Finite Group Databasebeta

← #214698

Subgroups — #214698

Order 512.

837 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35670C211yesmaximal, characteristic
2#17371C211yesmaximal, characteristic
3#17352C211yesmaximal, characteristic
4#9104C212yesmaximal, complemented
5#9100C212yesmaximal, complemented
6#8962C212yesmaximal, complemented
7#8958C212yesmaximal, complemented
8#8732C212yesmaximal
9#8327C212yesmaximal
10C2×#541C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2983C2212yes—
13#2982C2212yes—
14#2565C411yescharacteristic
15#2565C411yescharacteristic
16#2546C411yescharacteristic
17#2546C411yescharacteristic
18#2499C411yescharacteristic
19#2499C411yescharacteristic
20#2478—22no—
21#2478—22no—
22#2462—22no—
23#2462—22no—
24#2434C411yescharacteristic
25#2434C411yescharacteristic
26#2327C2211yescharacteristic
27#2327C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×C4⋊Q8C2211yescharacteristic
30#2298C2211yescharacteristic
31#1479C2214yes—
32#1479C2214yes—
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#541—44no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#126—21no—
51C2×#126—21no—
52C2×#125—21no—
53C2×#125—21no—
54C2×#125C2×C411yescharacteristic
55C2×C22⋊Q8C2×C411yescharacteristic
56C2×C22⋊Q8C2×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×Q8C2311yescharacteristic
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#398—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#397—22no—
81#397—22no—
82#395—22no—
83#395—22no—
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#391C2×C411yescharacteristic
90#391C2×C411yescharacteristic
91#391C2×C411yescharacteristic
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#388—21no—
97#388—21no—
98#387—22no—
99#387—22no—
100#387—21no—
101#387—21no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—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⋊C4—22no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C22⋊C4C2×C411yescharacteristic
119C4×C22⋊C4C2×C411yescharacteristic
120#376C2311yescharacteristic
121C43C2311yescharacteristic
122Q8⋊C8—44no—
123Q8⋊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—42no—
132C22×Q8—42no—
133C22×Q8C22⋊C412yes—
134C22×Q8C22×C411yescharacteristic
135C22×Q8C22×C411yescharacteristic
136C22×Q8C22×C411yescharacteristic
137C22×Q8C22×C411yescharacteristic
138C23×C4C22⋊C412yes—
139C23×C4C22×C411yescharacteristic
140C4⋊Q8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145#126—44no—
146#126—44no—
147#125—42no—
148#125—44no—
149#125—44no—
150C22⋊Q8—42no—
151C22⋊Q8—42no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—44no—
157C4×Q8—44no—
158C4×Q8—42no—
159C4×Q8—42no—
160#117—24no—
161#117—44no—
162#117—44no—
163#117—42no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C2×D411yescharacteristic
196C2×C22⋊C4C2×D411yescharacteristic
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42C22⋊C412yes—
202C2×C42C22⋊C412yes—
203C2×C42C22⋊C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—24no—
213C4⋊C8—24no—
214C4⋊C8—44no—
215C4⋊C8—44no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218#97—44no—
219C4×C8—24no—
220C4×C8—24no—
221C4×C8—44no—
222#95—21no—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238C24C2×C22⋊C411yescharacteristic
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—416no—
254C2×Q8#13114yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C2×C8—44no—
280C2×C8—44no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C42—22no—
315C42—24no—
316C42—24no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42H(5,𝔽2)14yes—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23C22×C22⋊C411yescharacteristic
335Q8—28no—
336Q8—48no—
337Q8—44no—
338Q8—44no—
339Q8—44no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—48no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
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—41no—
382C2×C4#40814yes—
383C2×C4#51812yes—
384C2×C4#53412yes—
385C2×C4#53412yes—
386C2×C4C2×#13112yes—
387C2×C4#56812yes—
388C8—84no—
389C8—84no—
390C22—21no—
391C22—21no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—81no—
396C22—81no—
397C22#230811yescharacteristic
398C22C2×#40811yescharacteristic
399C22#287911yescharacteristic
400C4—22no—
401C4—22no—
402C4—24no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—41no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—41no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4#297614yes—
418C2—21no—
419C2—21no—
420C2—81no—
421C2#2174111yescharacteristic
422C2#2178311yescharacteristic
423C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊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
4. #9104C2semidirect
5. #9100C2semidirect
6. #8962C2semidirect
7. #8958C2semidirect

Maximal subgroups

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

subgroupindex
#356702
#173712
#173522
#91042
#91002
#89622
#89582
#87322
#83272