Finite Group Databasebeta

← #110036

Subgroups — #110036

Order 512.

744 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20836C211yesmaximal, characteristic
2#20527C211yesmaximal, characteristic
3#10729C211yesmaximal, characteristic
4#10729C211yesmaximal, characteristic
5#8104C211yesmaximal, characteristic
6#8100C211yesmaximal, characteristic
7#7927C211yesmaximal, characteristic
8C23×M16C2211yescharacteristic
9C2×#424—21no—
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2280C2211yescharacteristic
14#2280C2211yescharacteristic
15#2280C2211yescharacteristic
16#1851—22no—
17#1850C414yescomplemented
18#1753—22no—
19#1752—21no—
20C8×C22⋊C4—21no—
21#1750—22no—
22C2×#344—22no—
23C2×#344—22no—
24#1729—21no—
25#1305C414yescomplemented
26#1304C414yes—
27#1282—22no—
28C24×C4C2311yescharacteristic
29C22×M16—21no—
30C22×M16—21no—
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—21no—
35C22×M16—21no—
36C22×M16—44no—
37C22×M16D412yes—
38C23×C8C2×C412yes—
39C23×C8D411yescharacteristic
40C23×C8D411yescharacteristic
41C2×#117—21no—
42C2×#117—21no—
43C2×#117D412yes—
44#424—44no—
45C2×C4⋊C8—24no—
46#407—21no—
47#407—21no—
48#407—21no—
49#407—21no—
50#407—21no—
51#407C2×C412yes—
52#407C2×C412yes—
53#407Q812yes—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8C2×C412yes—
63C2×C22⋊C8D412yes—
64C2×#97—22no—
65C2×C4×C8—22no—
66#381—21no—
67#381—21no—
68#381Q812yes—
69#380—21no—
70#380—21no—
71#380—21no—
72#380—21no—
73#380—21no—
74#380C2×C412yes—
75#379C2×C412yes—
76#379C2×C412yes—
77#379C2×C412yes—
78C4×C22⋊C4—21no—
79C4×C22⋊C4—21no—
80C4×C22⋊C4—21no—
81C4×C22⋊C4—21no—
82C4×C22⋊C4—21no—
83C4×C22⋊C4C2×C412yes—
84#344—44no—
85#344—44no—
86#336—22no—
87#336—22no—
88#336—24no—
89C25C22×C411yescharacteristic
90C23×C4—42no—
91C23×C4—42no—
92C23×C4C22×C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×Q811yescharacteristic
99C23×C4C2×Q811yescharacteristic
100C23×C4#4111yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—24no—
107C2×M16—24no—
108C2×M16—24no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—44no—
112C2×M16—44no—
113C2×M16—44no—
114C2×M16—42no—
115C2×M16—42no—
116C2×M16—42no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—42no—
123C2×M16—42no—
124C2×M16—42no—
125C2×M16—42no—
126C2×M16C4⋊C414yes—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—44no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—41no—
139C22×C8C22⋊C412yes—
140C22×C8C22⋊C412yes—
141#117—42no—
142#117—42no—
143#117—42no—
144#117—42no—
145#117—42no—
146#117—42no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—41no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4C4214yes—
158C2×C22⋊C4C4⋊C414yes—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—41no—
162C4⋊C8—48no—
163C22⋊C8—22no—
164C22⋊C8—22no—
165C22⋊C8—22no—
166C22⋊C8—22no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169C22⋊C8—22no—
170C22⋊C8—22no—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175#97—82no—
176C4×C8—82no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—41no—
182#95—41no—
183C24—42no—
184C24—42no—
185C24—81no—
186C24C2×C4211yescharacteristic
187C24C2×C4⋊C411yescharacteristic
188C24#11711yescharacteristic
189C24C4×D411yescharacteristic
190C24C4×D411yescharacteristic
191C24C4×D411yescharacteristic
192C24C4×Q811yescharacteristic
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C4⋊C411yescharacteristic
219C22×C4#11711yescharacteristic
220C22×C4#11711yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×Q811yescharacteristic
227C22×C4C4×Q811yescharacteristic
228C22×C4C4×Q811yescharacteristic
229C22×C4#12011yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12511yescharacteristic
241C22×C4#12511yescharacteristic
242C22×C4#12611yescharacteristic
243C22×C4#12611yescharacteristic
244C22×C4C4⋊Q811yescharacteristic
245M16—44no—
246M16—44no—
247M16—44no—
248M16—44no—
249M16—44no—
250M16—44no—
251M16—82no—
252M16—82no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—81no—
271C2×C8—81no—
272C2×C8—81no—
273C2×C8—81no—
274C4⋊C4—44no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—82no—
284C22⋊C4—82no—
285C22⋊C4—82no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—81no—
291C42—81no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—88no—
314C23C4×C22⋊C411yescharacteristic
315C23C4×C4⋊C411yescharacteristic
316C23C4×C4⋊C411yescharacteristic
317C23#38511yescharacteristic
318C23#38711yescharacteristic
319C23#38811yescharacteristic
320C23#38911yescharacteristic
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4#38511yescharacteristic
364C2×C4#38611yescharacteristic
365C2×C4#38711yescharacteristic
366C2×C4#38711yescharacteristic
367C2×C4#38711yescharacteristic
368C2×C4#38811yescharacteristic
369C2×C4#38911yescharacteristic
370C2×C4#39111yescharacteristic
371C2×C4B(4,𝔽2)12yes—
372C2×C4#45812yes—
373C8—44no—
374C8—44no—
375C8—42no—
376C8—42no—
377C8—82no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—44no—
398C22—44no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22#143611yescharacteristic
405C22#175712yes—
406C22#186712yes—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—84no—
414C4—84no—
415C4—162no—
416C4#186712yes—
417C4#188911yescharacteristic
418C4#189011yescharacteristic
419C2—22no—
420C2—22no—
421C2—41no—
422C2—41no—
423C2—41no—
424C2—41no—
425C2—41no—
426C2#1091411yescharacteristic
427C2#1091711yescharacteristic
428C2#1091811yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
socleC22#1436
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1850C4semidirect
25. #1305C4semidirect

Maximal subgroups

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

subgroupindex
#208362
#205272
#107292
#81042
#81002
#79272