Finite Group Databasebeta

← #214829

Subgroups — #214829

Order 512.

845 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35845C211yesmaximal, characteristic
2#20730C211yesmaximal, characteristic
3#17352C211yesmaximal, characteristic
4#9484C212yesmaximal, complemented
5#9392C212yesmaximal, complemented
6#9192C212yesmaximal
7#9034C212yesmaximal
8#8766C212yesmaximal, complemented
9#8392C212yesmaximal, complemented
10C2×#557C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2982C2212yes—
13#2921C2212yes—
14#2628C412yes—
15#2628C2211yescharacteristic
16#2581C412yes—
17#2581C2211yescharacteristic
18#2567C412yes—
19#2567C2211yescharacteristic
20#2453—21no—
21#2453—22no—
22#2441—21no—
23#2441—22no—
24#2396C412yes—
25#2396C2211yescharacteristic
26#2351C2211yescharacteristic
27Q8×C4⋊C4—21no—
28Q8×C22⋊C4—21no—
29#1822—21no—
30#1801C2212yes—
31#1801C2212yes—
32#1784—21no—
33C4×#101—21no—
34C4×#101—21no—
35#1548C2214yes—
36#1487C2214yes—
37#1461—22no—
38#1461C2212yes—
39#1458—22no—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C23×Q8D411yescharacteristic
44#557—44no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2311yescharacteristic
47C2×#125—22no—
48C2×#125—21no—
49#519—44no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D412yes—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×#117C2×C412yes—
61C2×#117D411yescharacteristic
62C2×#117D412yes—
63C2×#117C2311yescharacteristic
64C8×Q8—44no—
65#433—44no—
66#431—24no—
67#431C2314yes—
68#423C2314yes—
69C2×#101—22no—
70C2×#101—22no—
71#407—22no—
72C4×M16—22no—
73#400—22no—
74#400—21no—
75#400C2×C411yescharacteristic
76#400C2×C412yes—
77#399—22no—
78#399—21no—
79#399C2×C411yescharacteristic
80#398—22no—
81#398—21no—
82#396—22no—
83#396—21no—
84#395C2×C411yescharacteristic
85#395C2×C412yes—
86#395C2×C412yes—
87#395C2×C412yes—
88#395C2×C412yes—
89#393C2×C411yescharacteristic
90#393C2×C412yes—
91#391—22no—
92#391—22no—
93#391—21no—
94#391—21no—
95#389—22no—
96#389—21no—
97#389—21no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—21no—
104#387—22no—
105#387—22no—
106#387—22no—
107#384—22no—
108#384—21no—
109#382C2×C412yes—
110#382C2311yescharacteristic
111#380—22no—
112#380—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—22no—
118C4×C22⋊C4—21no—
119#343—22no—
120#343—22no—
121#333—44no—
122Q8⋊C8—44no—
123#324—24no—
124#324—22no—
125#323—22no—
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—42no—
129C22×Q8C22⋊C412yes—
130C22×Q8C22×C412yes—
131C22×Q8C22×C411yescharacteristic
132C22×Q8C2×D411yescharacteristic
133C22×Q8C2×D411yescharacteristic
134C23×C4C22⋊C412yes—
135C23×C4C2×D411yescharacteristic
136C2×M16—24no—
137C4⋊Q8—24no—
138C4⋊Q8—44no—
139#125—48no—
140#125—44no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—44no—
145C4×Q8—44no—
146C4×Q8—44no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×Q8—44no—
150C4×Q8—44no—
151#117—24no—
152#117—44no—
153#117—44no—
154#117—42no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4C22⋊C412yes—
184C2×C22⋊C4C22×C411yescharacteristic
185C2×C22⋊C4C2×D411yescharacteristic
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42C22⋊C412yes—
191C2×C42C22⋊C412yes—
192C2×C42C22⋊C412yes—
193C2×C42C22×C412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2411yescharacteristic
197C4⋊C8—24no—
198C4⋊C8—24no—
199C4⋊C8—24no—
200C4⋊C8—44no—
201#101—44no—
202#101—44no—
203C22⋊C8—24no—
204C22⋊C8—44no—
205#97—42no—
206C4×C8—24no—
207C4×C8—42no—
208C4×C8—44no—
209#95—21no—
210#95—22no—
211#95—21no—
212#95—22no—
213#95—22no—
214#95—21no—
215#95—22no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95C22×C411yescharacteristic
223#95C22×C412yes—
224#95C22×C411yescharacteristic
225C24C2×C22⋊C411yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×Q8—42no—
235C2×Q8—416no—
236C2×Q8—81no—
237C2×Q8—82no—
238C2×Q8#13514yes—
239C2×Q8#13714yes—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262M16—44no—
263M16—44no—
264C2×C8—44no—
265C2×C8—44no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—82no—
287C4⋊C4—81no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—81no—
296C22⋊C4—82no—
297C42—24no—
298C42—22no—
299C42—24no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—44no—
304C42—42no—
305C42—42no—
306C42—44no—
307C42H(5,𝔽2)14yes—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23C22×C22⋊C411yescharacteristic
314Q8—28no—
315Q8—28no—
316Q8—48no—
317Q8—44no—
318Q8—84no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—48no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4#41614yes—
357C2×C4#41914yes—
358C2×C4#51812yes—
359C2×C4#53412yes—
360C2×C4#53412yes—
361C2×C4C2×#13512yes—
362C2×C4C2×#13712yes—
363C8—84no—
364C8—84no—
365C22—21no—
366C22—21no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22#230811yescharacteristic
373C22C2×#41611yescharacteristic
374C22C2×#41911yescharacteristic
375C4—24no—
376C4—24no—
377C4—22no—
378C4—22no—
379C4—42no—
380C4—41no—
381C4—41no—
382C4—42no—
383C4—41no—
384C4—42no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4—82no—
389C4—81no—
390C4#319014yes—
391C2—21no—
392C2—21no—
393C2—81no—
394C2#2176211yescharacteristic
395C2#2178111yescharacteristic
396C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #9484C2semidirect
5. #9392C2semidirect
8. #8766C2semidirect
9. #8392C2semidirect

Maximal subgroups

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

subgroupindex
#358452
#207302
#173522
#94842
#93922
#91922
#90342
#87662
#83922