteristic
136C22×D4—21no—
137C22×D4—22no—
138C22×D4C22⋊C412yes—
139C22×D4C22×C411yescharacteristic
140C22×D4C22×C411yescharacteristic
141C23×C4—21no—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D412yes—
147C2×M16—44no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150#127—42no—
151#126—44no—
152#124—42no—
153#124—42no—
154#124—42no—
155#124—44no—
156#123—44no—
157C22⋊Q8—42no—
158C22⋊Q8—44no—
159#121—42no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—42no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—44no—
166C4×D4—42no—
167C4×D4—44no—
168#117—24no—
169#117—44no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C2×D412yes—
202C2×C22⋊C4C2×D411yescharacteristic
203C2×C22⋊C4C2×D411yescharacteristic
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42C22⋊C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2411yescharacteristic
215C4⋊C8—24no—
216C4⋊C8—24no—
217C4⋊C8—44no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C22⋊C8—44no—
221#97—44no—
222C4×C8—24no—
223C4×C8—24no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238C24—21no—
239C24C2×C22⋊C412yes—
240C24C2×C22⋊C411yescharacteristic
241#41—416no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—42no—
249C2×Q8—82no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—82no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294M16—164no—
295C2×C8—44no—
296C2×C8—44no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—24no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—44no—
320C22⋊C4—42no—
321C22⋊C4—44no—
322C22⋊C4—42no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C22⋊C4—82no—
329C42—22no—
330C42—24no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—42no—
336C42—42no—
337C42—81no—
338C42—81no—
339C42H(5,𝔽2)14yes—
340C23—21no—
341C23—22no—
342C23—22no—
343C23—21no—
344C23—21no—
345C23—22no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23C22×C22⋊C411yescharacteristic
356Q8—44no—
357Q8—44no—
358Q8—162no—
359D4—48no—
360D4—44no—
361D4—44no—
362D4—164no—
363D4—162no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—48no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—84no—
402C2×C4—162no—
403C2×C4—162no—
404C2×C4—162no—
405C2×C4#51812yes—
406C2×C4#53412yes—
407C2×C4#53412yes—
408C2×C4#56812yes—
409C2×C4#56812yes—
410C8—84no—
411C8—84no—
412C22—21no—
413C22—21no—
414C22—44no—
415C22—42no—
416C22—41no—
417C22—41no—
418C22—42no—
419C22—42no—
420C22—41no—
421C22—81no—
422C22—82no—
423C22—81no—
424C22—82no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22#230811yescharacteristic
430C22#287911yescharacteristic
431C22#287911yescharacteristic
432C4—22no—
433C4—22no—
434C4—42no—
435C4—41no—
436C4—42no—
437C4—41no—
438C4—41no—
439C4—42no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4#297814yes—
448C2—21no—
449C2—21no—
450C2—82no—
451C2—81no—
452C2—161no—
453C2#2174311yescharacteristic
454C2#2178411yescharacteristic
455C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2431C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17372C2semidirect
3. #17370C2semidirect
4. #9105C2semidirect
5. #9104C2semidirect
6. #8964C2semidirect
7. #8961C2semidirect
8. #8733C2semidirect
9. #8325C2semidirect

Maximal subgroups

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

subgroupindex
#356232
#173722
#173702
#91052
#91042
#89642
#89612
#87332
#83252