Finite Group Databasebeta

← #215669

Subgroups — #215669

Order 512.

764 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3#20778C211yesmaximal, characteristic
4#9895C212yesmaximal, complemented
5#9894C212yesmaximal, complemented
6#9893C214yesmaximal, complemented
7#8720C214yesmaximal
8#2864C2212yes—
9#2863—21no—
10#2863—21no—
11#2862C2211yescharacteristic
12#2859C2211yescharacteristic
13#2857C411yescharacteristic
14#2857C411yescharacteristic
15#2855—21no—
16#2855—21no—
17#2851C412yes—
18C4×C4⋊Q8C411yescharacteristic
19C4×C4⋊Q8C411yescharacteristic
20C4×#125C412yes—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8—21no—
23#2295—21no—
24#2295—21no—
25#2283C2212yes—
26C2×C43C2211yescharacteristic
27#1839—21no—
28#1839—21no—
29#1831—21no—
30#1831—21no—
31#1817—21no—
32#1817—21no—
33#1797—21no—
34#1797—21no—
35#1767C2212yes—
36#1767C2212yes—
37#1749C2212yes—
38#1749C2212yes—
39#1745C2212yes—
40#1745C2212yes—
41#1295C2218yes—
42#1295C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125C2×C412yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×#97C2314yes—
62C2×#97C2314yes—
63#391—21no—
64#391—21no—
65#391—21no—
66#391—21no—
67#391—21no—
68#391—21no—
69#391—21no—
70#391—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2×C412yes—
82#389C2×C412yes—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#385—22no—
92#385—22no—
93#384C2×C411yescharacteristic
94#384C2×C411yescharacteristic
95#383—22no—
96#382C2×C412yes—
97#382C2312yes—
98#382C2312yes—
99#381—22no—
100#380—21no—
101#380—21no—
102#380C2×C412yes—
103#380C2×C412yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C22⋊C4—22no—
111#376—22no—
112C43C2×C411yescharacteristic
113C43C2×C411yescharacteristic
114C43D411yescharacteristic
115C43D411yescharacteristic
116C43D411yescharacteristic
117C43D411yescharacteristic
118C43C2311yescharacteristic
119C43C2311yescharacteristic
120#336—22no—
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128C22×Q8—21no—
129C22×Q8—21no—
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C22×C8—24no—
138C22×C8—24no—
139C4⋊Q8—42no—
140C4⋊Q8—42no—
141#125—44no—
142C22⋊Q8—44no—
143C22⋊Q8—44no—
144C4×Q8—44no—
145C4×Q8—44no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C42—24no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—42no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2411yescharacteristic
201C4⋊C8—44no—
202C4⋊C8—44no—
203C22⋊C8—44no—
204C22⋊C8—44no—
205#97—28no—
206#97—28no—
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—42no—
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—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
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×C4C2×C22⋊C411yescharacteristic
264C22×C4C23×C411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C2×C8—44no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42—41no—
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—44no—
319C42—44no—
320C422−1+414yes—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23#42112yes—
328C23C22×C22⋊C411yescharacteristic
329Q8—84no—
330Q8—84no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
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—21no—
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—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
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#41312yes—
385C2×C4#41312yes—
386C2×C4#42112yes—
387C2×C4#42112yes—
388C2×C4#42112yes—
389C2×C4#51912yes—
390C2×C4#53612yes—
391C2×C4#53612yes—
392C8—84no—
393C8—84no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22#231011yescharacteristic
402C22C2×#41311yescharacteristic
403C22C2×#41311yescharacteristic
404C22C2×#42111yescharacteristic
405C22C2×#42111yescharacteristic
406C22C2×#42111yescharacteristic
407C22C2×#42111yescharacteristic
408C4—22no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—22no—
415C4—22no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—82no—
426C2—42no—
427C2#1765911yescharacteristic
428C2#1766811yescharacteristic
429C2#1767311yescharacteristic
430C2#1767711yescharacteristic
431C2#2183111yescharacteristic
432C2#2184511yescharacteristic
433C2#2184511yescharacteristic
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 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9895C2semidirect
5. #9894C2semidirect
6. #9893C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#207782
#98952
#98942
#98932
#87202