Finite Group Databasebeta

← #215166

Subgroups — #215166

Order 512.

1456 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1456 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2540C211yesmaximal, characteristic
2#21914C212yesmaximal, complemented
3#9653C214yesmaximal, complemented
4#9649C214yesmaximal, complemented
5#8305C212yesmaximal, complemented
6#8302C212yesmaximal, complemented
7C22×#124C2211yescharacteristic
8C22×C22⋊Q8C412yes—
9C22×#121C412yes—
10C22×C4×D4C2211yescharacteristic
11#2540—24no—
12#2540—24no—
13C2×#399C412yes—
14C2×#392C412yes—
15C2×#390C2214yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17#1917—24no—
18#1916—24no—
19#1817C2214yes—
20#1797C2214yes—
21#1749C2212yes—
22C4×C22⋊C8C2212yes—
23C2×#323C2212yes—
24C2×#323C2214yes—
25C2×#323C2212yes—
26C2×#323C2212yes—
27C2×#323C2214yes—
28C2×#323C2212yes—
29C23×Q8C2×C411yescharacteristic
30C23×D4C2×C411yescharacteristic
31C23×D4C2×C414yes—
32C23×D4C2×C411yescharacteristic
33C24×C4C2×C412yes—
34C2×#124—24no—
35C2×#124—42no—
36C2×#124—42no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—48no—
41C2×#121—22no—
42C2×#121—24no—
43C2×#121—22no—
44C2×#121—48no—
45C2×C4×D4—24no—
46C2×C4×D4—42no—
47C2×C4×D4—42no—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C414yes—
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2×C414yes—
53C22×C22⋊C4C2×C414yes—
54C22×C22⋊C4C2312yes—
55C22×C22⋊C4C2312yes—
56C22×C42C2311yescharacteristic
57C22×C42C2312yes—
58C2×C4⋊C8—24no—
59C2×C22⋊C8C2314yes—
60C2×C22⋊C8C2314yes—
61C2×#97—22no—
62C2×C4×C8—22no—
63#399—22no—
64#399—24no—
65#399—22no—
66#392—22no—
67#392—24no—
68#392—22no—
69#390—24no—
70#390—24no—
71#390—24no—
72#390—24no—
73C4×C4⋊C4D414yes—
74C4×C4⋊C4D414yes—
75C2×#95C2×C412yes—
76#336—24no—
77#336—28no—
78#336—22no—
79#336—22no—
80#323—42no—
81#323—44no—
82#323—42no—
83#323—42no—
84#323—44no—
85#323—42no—
86C25C22×C412yes—
87C25C22×C412yes—
88C22×Q8—22no—
89C22×Q8—41no—
90C22×Q8—41no—
91C22×Q8—44no—
92C22×D4—24no—
93C22×D4—28no—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—24no—
97C22×D4—24no—
98C22×D4—41no—
99C22×D4—41no—
100C22×D4—42no—
101C22×D4—41no—
102C22×D4—41no—
103C22×D4—44no—
104C22×D4—44no—
105C22×D4—48no—
106C22×D4—84no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—42no—
110C23×C4—42no—
111C23×C4—44no—
112C23×C4—44no—
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C22×C414yes—
116C23×C4C22×C412yes—
117C23×C4C22×C412yes—
118C23×C4C2411yescharacteristic
119C22×C8—24no—
120C22×C8—24no—
121#124—416no—
122C22⋊Q8—48no—
123C22⋊Q8—416no—
124C22⋊Q8—48no—
125#121—48no—
126#121—416no—
127#121—48no—
128C4×D4—416no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4C22⋊C412yes—
137C2×C4⋊C4C22⋊C414yes—
138C2×C4⋊C4C22⋊C412yes—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—48no—
156C2×C22⋊C4—48no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—82no—
160C2×C42C2×D414yes—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D414yes—
164C4⋊C8—84no—
165C22⋊C8—24no—
166C22⋊C8—24no—
167C22⋊C8—24no—
168C22⋊C8—24no—
169#97—82no—
170C4×C8—82no—
171#95C22⋊C412yes—
172#95C22⋊C414yes—
173#95C22⋊C412yes—
174C24—22no—
175C24—22no—
176C24—22no—
177C24—22no—
178C24—22no—
179C24—22no—
180C24—22no—
181C24—22no—
182C24—22no—
183C24—24no—
184C24—24no—
185C24—42no—
186C24—42no—
187C24—42no—
188C24—42no—
189C24—44no—
190C24—42no—
191C24—42no—
192C24#13114yes—
193C24C23×C411yescharacteristic
194C2×Q8—44no—
195C2×Q8—44no—
196C2×Q8—44no—
197C2×Q8—84no—
198C2×Q8—84no—
199C2×D4—48no—
200C2×D4—44no—
201C2×D4—48no—
202C2×D4—48no—
203C2×D4—44no—
204C2×D4—48no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—48no—
208C2×D4—44no—
209C2×D4—48no—
210C2×D4—44no—
211C2×D4—84no—
212C2×D4—84no—
213C2×D4—84no—
214C2×D4—84no—
215C2×D4—816no—
216C2×D4—816no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—28no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—41no—
234C22×C4—44no—
235C22×C4—41no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—88no—
246C22×C4—88no—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C414yes—
249C22×C4C2×C22⋊C414yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C22×D412yes—
252C22×C4H(5,𝔽2)12yes—
253C22×C4H(5,𝔽2)12yes—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—44no—
259C2×C8—44no—
260C4⋊C4—28no—
261C4⋊C4—24no—
262C4⋊C4—44no—
263C4⋊C4—48no—
264C4⋊C4—48no—
265C4⋊C4#10018yes—
266C22⋊C4—48no—
267C22⋊C4—48no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—48no—
275C22⋊C4—48no—
276C42—28no—
277C42—42no—
278C42—42no—
279C42—44no—
280C23—21no—
281C23—21no—
282C23—22no—
283C23—24no—
284C23—24no—
285C23—24no—
286C23—44no—
287C23—44no—
288C23—44no—
289C23—44no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—44no—
300C23—42no—
301C23—42no—
302C23—44no—
303C23—44no—
304C23—42no—
305C23—42no—
306C23—82no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23—84no—
315C23—84no—
316C23—82no—
317C23—82no—
318C23—84no—
319C23—84no—
320C23—82no—
321C23—82no—
322C23C22×C22⋊C411yescharacteristic
323C23#51811yescharacteristic
324C23#51811yescharacteristic
325C23C2×#13112yes—
326C23#56812yes—
327Q8—88no—
328D4—416no—
329D4—816no—
330D4—88no—
331D4—816no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—88no—
358C2×C4—88no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4C2×#10018yes—
364C2×C4C2×#10014yes—
365C2×C4#53514yes—
366C8—84no—
367C8—84no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—44no—
377C22—44no—
378C22—44no—
379C22—84no—
380C22—84no—
381C22—84no—
382C22—84no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22—84no—
388C22—84no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—164no—
393C22—162no—
394C22—162no—
395C22#230911yescharacteristic
396C22#288012yes—
397C22#288311yescharacteristic
398C22#288411yescharacteristic
399C22C22×#10012yes—
400C22#297612yes—
401C22#297711yescharacteristic
402C22#297811yescharacteristic
403C4—28no—
404C4—24no—
405C4—24no—
406C4—44no—
407C4—42no—
408C4—82no—
409C4—82no—
410C2—22no—
411C2—21no—
412C2—21no—
413C2—84no—
414C2—162no—
415C2#1742112yes—
416C2#1742512yes—
417C2#2179211yescharacteristic
418C2#2179611yescharacteristic
419C2#2179711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2540C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21914C2semidirect
3. #9653C2semidirect
4. #9649C2semidirect
5. #8305C2semidirect
6. #8302C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25402
#219142
#96532
#96492
#83052
#83022