Finite Group Databasebeta

← #110182

Subgroups — #110182

Order 512.

729 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21825C211yesmaximal, characteristic
2#21753C211yesmaximal, characteristic
3#20529C211yesmaximal, characteristic
4#17397C211yesmaximal, characteristic
5#8115C211yesmaximal, characteristic
6#8108C211yesmaximal, characteristic
7#8108C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13#2280C2211yescharacteristic
14#2280C2211yescharacteristic
15#1996—22no—
16#1996—22no—
17#1901—22no—
18#1901—22no—
19#1821C412yescomplemented
20#1819C414yes—
21#1784C412yescomplemented
22#1753—22no—
23#1753—22no—
24#1750—22no—
25#1472—22no—
26#1472—22no—
27#1463C414yescomplemented
28#1306—22no—
29#1305—22no—
30#1305—22no—
31C23×Q8C2×C411yescharacteristic
32C24×C4C2311yescharacteristic
33C22×M16—21no—
34C22×M16—21no—
35C22×M16—21no—
36C22×M16D412yes—
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⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×#117—21no—
50C2×#117—21no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C412yes—
53C22×C22⋊C4C2×C412yes—
54C2×#101—24no—
55#407—21no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407—21no—
62#407—21no—
63#407—21no—
64#407—21no—
65#407C2×C412yes—
66#407C2×C412yes—
67#407C2×C412yes—
68#407C2×C412yes—
69#407D412yes—
70#407D412yes—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8D412yes—
79C2×#97—22no—
80C2×C4×C8—22no—
81#381—21no—
82#381—21no—
83#380—21no—
84#380—21no—
85#380—21no—
86#380D412yes—
87#379C2×C412yes—
88#379C2×C412yes—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4D412yes—
93#343—24no—
94#342—28no—
95#336—24no—
96#324—28no—
97C25C22×C411yescharacteristic
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—82no—
105C23×C4—82no—
106C23×C4C4212yes—
107C23×C4C4212yes—
108C23×C4C22×C411yescharacteristic
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C23×C4#4111yescharacteristic
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—82no—
130C2×M16—82no—
131C2×M16—82no—
132C2×M16—82no—
133C2×M16C22⋊C414yes—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—41no—
137C22×C8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
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—
155#117—82no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—41no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C22⋊C414yes—
182C2×C42—22no—
183C2×C42—41no—
184#101—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—82no—
202C4×C8—82no—
203#95—22no—
204#95—22no—
205#95—41no—
206#95—41no—
207C24—42no—
208C24—81no—
209C24—81no—
210C24C2×C4211yescharacteristic
211C24C2×C22⋊C411yescharacteristic
212C24#11711yescharacteristic
213C24C4×D411yescharacteristic
214C24C4×D411yescharacteristic
215C24C4×D411yescharacteristic
216C24C4×D411yescharacteristic
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8—82no—
225C2×Q8—82no—
226C2×Q8—82no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4#11711yescharacteristic
254C22×C4#11711yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12311yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4#12311yescharacteristic
273C22×C4#12411yescharacteristic
274C22×C4#12411yescharacteristic
275C22×C4#12411yescharacteristic
276C22×C4#12611yescharacteristic
277C22×C4#12611yescharacteristic
278C22×C4#12711yescharacteristic
279M16—44no—
280M16—44no—
281M16—82no—
282M16—82no—
283M16—82no—
284M16—82no—
285M16—82no—
286M16—82no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—82no—
296C2×C8—82no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C42—42no—
318C42—42no—
319C42—82no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—88no—
342C23C4×C22⋊C411yescharacteristic
343C23C4×C22⋊C411yescharacteristic
344C23C4×C22⋊C411yescharacteristic
345C23#38611yescharacteristic
346C23#38811yescharacteristic
347C23#38811yescharacteristic
348C23#39011yescharacteristic
349Q8—82no—
350Q8—82no—
351Q8—82no—
352Q8—82no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
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—41no—
380C2×C4—41no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4#38611yescharacteristic
396C2×C4#38611yescharacteristic
397C2×C4#38811yescharacteristic
398C2×C4#38811yescharacteristic
399C2×C4#38811yescharacteristic
400C2×C4#38811yescharacteristic
401C2×C4#39011yescharacteristic
402C2×C4#39011yescharacteristic
403C8—44no—
404C8—44no—
405C8—82no—
406C8—82no—
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—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—44no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22#143711yescharacteristic
435C22#191512yes—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—84no—
444C4—82no—
445C4—82no—
446C4—162no—
447C4#191312yes—
448C2—22no—
449C2—41no—
450C2—41no—
451C2—41no—
452C2—41no—
453C2—41no—
454C2—41no—
455C2#1093811yescharacteristic
456C2#1094011yescharacteristic
457C2#1094111yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ 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
19. #1821C4semidirect
21. #1784C4semidirect
27. #1463C4semidirect

Maximal subgroups

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

subgroupindex
#218252
#217532
#205292
#173972
#81152
#81082