Finite Group Databasebeta

← #214700

Subgroups — #214700

Order 512.

857 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35669C211yesmaximal, characteristic
2#17371C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#9106C212yesmaximal, complemented
5#9100C212yesmaximal, complemented
6#8962C212yesmaximal, complemented
7#8960C212yesmaximal, complemented
8#8733C212yesmaximal, complemented
9#8325C212yesmaximal
10C2×#542C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2983C2212yes—
13#2982C2212yes—
14#2568C411yescharacteristic
15#2567C411yescharacteristic
16#2545C411yescharacteristic
17#2545C411yescharacteristic
18#2500C411yescharacteristic
19#2499C411yescharacteristic
20#2474—22no—
21#2470—22no—
22#2469—22no—
23#2465—22no—
24#2436C411yescharacteristic
25#2434C411yescharacteristic
26#2327C2211yescharacteristic
27#2326C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×#124C2211yescharacteristic
30#2298C2211yescharacteristic
31#1479C2214yes—
32#1478C2214yes—
33#1467—22no—
34#1464—22no—
35#1461—22no—
36#1461C2212yes—
37#1461C2212yes—
38#1458—22no—
39#1458C2212yes—
40#1458C2212yes—
41#1276C2214yes—
42#1274C2214yes—
43C23×Q8C2×C411yescharacteristic
44#542—44no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2×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#519—44no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D412yes—
61C2×C4×Q8D412yes—
62C2×C4×Q8C2311yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117D412yes—
68C2×#117D412yes—
69C2×#117C2311yescharacteristic
70#446—44no—
71C8×Q8—44no—
72#432—24no—
73#431—24no—
74#431C2314yes—
75#431C2314yes—
76#400—22no—
77#397—22no—
78#396—22no—
79#396—22no—
80#395—22no—
81#395—22no—
82#394—22no—
83#393—22no—
84#391—22no—
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#391C2×C411yescharacteristic
90#391C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#390C2×C411yescharacteristic
93#389—22no—
94#389—22no—
95#388—22no—
96#388—21no—
97#388—21no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#387—22no—
103#387—21no—
104#387—21no—
105#387—22no—
106#387—22no—
107#386—22no—
108#385—22no—
109#385—22no—
110#384C2×C411yescharacteristic
111#384C2×C411yescharacteristic
112#382C2×C411yescharacteristic
113#382C2×C411yescharacteristic
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C22⋊C4—22no—
118C4×C22⋊C4C2×C411yescharacteristic
119C4×C22⋊C4C2×C411yescharacteristic
120#376C2311yescharacteristic
121C43C2311yescharacteristic
122Q8⋊C8—44no—
123D4⋊C8—44no—
124#324—24no—
125#324—22no—
126#324—22no—
127#324—24no—
128#323—22no—
129#323—22no—
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8C22⋊C412yes—
135C22×Q8C22×C411yescharacteristic
136C22×Q8C22×C411yescharacteristic
137C22×Q8C22×C411yescharacteristic
138C22×D4—21no—
139C22×D4C22×C411yescharacteristic
140C23×C4—21no—
141C23×C4C22⋊C412yes—
142C23×C4C22×C411yescharacteristic
143C4⋊Q8—24no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146#126—44no—
147#124—24no—
148#124—42no—
149#124—42no—
150#124—44no—
151#123—44no—
152C22⋊Q8—42no—
153C22⋊Q8—44no—
154#121—42no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—44no—
159C4×Q8—44no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×D4—24no—
163#117—24no—
164#117—44no—
165#117—44no—
166#117—42no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C2×D411yescharacteristic
199C2×C22⋊C4C2×D411yescharacteristic
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22⋊C412yes—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2411yescharacteristic
214C4⋊C8—24no—
215C4⋊C8—24no—
216C4⋊C8—44no—
217C4⋊C8—44no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220#97—44no—
221C4×C8—24no—
222C4×C8—24no—
223C4×C8—44no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
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⋊C411yescharacteristic
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—416no—
253C2×Q8—82no—
254C2×Q8#13114yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—42no—
259C2×D4—82no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—82no—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C2×C8—44no—
286C2×C8—44no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—81no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—44no—
318C22⋊C4—82no—
319C22⋊C4—81no—
320C42—22no—
321C42—24no—
322C42—24no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—81no—
333C42H(5,𝔽2)14yes—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—41no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23C22×C22⋊C411yescharacteristic
345Q8—28no—
346Q8—48no—
347Q8—44no—
348Q8—44no—
349Q8—162no—
350D4—44no—
351D4—162no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—48no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—162no—
392C2×C4#40814yes—
393C2×C4#51812yes—
394C2×C4#53412yes—
395C2×C4#53412yes—
396C2×C4C2×#13112yes—
397C2×C4#56812yes—
398C8—84no—
399C8—84no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22#230811yescharacteristic
413C22C2×#40811yescharacteristic
414C22#287911yescharacteristic
415C4—22no—
416C4—22no—
417C4—24no—
418C4—42no—
419C4—41no—
420C4—42no—
421C4—42no—
422C4—41no—
423C4—42no—
424C4—41no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4—81no—
430C4—81no—
431C4#297614yes—
432C2—21no—
433C2—21no—
434C2—81no—
435C2—161no—
436C2#2174111yescharacteristic
437C2#2178311yescharacteristic
438C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
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. #17371C2semidirect
3. #17352C2semidirect
4. #9106C2semidirect
5. #9100C2semidirect
6. #8962C2semidirect
7. #8960C2semidirect
8. #8733C2semidirect

Maximal subgroups

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

subgroupindex
#356692
#173712
#173522
#91062
#91002
#89622
#89602
#87332
#83252