Finite Group Databasebeta

← #162087

Subgroups — #162087

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20772C211yesmaximal, characteristic
3C2×#1839C211yesmaximal, characteristic
4#7729C214yesmaximal, complemented
5#7681C214yesmaximal
6#7605C212yesmaximal, complemented
7#7604C212yesmaximal, complemented
8#2864—21no—
9#2863—21no—
10#2863C2212yes—
11#2862C411yescharacteristic
12#2859C411yescharacteristic
13#2857C411yescharacteristic
14#2857C411yescharacteristic
15#2855—21no—
16#2855C2212yes—
17#2851—21no—
18C4×C4⋊Q8C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C4×#125—21no—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8C412yes—
23#2295—21no—
24#2295C412yes—
25#2283—21no—
26C2×C43C2211yescharacteristic
27#1839—21no—
28#1839—21no—
29#1839—21no—
30#1839—21no—
31#1839C2212yes—
32#1839C2212yes—
33#1839C2212yes—
34#1839C2212yes—
35#1728—21no—
36#1728—21no—
37#1728C2212yes—
38C2×#336—21no—
39C2×#336—21no—
40C2×#336C2212yes—
41#1272C2218yes—
42#1271C2218yes—
43C22×M16—22no—
44C23×C8—22no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C412yes—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59#391—21no—
60#391—21no—
61#391—21no—
62#391—21no—
63#391C2×C412yes—
64#391C2×C412yes—
65#391C2×C412yes—
66#391C2312yes—
67#389—21no—
68#389—21no—
69#389—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389C2×C412yes—
76#389C2×C412yes—
77#389C2×C412yes—
78#389C2312yes—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#385—22no—
88#385—22no—
89#384C2×C411yescharacteristic
90#384C2×C411yescharacteristic
91#383—22no—
92#382—21no—
93#382—21no—
94#382—21no—
95#381—22no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380C2×C412yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2×C412yes—
106C4×C22⋊C4—22no—
107#376—22no—
108C43C2×C411yescharacteristic
109C43C2×C411yescharacteristic
110C43D411yescharacteristic
111C43D411yescharacteristic
112C43D411yescharacteristic
113C43D411yescharacteristic
114C43C2311yescharacteristic
115C43C2311yescharacteristic
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336C2314yes—
127#336C2314yes—
128C22×Q8—21no—
129C22×Q8C22×C412yes—
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×M16—44no—
138C2×M16—44no—
139C2×M16—44no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—44no—
143C22×C8—44no—
144C22×C8—44no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147#125—44no—
148C22⋊Q8—44no—
149C22⋊Q8—44no—
150C4×Q8—44no—
151C4×Q8—44no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42C22⋊C411yescharacteristic
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2411yescharacteristic
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221C24C2×C22⋊C411yescharacteristic
222C2×Q8—24no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4#13714yes—
264C22×C4C23×C411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267M16—88no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C2×C8—88no—
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—41no—
307C42—41no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42—44no—
321C42—44no—
322C42#13114yes—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23#41012yes—
328C23#41912yes—
329C23#42112yes—
330C23C22×C22⋊C411yescharacteristic
331Q8—84no—
332Q8—84no—
333C2×C4—28no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—28no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4#40812yes—
387C2×C4#40812yes—
388C2×C4#41312yes—
389C2×C4#41912yes—
390C2×C4#42112yes—
391C2×C4C2×#13112yes—
392C2×C4C2×#13712yes—
393C2×C4#57512yes—
394C8—84no—
395C8—84no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22#151514yes—
403C22C2×#40811yescharacteristic
404C22C2×#41011yescharacteristic
405C22C2×#41311yescharacteristic
406C22C2×#41911yescharacteristic
407C22#289611yescharacteristic
408C22C2×#42111yescharacteristic
409C22C2×#42111yescharacteristic
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—82no—
425C4—82no—
426C4—82no—
427C4—82no—
428C2—24no—
429C2#1738311yescharacteristic
430C2#1746511yescharacteristic
431C2#1747911yescharacteristic
432C2C2×#151511yescharacteristic
433C2#1759311yescharacteristic
434C2#1766211yescharacteristic
435C2#1774811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7729C2semidirect
6. #7605C2semidirect
7. #7604C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#207722
C2×#18392
#77292
#76812
#76052
#76042