Finite Group Databasebeta

← #213588

Subgroups — #213588

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2#21820C212yesmaximal, complemented
3#9084C214yesmaximal, complemented
4#9080C214yesmaximal, complemented
5#8703C212yesmaximal, complemented
6#8302C212yesmaximal
7C22×C22⋊Q8C412yes—
8C22×C4×Q8C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#432C2212yes—
11C2×#431C2212yes—
12#2536—22no—
13#2536—22no—
14#2536—22no—
15#2536—22no—
16C2×#398C412yes—
17C2×#394C412yes—
18C2×#388C2214yes—
19C2×#382C2211yescharacteristic
20C2×#380C412yes—
21#1985—22no—
22#1985—22no—
23#1984—22no—
24#1984—22no—
25#1817C2214yes—
26#1817C2214yes—
27C2×#323C2212yes—
28C2×#323C2214yes—
29C2×#323C2212yes—
30#1280C2212yes—
31#1280C2214yes—
32#1280C2212yes—
33C23×Q8C2×C411yescharacteristic
34C23×D4C2×C411yescharacteristic
35C24×C4C2×C412yes—
36C2×C22⋊Q8—24no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—48no—
40C2×C4×Q8—41no—
41C2×C4×Q8—42no—
42C2×C4×Q8—41no—
43C2×C4×Q8D412yes—
44C2×C4×Q8D412yes—
45C2×C4×Q8D412yes—
46C2×C4×Q8D412yes—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—42no—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C414yes—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C414yes—
58C22×C22⋊C4C2×C414yes—
59C22×C22⋊C4C2312yes—
60C22×C42C2311yescharacteristic
61C22×C42C2312yes—
62#432—42no—
63#432—42no—
64#431—42no—
65#431—42no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8C2314yes—
69C2×C22⋊C8C2314yes—
70C2×#97—22no—
71C2×C4×C8—22no—
72#398—22no—
73#398—22no—
74#398—22no—
75#398—22no—
76#394—22no—
77#394—22no—
78#394—22no—
79#394—22no—
80#388—24no—
81#388—24no—
82#388—24no—
83#388—24no—
84#382—22no—
85#382—22no—
86#380—22no—
87#380—22no—
88#380—24no—
89C2×#95C2×C412yes—
90C2×#95C2×C414yes—
91C2×#95C2312yes—
92#336—24no—
93#336—24no—
94#336—24no—
95#336—24no—
96#323—42no—
97#323—44no—
98#323—42no—
99C25C22×C412yes—
100C22×Q8—24no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—42no—
104C22×Q8C22⋊C412yes—
105C22×Q8C22⋊C411yescharacteristic
106C22×Q8C22⋊C411yescharacteristic
107C22×D4—24no—
108C22×D4—22no—
109C22×D4—41no—
110C22×D4—41no—
111C22×D4—42no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—44no—
117C23×C4—44no—
118C23×C4C22×C412yes—
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C412yes—
122C23×C4C22×C414yes—
123C23×C4C22×C412yes—
124C23×C4C2411yescharacteristic
125C22×C8—24no—
126C22×C8—24no—
127C22⋊Q8—48no—
128C22⋊Q8—416no—
129C22⋊Q8—48no—
130C4×Q8—42no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×D4—44no—
139C4×D4—44no—
140C4×D4—44no—
141C4×D4—44no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—48no—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22⋊C411yescharacteristic
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C412yes—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—48no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C4⋊C8—44no—
187C4⋊C8—44no—
188C22⋊C8—24no—
189C22⋊C8—24no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192#97—44no—
193C4×C8—44no—
194#95—24no—
195#95—24no—
196#95—22no—
197#95—22no—
198#95—24no—
199C24—22no—
200C24—22no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24—42no—
206C24—42no—
207C24—42no—
208C24—42no—
209C24#13114yes—
210C24C23×C411yescharacteristic
211C2×Q8—24no—
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—41no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—48no—
220C2×Q8—48no—
221C2×Q8#10014yes—
222C2×D4—48no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—48no—
226C2×D4—44no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—88no—
261C22×C4—88no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C22×D412yes—
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C4H(5,𝔽2)11yescharacteristic
272C22×C4H(5,𝔽2)12yes—
273C2×C8—48no—
274C2×C8—48no—
275C2×C8—44no—
276C2×C8—44no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—48no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C22⋊C4—48no—
290C22⋊C4—48no—
291C22⋊C4—48no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—48no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C23—22no—
308C23—21no—
309C23—21no—
310C23—24no—
311C23—24no—
312C23—24no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—82no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23—82no—
330C23—82no—
331C23—82no—
332C23—82no—
333C23C22×C22⋊C411yescharacteristic
334C23#51811yescharacteristic
335C23#51811yescharacteristic
336C23C2×#13112yes—
337C23#56812yes—
338Q8—44no—
339Q8—48no—
340Q8—44no—
341D4—416no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—88no—
378C2×C4—88no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4C2×#10014yes—
384C2×C4C2×#10012yes—
385C2×C4#41312yes—
386C2×C4#53412yes—
387C2×C4#53412yes—
388C8—84no—
389C8—84no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—44no—
399C22—44no—
400C22—44no—
401C22—84no—
402C22—84no—
403C22—84no—
404C22—84no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22#230811yescharacteristic
410C22#288012yes—
411C22#288412yes—
412C22C22×#10011yescharacteristic
413C22C2×#41311yescharacteristic
414C22#297612yes—
415C22#297711yescharacteristic
416C22#297811yescharacteristic
417C4—22no—
418C4—22no—
419C4—22no—
420C4—22no—
421C4—24no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—84no—
427C4—82no—
428C2—22no—
429C2—21no—
430C2—21no—
431C2—84no—
432C2#1742312yes—
433C2#1742512yes—
434C2#2174411yescharacteristic
435C2#2175111yescharacteristic
436C2#2175211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21820C2semidirect
3. #9084C2semidirect
4. #9080C2semidirect
5. #8703C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
#218202
#90842
#90802
#87032
#83022