Finite Group Databasebeta

← #197859

Subgroups — #197859

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic
2C2×#2982C211yesmaximal, characteristic
3#21864C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5C2×#1544C211yesmaximal, characteristic
6#9220C214yesmaximal, complemented
7#9218C214yesmaximal, complemented
8#8046C211yesmaximal, characteristic
9#7878C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3113—24no—
13#3113C418yescomplemented
14C2×#483—21no—
15C2×#475C2212yes—
16C2×#462C2212yes—
17C2×#451—21no—
18#2982—24no—
19#2982—24no—
20C2×C8×Q8—22no—
21C2×C8×Q8C2214yes—
22C2×#441—22no—
23C2×#432C2211yescharacteristic
24C2×#431—22no—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28#2285C2211yescharacteristic
29#1990C2212yes—
30#1956C2212yes—
31#1873—22no—
32#1824—21no—
33#1805—22no—
34#1803C2214yes—
35#1787—21no—
36C4×#103—22no—
37#1544—24no—
38C2×#332C2212yes—
39C2×#332C2212yes—
40#1296C2214yes—
41#1277C2214yes—
42C23×Q8D411yescharacteristic
43C22×Q16—22no—
44#533—22no—
45#533—22no—
46#533—22no—
47#533C2×C414yes—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C414yes—
60C2×C4×Q8D414yes—
61C2×C4×Q8D412yes—
62C2×C4×Q8C2312yes—
63C2×#117—21no—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#483—28no—
70#475—24no—
71#475C2×C418yes—
72#462—24no—
73#462C2×C418yes—
74#451—28no—
75C8×Q8—216no—
76C8×Q8—28no—
77C8×Q8—28no—
78#441—216no—
79#432—22no—
80#432C2×C414yes—
81#431—216no—
82#431—22no—
83#431—22no—
84#431—22no—
85#431—22no—
86#431—22no—
87#431D414yes—
88C2×#106—22no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8D412yes—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—24no—
97C2×#103—24no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8D412yes—
102C2×#97—21no—
103C2×C4×C8—24no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2312yes—
107#384C2312yes—
108#382—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113#340—24no—
114#340—24no—
115#332—24no—
116#332—24no—
117C22×Q8—28no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8C22⋊C414yes—
123C22×Q8C2×D412yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4111yescharacteristic
129C2×Q16—28no—
130C2×Q16—48no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8C2×D414yes—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8C22×C412yes—
142C4⋊Q8C22×C412yes—
143#125—24no—
144#125—42no—
145C22⋊Q8—24no—
146C22⋊Q8—24no—
147C22⋊Q8—42no—
148C22⋊Q8—42no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×Q8C22⋊C418yes—
161C4×Q8C22×C414yes—
162C4×Q8C2×D414yes—
163#117—24no—
164#117—42no—
165#117—82no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4C22⋊C414yes—
182C2×C4⋊C4C2×D412yes—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42C22×C412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42#4111yescharacteristic
197C2×C42#4112yes—
198C2×C42C2411yescharacteristic
199#106—28no—
200C4⋊C8—216no—
201C4⋊C8—24no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8C22×C414yes—
207C4⋊C8C2×D414yes—
208#103—28no—
209#103—28no—
210#103—48no—
211#103—84no—
212C22⋊C8—24no—
213C22⋊C8—24no—
214C22⋊C8—24no—
215C22⋊C8—42no—
216#97—24no—
217C4×C8—216no—
218C4×C8—24no—
219C4×C8—22no—
220C4×C8—22no—
221C4×C8—42no—
222#95—41no—
223#95—41no—
224C24#12111yescharacteristic
225C2×Q8—216no—
226C2×Q8—28no—
227C2×Q8—216no—
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—44no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8C2×C22⋊C414yes—
239C2×Q8#13518yes—
240C22×C4—28no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—82no—
253C22×C4C2×C22⋊C412yes—
254C22×C4C4×D412yes—
255C22×C4C4×D412yes—
256C22×C4#12012yes—
257C22×C4#12112yes—
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12312yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C2×#4111yescharacteristic
265Q16—416no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—44no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—44no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—28no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—82no—
294C4⋊C4C2×C22⋊C414yes—
295C22⋊C4—44no—
296C22⋊C4—82no—
297C42—24no—
298C42—24no—
299C42—28no—
300C42—44no—
301C42—82no—
302C42C2×C22⋊C412yes—
303C42C4×D412yes—
304C42C4×D412yes—
305C42#12012yes—
306C42#12112yes—
307C42#12312yes—
308C42C23×C411yescharacteristic
309C42C22×D411yescharacteristic
310C42C22×D411yescharacteristic
311C42C2×#4111yescharacteristic
312C23—21no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23#38612yes—
317C23C2×#12111yescharacteristic
318Q8—216no—
319Q8—24no—
320Q8—28no—
321Q8—44no—
322Q8—44no—
323Q8—82no—
324Q8—82no—
325Q8#41618yes—
326C2×C4—216no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—28no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—84no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4#38612yes—
352C2×C4#38612yes—
353C2×C4#38612yes—
354C2×C4C22×C22⋊C411yescharacteristic
355C2×C4C2×C4×D411yescharacteristic
356C2×C4C2×C4×D411yescharacteristic
357C2×C4C2×#12011yescharacteristic
358C2×C4C2×#12111yescharacteristic
359C2×C4C2×#12311yescharacteristic
360C2×C4C2×#13514yes—
361C2×C4#57712yes—
362C2×C4#57812yes—
363C8—44no—
364C8—44no—
365C8—44no—
366C8—82no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22C2×#38611yescharacteristic
373C22#304311yescharacteristic
374C22#304411yescharacteristic
375C22#305012yes—
376C22#306912yes—
377C4—21no—
378C4—21no—
379C4—22no—
380C4—22no—
381C4—28no—
382C4—22no—
383C4—44no—
384C4—82no—
385C4—82no—
386C4—162no—
387C4C2×#41612yes—
388C4#294712yes—
389C4#301312yes—
390C4#305012yes—
391C2—42no—
392C2#1940012yes—
393C2#1940712yes—
394C2#2093011yescharacteristic
395C2#2096711yescharacteristic
396C2#2100611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9220C2semidirect
7. #9218C2semidirect
13. #3113C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
C2×#29822
#218642
#207342
C2×#15442
#92202
#92182
#80462
#78782