Finite Group Databasebeta

← #413091

Subgroups — #413091

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35317C211yesmaximal, characteristic
2C2×#2038C214yesmaximal, complemented
3C2×#1966C211yesmaximal, characteristic, complemented
4C2×#1963C211yesmaximal, characteristic
5#21201C211yesmaximal, characteristic
6#21172C211yesmaximal, characteristic, complemented
7#14274C211yesmaximal, characteristic, complemented
8#10998C214yesmaximal, complemented
9#10701C211yesmaximal, characteristic, complemented
10C2×#536C2212yes—
11C2×#536C2212yes—
12C2×#532C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#428C2212yes—
15C2×#423C2211yescharacteristic
16C22×#104C2214yes—
17C2×#417C2214yes—
18C22×#101C2214yes—
19C2×C4×M16C2211yescharacteristic
20#2868—21no—
21#2858—21no—
22#2658—22no—
23#2542—24no—
24#2501—24no—
25C2×#391C2214yes—
26C2×#382C2211yescharacteristic
27C4×#117C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#2038—216no—
30#1966—24no—
31#1963—24no—
32#1961—21no—
33#1960—21no—
34#1958—22no—
35#1949—24no—
36#1736C2212yes—
37C2×#339C2212yes—
38C2×#339C2212yes—
39#1445—24no—
40C22×#41C2312yes—
41C23×Q8C2312yes—
42C22×M16C2312yes—
43C22×M16C2312yes—
44#536—28no—
45#536—28no—
46#536—44no—
47#536—44no—
48#532—22no—
49#532—22no—
50#532—22no—
51#532—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8D411yescharacteristic
62C2×#127D411yescharacteristic
63C2×#124—22no—
64C2×#124—22no—
65C2×#124D412yes—
66C2×#123—24no—
67C2×#123—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×#121—22no—
71C2×C4×D4—22no—
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C22×C4⋊C4D414yes—
75C22×C4⋊C4C2312yes—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#428—28no—
79C2×#107—22no—
80C2×#106—22no—
81#423—22no—
82#423D414yes—
83C2×C4⋊C8—22no—
84C2×#104—28no—
85C2×#104—28no—
86C2×#104—44no—
87#417—28no—
88#417—44no—
89C2×#103—24no—
90C2×#102—24no—
91C2×#101—28no—
92C2×#101—28no—
93C2×#101—44no—
94C4×M16—22no—
95C4×M16—22no—
96C2×#97—21no—
97C2×C4×C8—21no—
98#399—216no—
99#391—28no—
100#391—44no—
101#390—28no—
102#384—28no—
103#382—22no—
104#382—21no—
105#382—21no—
106C4×C4⋊C4D414yes—
107C4×C22⋊C4—22no—
108#376—21no—
109#376—22no—
110#376—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—28no—
114C43—21no—
115#340—24no—
116#339—24no—
117#339D418yescomplemented
118C2×#41—22no—
119C2×#41—24no—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41—22no—
123C2×#41—24no—
124C2×#41—24no—
125C2×#41—24no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—24no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—24no—
134C22×Q8—24no—
135C22×Q8—24no—
136C22×Q8C2×D412yes—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4C2×D412yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C2×D414yes—
144C23×C4C2×D412yes—
145C23×C4C2411yescharacteristic
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—24no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—24no—
154C2×M16C2×D414yes—
155C2×M16C2×D414yes—
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—48no—
161C4⋊Q8—48no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—44no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—44no—
168#127—22no—
169#127—22no—
170#124—24no—
171#124—24no—
172#124—48no—
173#124—48no—
174#123—416no—
175#123—416no—
176C22⋊Q8—48no—
177C22⋊Q8—48no—
178C22⋊Q8—48no—
179C22⋊Q8—48no—
180#121—44no—
181#121—44no—
182C4×D4—44no—
183C4×D4—44no—
184#117—22no—
185#117—22no—
186#117C2×D414yes—
187#117#4114yes—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—24no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42C2×D414yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222#107—44no—
223#106—44no—
224C4⋊C8—28no—
225#104—416no—
226#103—48no—
227#102—48no—
228#101—416no—
229#97—42no—
230C4×C8—42no—
231#95—24no—
232#95—416no—
233#95—42no—
234#95—42no—
235#95—44no—
236C24—22no—
237C24C22×D411yescharacteristic
238#41—44no—
239#41—44no—
240#41—44no—
241#41—44no—
242#41—44no—
243#41—44no—
244#41—44no—
245#41—44no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—48no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—44no—
288C22×C4—48no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—48no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4#12012yes—
301C22×C4#12012yes—
302C22×C4#12114yes—
303C22×C4#12314yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D412yes—
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4112yes—
312C22×C4H(5,𝔽2)12yes—
313C22×C4H(5,𝔽2)12yes—
314M16—24no—
315M16—24no—
316M16—28no—
317M16—42no—
318M16—42no—
319M16—44no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—416no—
329C4⋊C4—416no—
330C4⋊C4—48no—
331C4⋊C4—48no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4#12414yes—
337C4⋊C4#12714yes—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—416no—
341C22⋊C4—416no—
342C22⋊C4—48no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—24no—
347C42—48no—
348C42—42no—
349C42—42no—
350C42#12112yes—
351C42#12112yes—
352C42#12114yes—
353C42#12312yes—
354C42#12312yes—
355C42#12414yes—
356C23—24no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—48no—
361C23C2×#12011yescharacteristic
362C23C2×#12111yescharacteristic
363C23C2×#12311yescharacteristic
364C23D4212yes—
365C23#54612yes—
366Q8—44no—
367Q8—44no—
368Q8—82no—
369Q8—82no—
370Q8—84no—
371Q8—84no—
372Q8—82no—
373Q8—82no—
374D4—44no—
375D4—44no—
376D4—88no—
377D4—84no—
378D4—84no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—48no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—48no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—88no—
404C2×C4—82no—
405C2×C4—84no—
406C2×C4#39414yes—
407C2×C4#39414yes—
408C2×C4C2×#12111yescharacteristic
409C2×C4C2×#12111yescharacteristic
410C2×C4C2×#12311yescharacteristic
411C2×C4C2×#12411yescharacteristic
412C2×C4C2×#12411yescharacteristic
413C2×C4C2×#12711yescharacteristic
414C2×C4#53212yes—
415C2×C4#53212yes—
416C2×C4#53511yescharacteristic
417C2×C4#53511yescharacteristic
418C2×C4#54012yes—
419C2×C4#54012yes—
420C2×C4D4212yes—
421C2×C4D4212yes—
422C2×C4#54612yes—
423C2×C4#55012yes—
424C2×C4#55012yes—
425C2×C4#55912yes—
426C8—44no—
427C8—44no—
428C22—24no—
429C22—24no—
430C22—22no—
431C22—22no—
432C22—84no—
433C22—82no—
434C22—84no—
435C22—84no—
436C22C2×#39411yescharacteristic
437C22#242812yes—
438C22#243311yescharacteristic
439C22#243511yescharacteristic
440C22#244512yes—
441C22#302014yes—
442C4—24no—
443C4—24no—
444C4—44no—
445C4—44no—
446C4—84no—
447C4—84no—
448C4—82no—
449C4—162no—
450C4#242812yes—
451C4#244512yes—
452C4#254111yescharacteristic
453C4#254811yescharacteristic
454C4#264111yescharacteristic
455C4#285811yescharacteristic
456C2—22no—
457C2—22no—
458C2—162no—
459C2#1514011yescharacteristic
460C2#2349412yes—
461C2#2575012yes—
462C2#2587912yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2038C2semidirect
3. C2×#1966C2semidirect
6. #21172C2semidirect
7. #14274C2semidirect
8. #10998C2semidirect
9. #10701C2semidirect
117. #339D4semidirect

Maximal subgroups

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

subgroupindex
#353172
C2×#20382
C2×#19662
C2×#19632
#212012
#211722
#142742
#109982
#107012