Finite Group Databasebeta

← #165061

Subgroups — #165061

Order 512.

697 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36535C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic
3#22734C211yesmaximal, characteristic
4#7717C212yesmaximal
5#7677C212yesmaximal
6#7663C211yesmaximal, characteristic
7#7660C211yesmaximal, characteristic
8#7652C211yesmaximal, characteristic
9#7619C212yesmaximal
10#7585C212yesmaximal
11#7570C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2864C2212yes—
14#2859C2211yescharacteristic
15#2590—22no—
16#2588—22no—
17#2581—21no—
18#2571C411yescharacteristic
19#2570C412yes—
20#2565—21no—
21#2552C411yescharacteristic
22#2550C412yes—
23#2370—21no—
24#2355—21no—
25#2355C411yescharacteristic
26#2355C411yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×#125C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—22no—
32#1939—21no—
33#1917—22no—
34#1839C2212yes—
35#1831C2212yes—
36#1831C2211yescharacteristic
37#1831C2211yescharacteristic
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2212yes—
42#1745C2212yes—
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2211yescharacteristic
49#1271C2211yescharacteristic
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×#125C2×C412yes—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C8×Q8—44no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×#97—22no—
68C2×C4×C8—22no—
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71#400—22no—
72#400—22no—
73#398—22no—
74#398—22no—
75#398—22no—
76#395—22no—
77#391—21no—
78#391—21no—
79#391—21no—
80#391—22no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—22no—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2311yescharacteristic
91#389C2312yes—
92#389C2311yescharacteristic
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#384—21no—
104#384C2×C411yescharacteristic
105#384C2×C411yescharacteristic
106#384C2×C411yescharacteristic
107#382—21no—
108#382—21no—
109#382C2×C412yes—
110#382C2×C412yes—
111#382C2×C412yes—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C43C2312yes—
128C43C2311yescharacteristic
129#336—21no—
130#336—22no—
131#336—21no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141C22×Q8—21no—
142C22×Q8C22⋊C411yescharacteristic
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C4⋊Q8—22no—
150C4⋊Q8—41no—
151#125—24no—
152#125—42no—
153#125—42no—
154#125—42no—
155C4×Q8—24no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—42no—
161C4×Q8—42no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42C22×C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C412yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2411yescharacteristic
206C4⋊C8—44no—
207C4⋊C8—42no—
208C4⋊C8—42no—
209C4⋊C8—44no—
210C4⋊C8—42no—
211C4⋊C8—42no—
212#97—44no—
213C4×C8—44no—
214C4×C8—42no—
215C4×C8—42no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—21no—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—22no—
231#95—21no—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—81no—
240C2×Q8#10114yes—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—41no—
276C4⋊C4—21no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—81no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4#13112yes—
309C42—22no—
310C42—21no—
311C42—22no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—24no—
318C42—22no—
319C42—42no—
320C42—41no—
321C42—41no—
322C42—42no—
323C42—41no—
324C42—42no—
325C42—42no—
326C42—82no—
327C42—82no—
328C42#13512yes—
329C42#13512yes—
330C42AGL(1,ℤ/8)12yes—
331C42#13712yes—
332C23C22×C22⋊C411yescharacteristic
333Q8—28no—
334Q8—84no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4C2×#10114yes—
378C2×C4C2×#10112yes—
379C2×C4#41612yes—
380C2×C4#41612yes—
381C2×C4#41712yes—
382C2×C4#41712yes—
383C2×C4C2×#13111yescharacteristic
384C2×C4#56811yescharacteristic
385C2×C4C2×#13511yescharacteristic
386C2×C4C2×#13511yescharacteristic
387C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
388C2×C4C2×#13711yescharacteristic
389C8—42no—
390C8—42no—
391C8—81no—
392C8—82no—
393C8—82no—
394C8—81no—
395C22#288011yescharacteristic
396C22#288411yescharacteristic
397C22C22×#10111yescharacteristic
398C22C2×#41611yescharacteristic
399C22C2×#41711yescharacteristic
400C22#290111yescharacteristic
401C22#290111yescharacteristic
402C4—21no—
403C4—21no—
404C4—24no—
405C4—22no—
406C4—22no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—42no—
412C4—41no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—82no—
417C4—81no—
418C4—82no—
419C4—81no—
420C4—81no—
421C4#155214yes—
422C2#1743011yescharacteristic
423C2#1751011yescharacteristic
424C2#1751211yescharacteristic
425C2#1761611yescharacteristic
426C2#1762811yescharacteristic
427C2#1769211yescharacteristic
428C2C2×#155211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#365352
#228322
#227342
#77172
#76772
#76632
#76602
#76522
#76192
#75852
#75702