Finite Group Databasebeta

← #208194

Subgroups — #208194

Order 512.

1070 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic
3C2×C4×C4⋊C8C211yesmaximal, characteristic
4#9898C212yesmaximal, complemented
5#9691C212yesmaximal, complemented
6#9127C214yesmaximal
7#8235C214yesmaximal, complemented
8C2×#533C412yes—
9C2×#432—21no—
10C22×C4⋊C8C2212yes—
11C22×C4×C8—21no—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14#2862C412yes—
15#2859—21no—
16#2859—21no—
17#2857—21no—
18#2857—21no—
19#2857C412yes—
20#2857C2212yes—
21C4×C4⋊Q8—21no—
22C4×C4⋊Q8—21no—
23C4×C4⋊Q8C412yes—
24C4×C4⋊Q8C2212yes—
25#2285—21no—
26#2285—21no—
27C2×C43C2211yescharacteristic
28#1842C2212yes—
29#1831—21no—
30#1797—21no—
31#1797C2212yes—
32#1767—21no—
33#1767C2212yes—
34C4×C4⋊C8—21no—
35C4×C4⋊C8—21no—
36C4×C4⋊C8C2212yes—
37C4×C4⋊C8C2212yes—
38C2×#336—21no—
39C2×#333C2218yes—
40#1296C2218yes—
41C23×C8—22no—
42#533—48no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2312yes—
49C2×#125—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C414yes—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C414yes—
54C2×#117—22no—
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42C2311yescharacteristic
62#432—44no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—28no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8C2314yes—
68C2×C4⋊C8C2314yes—
69C2×C22⋊C8—22no—
70C2×#97—22no—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8—44no—
74#391—22no—
75#391—22no—
76#391C2×C414yes—
77#391C2×C414yes—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384C2×C412yes—
93#384C2312yes—
94#382—22no—
95#382—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C43C2×C411yescharacteristic
105C43C2×C411yescharacteristic
106C43D411yescharacteristic
107C43D411yescharacteristic
108C43D411yescharacteristic
109C43D411yescharacteristic
110C43C2311yescharacteristic
111C43C2311yescharacteristic
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#333—216no—
117C22×Q8—22no—
118C22×Q8C22×C414yes—
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C22×C8—28no—
127C22×C8—24no—
128C22×C8—24no—
129C22×C8—44no—
130C22×C8—44no—
131C4⋊Q8—28no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135C4⋊Q8—44no—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138#125—48no—
139C22⋊Q8—48no—
140C22⋊Q8—48no—
141C4×Q8—48no—
142C4×Q8—48no—
143#117—48no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C22×C414yes—
158C2×C22⋊C4—24no—
159C2×C42—24no—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—24no—
165C2×C42—24no—
166C2×C42—42no—
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
179C2×C42SD1614yes—
180C2×C42Q1614yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2411yescharacteristic
197C4⋊C8—28no—
198C4⋊C8—24no—
199C4⋊C8—28no—
200C4⋊C8—44no—
201C4⋊C8—44no—
202C4⋊C8#4118yes—
203C22⋊C8—44no—
204#97—44no—
205C4×C8—44no—
206C4×C8—44no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211C24C2×C22⋊C411yescharacteristic
212C2×Q8—28no—
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—44no—
216C2×Q8—44no—
217C2×Q8—44no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4#10014yes—
238C22×C4#10314yes—
239C22×C4#10314yes—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×SD1612yes—
252C22×C4C2×Q1612yes—
253C22×C4#13714yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C2×C8—28no—
258C2×C8—416no—
259C2×C8—44no—
260C2×C8—44no—
261C2×C8—44no—
262C2×C8—44no—
263C2×C8—44no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—48no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C22⋊C4—48no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—21no—
285C42—22no—
286C42—22no—
287C42—21no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—41no—
301C42—41no—
302C42—48no—
303C42#10314yes—
304C42#10314yes—
305C42C2×SD1612yes—
306C42C2×SD1612yes—
307C42C2×Q1612yes—
308C42C2×Q1612yes—
309C42#13514yes—
310C42AGL(1,ℤ/8)12yes—
311C42#13712yes—
312C42C2×#4114yes—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23C2×#10012yes—
318C23C2×#10312yes—
319C23#41912yes—
320C23C22×C22⋊C411yescharacteristic
321Q8—88no—
322Q8—88no—
323C2×C4—28no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
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—28no—
341C2×C4—28no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4C2×#10012yes—
361C2×C4C2×#10012yes—
362C2×C4#41312yes—
363C2×C4C2×#10312yes—
364C2×C4C2×#10312yes—
365C2×C4C2×#10312yes—
366C2×C4C2×#10312yes—
367C2×C4C2×#10312yes—
368C2×C4#41612yes—
369C2×C4#41612yes—
370C2×C4#41712yes—
371C2×C4#41712yes—
372C2×C4#41912yes—
373C2×C4C4×#4112yes—
374C2×C4#52512yes—
375C2×C4#53212yes—
376C2×C4C22×SD1611yescharacteristic
377C2×C4C22×Q1611yescharacteristic
378C2×C4C2×#13512yes—
379C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
380C2×C4C2×#13712yes—
381C2×C4C2×#13711yescharacteristic
382C8—48no—
383C8—84no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22#154814yes—
391C22#229711yescharacteristic
392C22C22×#10011yescharacteristic
393C22C2×#41311yescharacteristic
394C22C22×#10311yescharacteristic
395C22C2×#41611yescharacteristic
396C22C2×#41911yescharacteristic
397C22C2×#41711yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—22no—
404C4—22no—
405C4—22no—
406C4—41no—
407C4—41no—
408C4—84no—
409C4—84no—
410C4#155014yes—
411C4#155114yes—
412C4#311012yes—
413C4#311312yes—
414C4#316712yes—
415C4#317012yes—
416C2—24no—
417C2C2×#154811yescharacteristic
418C2C2×#155011yescharacteristic
419C2C2×#155111yescharacteristic
420C2#1771811yescharacteristic
421C2#2139411yescharacteristic
422C2#2140011yescharacteristic
423C2#2142411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9898C2semidirect
5. #9691C2semidirect
7. #8235C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#208102
C2×C4×C4⋊C82
#98982
#96912
#91272
#82352