Finite Group Databasebeta

← #154181

Subgroups — #154181

Order 512.

2862 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1442C214yesmaximal
2C2×C4⋊#95C211yesmaximal, characteristic
3C2×C4⋊#95C211yesmaximal, characteristic
4C2×C4×#95C211yesmaximal, characteristic
5#7557C214yesmaximal
6#7554C214yesmaximal
7C23×C4⋊C4C2212yes—
8C23×C4⋊C4C2212yes—
9C23×C42C2212yes—
10C23×C42C2211yescharacteristic
11C22×#95C2212yes—
12C22×#95C2212yes—
13C22×#95C2212yes—
14C22×#95C2212yes—
15C22×#95C2218yes—
16C22×#95C2218yes—
17C22×#95C2212yes—
18C22×#95C2212yes—
19#1442—216no—
20C4⋊#95—24no—
21C4⋊#95—24no—
22C4×#95C418yescomplemented
23C24×C4C2314yes—
24C24×C4C2314yes—
25C24×C4C2314yes—
26C24×C4C2311yescharacteristic
27C24×C4C2311yescharacteristic
28C24×C4C2311yescharacteristic
29C22×C4⋊C4—24no—
30C22×C4⋊C4—24no—
31C22×C4⋊C4—28no—
32C22×C4⋊C4—28no—
33C22×C4⋊C4—28no—
34C22×C4⋊C4—28no—
35C22×C4⋊C4—44no—
36C22×C4⋊C4—44no—
37C22×C42—24no—
38C22×C42—24no—
39C22×C42—44no—
40C22×C42—44no—
41C22×C42—44no—
42C22×C42C2×C418yes—
43C22×C42C2×C414yes—
44C22×C42D414yes—
45C22×C42Q814yes—
46C2×#95—24no—
47C2×#95—24no—
48C2×#95—24no—
49C2×#95—24no—
50C2×#95—24no—
51C2×#95—24no—
52C2×#95—216no—
53C2×#95—216no—
54C2×#95—28no—
55C2×#95—216no—
56C2×#95—216no—
57C2×#95—28no—
58C2×#95—28no—
59C2×#95—28no—
60C2×#95—48no—
61C2×#95—48no—
62C2×#95C2×C418yes—
63C2×#95C2×C418yes—
64C2×#95D418yes—
65C2×#95Q818yes—
66C25C2411yescharacteristic
67C23×C4—24no—
68C23×C4—24no—
69C23×C4—24no—
70C23×C4—24no—
71C23×C4—24no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—24no—
75C23×C4—24no—
76C23×C4—28no—
77C23×C4—28no—
78C23×C4—24no—
79C23×C4—24no—
80C23×C4—28no—
81C23×C4—28no—
82C23×C4—24no—
83C23×C4—24no—
84C23×C4—28no—
85C23×C4—24no—
86C23×C4—24no—
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—28no—
90C23×C4—28no—
91C23×C4—28no—
92C23×C4—28no—
93C23×C4—28no—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—28no—
97C23×C4—28no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4C22×C418yes—
103C23×C4C22×C412yes—
104C23×C4C22×C412yes—
105C23×C4C22×C412yes—
106C23×C4C2×D418yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D414yes—
110C23×C4C2×D412yes—
111C23×C4C2×Q818yes—
112C23×C4C2×Q812yes—
113C23×C4C2×Q812yes—
114C23×C4C2×Q814yes—
115C23×C4C2×Q812yes—
116C23×C4#4118yes—
117C23×C4#4118yes—
118C23×C4#4114yes—
119C23×C4#4114yes—
120C23×C4#4114yes—
121C23×C4#4114yes—
122C2×C4⋊C4—216no—
123C2×C4⋊C4—216no—
124C2×C4⋊C4—432no—
125C2×C4⋊C4—432no—
126C2×C4⋊C4—48no—
127C2×C4⋊C4—48no—
128C2×C4⋊C4—48no—
129C2×C4⋊C4—48no—
130C2×C42—216no—
131C2×C42—48no—
132C2×C42—48no—
133C2×C42—48no—
134C2×C42—48no—
135C2×C42—48no—
136C2×C42—416no—
137C2×C42—416no—
138C2×C42C4⋊C4116yes—
139#95—216no—
140#95—432no—
141#95—432no—
142#95—48no—
143#95—48no—
144#95—48no—
145#95—48no—
146#95C4⋊C4132yescomplemented
147C24C23×C411yescharacteristic
148C24C22×D411yescharacteristic
149C24C22×D412yes—
150C24C22×Q811yescharacteristic
151C24C22×Q812yes—
152C24C2×#4112yes—
153C24C2×#4112yes—
154C24C2×#4112yes—
155C24C2×#4112yes—
156C24C2×#4114yes—
157C24C2×#4114yes—
158C24H(5,𝔽2)14yes—
159C24H(5,𝔽2)12yes—
160C242−1+412yes—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—28no—
177C22×C4—28no—
178C22×C4—28no—
179C22×C4—28no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—216no—
185C22×C4—216no—
186C22×C4—216no—
187C22×C4—216no—
188C22×C4—28no—
189C22×C4—28no—
190C22×C4—28no—
191C22×C4—28no—
192C22×C4—28no—
193C22×C4—28no—
194C22×C4—28no—
195C22×C4—28no—
196C22×C4—28no—
197C22×C4—28no—
198C22×C4—216no—
199C22×C4—216no—
200C22×C4—216no—
201C22×C4—216no—
202C22×C4—28no—
203C22×C4—28no—
204C22×C4—216no—
205C22×C4—216no—
206C22×C4—28no—
207C22×C4—216no—
208C22×C4—28no—
209C22×C4—216no—
210C22×C4—28no—
211C22×C4—28no—
212C22×C4—48no—
213C22×C4—48no—
214C22×C4—48no—
215C22×C4—48no—
216C22×C4—48no—
217C22×C4—48no—
218C22×C4—48no—
219C22×C4—48no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—48no—
227C22×C4—48no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—48no—
233C22×C4—48no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—48no—
240C22×C4—48no—
241C22×C4—48no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—44no—
246C22×C4—48no—
247C22×C4—44no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—48no—
253C22×C4—48no—
254C22×C4—48no—
255C22×C4—48no—
256C22×C4C2×C4⋊C4116yes—
257C22×C4C2×C4⋊C414yes—
258C22×C4C2×C4⋊C414yes—
259C22×C4C2×C4⋊C418yes—
260C22×C4C2×C4⋊C414yes—
261C22×C4#11718yes—
262C22×C4#11718yes—
263C22×C4#12414yes—
264C22×C4#12414yes—
265C22×C4#12414yes—
266C22×C4#12414yes—
267C22×C4#12518yes—
268C22×C4#12518yes—
269C22×C4#12714yes—
270C22×C4C4⋊Q814yes—
271C22×C4C4⋊Q818yes—
272C4⋊C4—432no—
273C4⋊C4—432no—
274C42—416no—
275C42—432no—
276C23C22×C4⋊C411yescharacteristic
277C23C22×C4⋊C412yes—
278C23C2×#11712yes—
279C23C2×#11712yes—
280C23C4×#4112yes—
281C23C4×#4112yes—
282C23#51811yescharacteristic
283C23#51812yes—
284C23#51911yescharacteristic
285C23#52512yes—
286C23#52512yes—
287C23C2×#12411yescharacteristic
288C23C2×#12411yescharacteristic
289C23C2×#12411yescharacteristic
290C23C2×#12411yescharacteristic
291C23C2×#12512yes—
292C23C2×#12512yes—
293C23#52918yes—
294C23#52918yes—
295C23C2×#12711yescharacteristic
296C23C2×C4⋊Q811yescharacteristic
297C23C2×C4⋊Q812yes—
298C23#53214yes—
299C23#53214yes—
300C23#53312yes—
301C23#53312yes—
302C23#53314yes—
303C23#53314yes—
304C23#53412yes—
305C23#53712yes—
306C23#53814yes—
307C23#53814yes—
308C23#53914yes—
309C23#53914yes—
310C23#54312yes—
311C23#54412yes—
312C23#54618yes—
313C23#54614yes—
314C23#54814yes—
315C23#55118yes—
316C23#55214yes—
317C23#55314yes—
318C23#55318yes—
319C23#55418yes—
320C23#55414yes—
321C23#55614yes—
322C23#55714yes—
323C23#55914yes—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—216no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—216no—
335C2×C4—216no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—216no—
339C2×C4—28no—
340C2×C4—216no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—216no—
344C2×C4—216no—
345C2×C4—216no—
346C2×C4—48no—
347C2×C4—416no—
348C2×C4—416no—
349C2×C4—48no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—416no—
353C2×C4—416no—
354C2×C4—416no—
355C2×C4—48no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—416no—
362C2×C4—416no—
363C2×C4—416no—
364C2×C4—88no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—88no—
374C2×C4—88no—
375C2×C4—88no—
376C2×C4#38218yes—
377C2×C4#38218yes—
378C2×C4#38418yes—
379C22C2×#38211yescharacteristic
380C22C2×#38211yescharacteristic
381C22C2×#38411yescharacteristic
382C22#228514yes—
383C22#229611yescharacteristic
384C22#229611yescharacteristic
385C22#229814yes—
386C22#231212yes—
387C22#231312yes—
388C22#231511yescharacteristic
389C22#231611yescharacteristic
390C22#234912yes—
391C22#234914yes—
392C22#235112yes—
393C22#235314yes—
394C22#235612yes—
395C22#235712yes—
396C22#239812yes—
397C22#239812yes—
398C22#240214yes—
399C22#253912yes—
400C22#254012yes—
401C22#254914yes—
402C22#255014yes—
403C22#255718yes—
404C22#255818yes—
405C22#255918yes—
406C22#256018yes—
407C22#256112yes—
408C22#256212yes—
409C22#256914yes—
410C22#257014yes—
411C22#258312yes—
412C22#258412yes—
413C22#258718yes—
414C22#258818yes—
415C22#258918yes—
416C22#259018yes—
417C22#259112yes—
418C22#259212yes—
419C22#263912yes—
420C22#263912yes—
421C22#264614yes—
422C22#285811yescharacteristic
423C22#285811yescharacteristic
424C22#285914yes—
425C22#286211yescharacteristic
426C22#286211yescharacteristic
427C4—28no—
428C4—28no—
429C4—28no—
430C4—416no—
431C4—88no—
432C4—88no—
433C2#1385411yescharacteristic
434C2#1394814yes—
435C2#1394914yes—
436C2#1395411yescharacteristic
437C2#1395511yescharacteristic
438C2#1396511yescharacteristic
439C2#1396611yescharacteristic
440C2#1434711yescharacteristic
441C2#1468711yescharacteristic
442C2#1516712yes—
443C2#1516818yes—
444C2#1517012yes—
445C2#1520712yes—
446C2#1520912yes—
subgroupG/H
center, Frattini, socle, prime radical, cosocleC25C24
commutatorC24C23×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C1

Complements

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

NH ≅ G/Ntype
22. C4×#95C4semidirect
146. #95C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14422
C2×C4⋊#952
C2×C4×#952
#75572
#75542