Finite Group Databasebeta

← #355129

Subgroups — #355129

Order 512.

1970 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2C2×#1823C212yesmaximal
3C2×C4×C22⋊C8C211yesmaximal, characteristic
4C4×#380C211yesmaximal, characteristic
5#10884C214yesmaximal, complemented
6#10880C214yesmaximal, complemented
7#10730C212yesmaximal
8C23×C4⋊C4—21no—
9C23×C4⋊C4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13C22×C4×C8C2212yes—
14C2×#389—24no—
15C2×#389—21no—
16C2×#389—21no—
17C2×#389C418yescomplemented
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#380C2212yes—
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C22⋊C4—21no—
23C22×#95—21no—
24#1823C4116yescomplemented
25C4×C22⋊C8—24no—
26C2×#340C2218yes—
27C2×#340C2218yes—
28C2×#336C2212yes—
29C4⋊#95C2212yes—
30#1434—21no—
31C22⋊#95—21no—
32C4×#95—21no—
33C24×C4—21no—
34C24×C4—21no—
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4C2311yescharacteristic
38C23×C8C2314yes—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4—28no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—44no—
48C22×C4⋊C4—44no—
49C22×C4⋊C4—44no—
50C22×C4⋊C4C2×C418yes—
51C22×C4⋊C4C2×C414yes—
52C22×C4⋊C4C2×C414yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2314yes—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C42—22no—
58C22×C42—44no—
59C22×C42C2×C414yes—
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D414yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C22⋊C8—24no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8C2×C418yes—
68C2×C4×C8—24no—
69C2×C4×C8—44no—
70#389—28no—
71#389—28no—
72#389—24no—
73#389—24no—
74#389C2×C418yes—
75#389C2×C418yes—
76#380—24no—
77#380C2×C418yes—
78C4×C4⋊C4—44no—
79C4×C22⋊C4—44no—
80C2×#95—28no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87#340—48no—
88#340C2×C4132yescomplemented
89#336—24no—
90C25C2×D411yescharacteristic
91C23×C4—24no—
92C23×C4—24no—
93C23×C4—24no—
94C23×C4—28no—
95C23×C4—24no—
96C23×C4—22no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—22no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—44no—
107C23×C4—44no—
108C23×C4—44no—
109C23×C4—42no—
110C23×C4—42no—
111C23×C4C22⋊C414yes—
112C23×C4C22⋊C414yes—
113C23×C4D814yes—
114C23×C4SD1614yes—
115C23×C4C22×C412yes—
116C23×C4C22×C414yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4#4114yes—
124C23×C4C2411yescharacteristic
125C22×C8—24no—
126C22×C8—24no—
127C22×C8—28no—
128C22×C8—28no—
129C22×C8C22×C418yes—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—216no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—48no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—48no—
147C2×C4⋊C4—432no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4C42116yescomplemented
153C2×C4⋊C4C22×C414yes—
154C2×C4⋊C4C22×C418yes—
155C2×C4⋊C4C22×C418yes—
156C2×C4⋊C4C22×C414yes—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—44no—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—48no—
163C2×C42—48no—
164C2×C42—42no—
165C2×C42—42no—
166C2×C42—44no—
167C2×C42—44no—
168C2×C42—44no—
169C2×C42C22⋊C414yes—
170C2×C42C22⋊C414yes—
171C2×C42C22×C412yes—
172C2×C42C22×C412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C22⋊C8—216no—
176C4×C8—48no—
177#95—28no—
178#95—28no—
179#95—28no—
180#95—44no—
181#95—44no—
182#95—44no—
183C24—22no—
184C24—22no—
185C24—22no—
186C24—22no—
187C24C2×C22⋊C412yes—
188C24C4×D414yes—
189C24C2×D812yes—
190C24C2×SD1612yes—
191C24#13514yes—
192C24C22×D411yescharacteristic
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—216no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—216no—
207C22×C4—24no—
208C22×C4—28no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—216no—
213C22×C4—28no—
214C22×C4—28no—
215C22×C4—24no—
216C22×C4—28no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—48no—
221C22×C4—48no—
222C22×C4—416no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—48no—
227C22×C4—48no—
228C22×C4—416no—
229C22×C4—416no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—48no—
243C22×C4—88no—
244C22×C4—88no—
245C22×C4#102116yes—
246C22×C4C2×C4214yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4#11714yes—
253C22×C4C4×D418yes—
254C22×C4C4×D414yes—
255C22×C4C2×D812yes—
256C22×C4C2×D812yes—
257C22×C4C2×SD1612yes—
258C22×C4C2×SD1612yes—
259C22×C4#13714yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C23×C412yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C2×#4112yes—
264C22×C4H(5,𝔽2)14yes—
265C22×C42−1+414yes—
266C2×C8—28no—
267C2×C8—216no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—48no—
271C4⋊C4—24no—
272C4⋊C4—28no—
273C4⋊C4—216no—
274C4⋊C4—28no—
275C4⋊C4—216no—
276C4⋊C4—48no—
277C4⋊C4—44no—
278C4⋊C4—48no—
279C4⋊C4—416no—
280C4⋊C4—416no—
281C4⋊C4C2×C4218yes—
282C22⋊C4—416no—
283C42—24no—
284C42—24no—
285C42—24no—
286C42—24no—
287C42—44no—
288C42—44no—
289C42—88no—
290C23—22no—
291C23—22no—
292C23—28no—
293C23—22no—
294C23—22no—
295C23—24no—
296C23—24no—
297C23—28no—
298C23—28no—
299C23—24no—
300C23—24no—
301C23—28no—
302C23C4×C22⋊C414yes—
303C23C2×#10214yes—
304C23#41614yes—
305C23C4×D818yes—
306C23C4×SD1618yes—
307C23C22×C22⋊C411yescharacteristic
308C23C2×C4×D412yes—
309C23#53412yes—
310C23#53612yes—
311C23C22×D811yescharacteristic
312C23C22×SD1611yescharacteristic
313C23C2×#13512yes—
314C23C2×#13712yes—
315C23#57512yes—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—28no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—28no—
330C2×C4—28no—
331C2×C4—216no—
332C2×C4—28no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—28no—
337C2×C4—416no—
338C2×C4—416no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—432no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—48no—
350C2×C4—48no—
351C2×C4—816no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4C4×C22⋊C414yes—
359C2×C4C2×#10214yes—
360C2×C4C2×#10214yes—
361C2×C4#41914yes—
362C2×C4C22×C4211yescharacteristic
363C2×C4C2×#11711yescharacteristic
364C2×C4C2×C4×D412yes—
365C2×C4#51812yes—
366C2×C4#51912yes—
367C2×C4#52018yes—
368C2×C4#53412yes—
369C2×C4#53612yes—
370C2×C4#53918yes—
371C8—48no—
372C22—24no—
373C22—216no—
374C22—216no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—24no—
380C22C4×#10218yes—
381C22C2×C4×C22⋊C411yescharacteristic
382C22#230811yescharacteristic
383C22#231011yescharacteristic
384C22#232214yes—
385C22C22×#10211yescharacteristic
386C22C2×#41611yescharacteristic
387C22C2×#41911yescharacteristic
388C22#289611yescharacteristic
389C22C2×C4×D812yes—
390C22C2×C4×SD1612yes—
391C22C4×#13714yes—
392C22#318612yes—
393C22#318712yes—
394C22#319014yes—
395C22#322612yes—
396C22#322912yes—
397C22#323214yes—
398C4—24no—
399C4—24no—
400C4—24no—
401C4—24no—
402C4—42no—
403C4—48no—
404C4—88no—
405C4—84no—
406C4#227412yes—
407C4#231312yes—
408C4#232214yes—
409C2—28no—
410C2#1377611yescharacteristic
411C2C2×C4×#10211yescharacteristic
412C2C4×#41911yescharacteristic
413C2#2175811yescharacteristic
414C2#2176211yescharacteristic
415C2#2183311yescharacteristic
416C2#2183611yescharacteristic
417C2#2207514yes—
418C2#2207814yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10884C2semidirect
6. #10880C2semidirect
17. C2×#389C4semidirect
24. #1823C4semidirect
88. #340C2×C4semidirect
152. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
C2×#18232
C2×C4×C22⋊C82
C4×#3802
#108842
#108802
#107302