Finite Group Databasebeta

← #110956

Subgroups — #110956

Order 512.

775 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21762C211yesmaximal, characteristic
2#20526C211yesmaximal, characteristic
3C2×#1508C211yesmaximal, characteristic
4C2×#1472C211yesmaximal, characteristic
5#8051C211yesmaximal, characteristic
6#8008C211yesmaximal, characteristic
7#7925C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1892—22no—
16#1873—22no—
17#1823C412yescomplemented
18#1787C412yescomplemented
19#1752—22no—
20#1752—22no—
21#1749—22no—
22#1508—22no—
23#1508—22no—
24C2×#328C414yescomplemented
25#1472—22no—
26#1472C414yescomplemented
27C2×#324—22no—
28#1297—22no—
29#1293—22no—
30#1282—22no—
31C23×Q8C2×C411yescharacteristic
32C24×C4C2311yescharacteristic
33C22×M16—21no—
34C22×M16—21no—
35C23×C8—21no—
36C23×C8D412yes—
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×C22⋊Q8C2×C412yes—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C42—21no—
55C2×#103—24no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407—21no—
62#407—21no—
63#407C2×C412yes—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8C2×C412yes—
75C2×C22⋊C8C2×C412yes—
76C2×C22⋊C8C2×C412yes—
77C2×C22⋊C8D412yes—
78C2×C22⋊C8D412yes—
79C2×C22⋊C8D412yes—
80C2×#97—24no—
81#381—21no—
82#380—21no—
83#380C2×C412yes—
84#379—21no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4D412yes—
90C4×C22⋊C4D412yes—
91C2×#95C2×C412yes—
92#340—24no—
93#336—24no—
94#328—28no—
95#324—28no—
96C25#4111yescharacteristic
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—82no—
103C22×Q8C4212yes—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—82no—
108C23×C4C22×C411yescharacteristic
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C23×C4#4111yescharacteristic
122C2×M16—22no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—82no—
127C2×M16—82no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—41no—
135C22×C8—82no—
136C22×C8—82no—
137C22×C8C22⋊C414yes—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154#117—82no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4C4212yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—41no—
183C2×C42—82no—
184#103—84no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8C22⋊C414yes—
201#97—84no—
202#95—22no—
203#95—22no—
204#95—41no—
205#95C22⋊C414yes—
206C24—24no—
207C24—42no—
208C24—81no—
209C24#11711yescharacteristic
210C24C4×D411yescharacteristic
211C24C4×D411yescharacteristic
212C24#12111yescharacteristic
213C24#12311yescharacteristic
214C24#12411yescharacteristic
215C24#12611yescharacteristic
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—44no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8—82no—
225C2×Q8—82no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4C2×C4211yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4#11711yescharacteristic
258C22×C4#11711yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4C4×D411yescharacteristic
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12311yescharacteristic
276C22×C4#12311yescharacteristic
277C22×C4#12411yescharacteristic
278C22×C4#12411yescharacteristic
279C22×C4#12611yescharacteristic
280C22×C4#12711yescharacteristic
281M16—82no—
282M16—82no—
283M16—82no—
284M16—82no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—82no—
293C2×C8—82no—
294C2×C8—82no—
295C2×C8—82no—
296C2×C8—82no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—44no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C42—44no—
316C42—82no—
317C42—82no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—88no—
339C23C4×C22⋊C411yescharacteristic
340C23#38611yescharacteristic
341C23#38811yescharacteristic
342C23#38811yescharacteristic
343C23#38811yescharacteristic
344C23#38811yescharacteristic
345C23#39011yescharacteristic
346C23#44414yes—
347Q8—44no—
348Q8—82no—
349Q8—82no—
350Q8—82no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—88no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4C4×C22⋊C411yescharacteristic
395C2×C4C4×C22⋊C411yescharacteristic
396C2×C4#38611yescharacteristic
397C2×C4#38611yescharacteristic
398C2×C4#38811yescharacteristic
399C2×C4#38811yescharacteristic
400C2×C4#39011yescharacteristic
401C2×C4#39011yescharacteristic
402C8—44no—
403C8—44no—
404C8—82no—
405C8—82no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—44no—
424C22—44no—
425C22—44no—
426C22—44no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22#143711yescharacteristic
431C22#176512yes—
432C22#190212yes—
433C22#193812yes—
434C22#197512yes—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—84no—
443C4—82no—
444C4—82no—
445C4—162no—
446C4#191312yes—
447C2—22no—
448C2—22no—
449C2—22no—
450C2—22no—
451C2—41no—
452C2—41no—
453C2—41no—
454C2#1093011yescharacteristic
455C2#1093811yescharacteristic
456C2#1095211yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1823C4semidirect
18. #1787C4semidirect
24. C2×#328C4semidirect
26. #1472C4semidirect

Maximal subgroups

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

subgroupindex
#217622
#205262
C2×#15082
C2×#14722
#80512
#80082
#79252