Finite Group Databasebeta

← #165534

Subgroups — #165534

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36508C211yesmaximal, characteristic
2#22022C212yesmaximal, complemented
3#7722C212yesmaximal, complemented
4#7696C212yesmaximal
5#7656C212yesmaximal
6#7649C211yesmaximal, characteristic
7#7624C212yesmaximal, complemented
8#7601C211yesmaximal, characteristic
9#7590C212yesmaximal, complemented
10C2×#557C411yescharacteristic
11C2×#556C411yescharacteristic
12#2862C2211yescharacteristic
13#2861C2212yes—
14#2565C412yes—
15#2546C411yescharacteristic
16#2545C412yes—
17#2542C411yescharacteristic
18#2519—22no—
19#2519—22no—
20#2429—22no—
21#2362—22no—
22C4×C4⋊Q8C2212yes—
23C4×#124C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25#1984—22no—
26#1939—22no—
27#1917—22no—
28#1917—22no—
29#1839C2212yes—
30#1839C2212yes—
31#1831C2212yes—
32#1831C2212yes—
33#1767C2212yes—
34#1744C2212yes—
35#1272C2212yes—
36#1272C2212yes—
37#1272C2212yes—
38#1272C2212yes—
39#1272C2211yescharacteristic
40#1272C2212yes—
41#1272C2211yescharacteristic
42#1270C2211yescharacteristic
43#1270C2212yes—
44#1270C2211yescharacteristic
45#557—44no—
46#556—44no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#126—22no—
52C2×#125—22no—
53C2×#125C2×C412yes—
54C2×#124C2×C411yescharacteristic
55C2×#124C2×C412yes—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—22no—
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8D412yes—
64C2×C4×D4C2×C411yescharacteristic
65C2×#117—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×#97—22no—
70C2×C4×C8—22no—
71#399—21no—
72#399—21no—
73#398—22no—
74#398—22no—
75#398—21no—
76#398—21no—
77#396—22no—
78#396—22no—
79#391—22no—
80#391C2×C412yes—
81#391C2×C412yes—
82#391C2311yescharacteristic
83#391C2312yes—
84#390C2×C412yes—
85#390C2311yescharacteristic
86#389—22no—
87#389—22no—
88#389—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#387—22no—
97#387—22no—
98#384C2×C412yes—
99#382—22no—
100#382C2×C411yescharacteristic
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C22⋊C4—22no—
109C43C2312yes—
110C43C2311yescharacteristic
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121#336C2312yes—
122C22×Q8—21no—
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22×C412yes—
125C22×Q8C22×C411yescharacteristic
126C22×D4C22×C411yescharacteristic
127C23×C4—21no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C4⋊Q8—24no—
133C4⋊Q8—42no—
134C4⋊Q8—42no—
135C4⋊Q8—42no—
136C4⋊Q8—44no—
137#126—48no—
138#125—44no—
139#125—48no—
140#124—22no—
141#124—41no—
142#124—44no—
143C22⋊Q8—48no—
144C22⋊Q8—44no—
145C4×Q8—24no—
146C4×Q8—44no—
147C4×Q8—44no—
148C4×Q8—44no—
149C4×Q8—42no—
150C4×Q8—44no—
151C4×D4—42no—
152#117—44no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C412yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C412yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2411yescharacteristic
194C4⋊C8—44no—
195C4⋊C8—82no—
196C4⋊C8—82no—
197#97—82no—
198C4×C8—44no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95C22⋊C412yes—
208#95C22⋊C412yes—
209#95C22⋊C411yescharacteristic
210#95C22⋊C411yescharacteristic
211C24—21no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—42no—
216C2×Q8—44no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8#10014yes—
223C2×Q8#13112yes—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—41no—
227C2×D4—41no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D412yes—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C4⋊C4—22no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—42no—
279C22⋊C4—44no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C42—21no—
285C42—22no—
286C42—21no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—24no—
296C42—24no—
297C42—42no—
298C42—42no—
299C42—41no—
300C42—44no—
301C42—44no—
302C42—42no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—42no—
307C23—42no—
308C23C22×C22⋊C411yescharacteristic
309Q8—44no—
310Q8—42no—
311Q8—42no—
312Q8—82no—
313Q8—82no—
314Q8—84no—
315D4—81no—
316D4—81no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—84no—
351C2×C4C2×#10012yes—
352C2×C4C2×#10014yes—
353C2×C4#42114yes—
354C2×C4C2×#13111yescharacteristic
355C2×C4#56811yescharacteristic
356C2×C4#57712yes—
357C2×C4#57812yes—
358C8—82no—
359C8—82no—
360C8—82no—
361C8—82no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—82no—
366C22—82no—
367C22#288011yescharacteristic
368C22#288411yescharacteristic
369C22C22×#10011yescharacteristic
370C22#289812yes—
371C22C2×#42112yes—
372C4—22no—
373C4—24no—
374C4—21no—
375C4—21no—
376C4—24no—
377C4—22no—
378C4—22no—
379C4—42no—
380C4—42no—
381C4—41no—
382C4—42no—
383C4—42no—
384C4—81no—
385C4—82no—
386C4—82no—
387C4—81no—
388C2—161no—
389C2#1742511yescharacteristic
390C2#1752412yes—
391C2#1760112yes—
392C2#1767511yescharacteristic
393C2#1777311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22022C2semidirect
3. #7722C2semidirect
7. #7624C2semidirect
9. #7590C2semidirect

Maximal subgroups

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

subgroupindex
#365082
#220222
#77222
#76962
#76562
#76492
#76242
#76012
#75902