Finite Group Databasebeta

← #216197

Subgroups — #216197

Order 512.

924 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2336C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic
3#21912C211yesmaximal, characteristic
4#9880C212yesmaximal
5#9650C212yesmaximal
6#9082C212yesmaximal
7#9082C212yesmaximal
8#8560C212yesmaximal
9#8558C212yesmaximal
10C22×#125C411yescharacteristic
11C2×#431C2212yes—
12#2336—24no—
13#2336—24no—
14C2×#389C411yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#387C411yescharacteristic
17C2×#387C411yescharacteristic
18C2×#387C411yescharacteristic
19C2×#387C411yescharacteristic
20C2×#387C411yescharacteristic
21C2×#387C2211yescharacteristic
22C2×#387C2211yescharacteristic
23C2×#387C2211yescharacteristic
24C2×#383C411yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1939—24no—
29#1916—24no—
30#1817C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1797C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2214yes—
36C2×#324C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C2×#125—24no—
40C2×#125—24no—
41C2×#125C818yescomplemented
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61#431—44no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C22⋊C8C2314yes—
67C2×C22⋊C8C2314yes—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70#389—24no—
71#389—22no—
72#389—22no—
73#387—24no—
74#387—24no—
75#387—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—24no—
82#387—24no—
83#387—24no—
84#383—24no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4D414yes—
90C4×C4⋊C4D414yes—
91C2×#95C2×C411yescharacteristic
92C2×#95C2×C411yescharacteristic
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C411yescharacteristic
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2311yescharacteristic
106C2×#95C2311yescharacteristic
107#336—22no—
108#336—22no—
109#336—22no—
110#336—24no—
111#336—22no—
112#324—44no—
113#324—44no—
114C23×C4C22×C411yescharacteristic
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—24no—
130C22×C8—24no—
131#125—232no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—81no—
156C2×C4⋊C4—81no—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C22⋊C414yes—
159C2×C4⋊C4C2×C814yes—
160C2×C4⋊C4C2×C814yes—
161C2×C4⋊C4C2×C814yes—
162C2×C4⋊C4C2×C814yes—
163C2×C4⋊C4C2×C814yes—
164C2×C4⋊C4C2×C814yes—
165C2×C4⋊C4M1614yes—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—42no—
174C2×C42—81no—
175C2×C42—81no—
176C2×C42—81no—
177C2×C42—81no—
178C2×C42C2×C814yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C4⋊C8—44no—
186C4⋊C8—82no—
187C4⋊C8—82no—
188C4⋊C8—82no—
189C22⋊C8—24no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192C22⋊C8—24no—
193C4×C8—44no—
194C4×C8—82no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95C22⋊C414yes—
212#95C22⋊C414yes—
213C24C23×C411yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C22×C812yes—
270C22×C4C22×C812yes—
271C22×C4C22×C812yes—
272C22×C4C22×C812yes—
273C22×C4C22×C812yes—
274C22×C4C22×C812yes—
275C22×C4C22×C812yes—
276C22×C4C2×M1612yes—
277C22×C4C2×M1612yes—
278C22×C4C2×M1612yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4H(5,𝔽2)12yes—
282C22×C42−1+412yes—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—44no—
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—44no—
289C4⋊C4—28no—
290C4⋊C4—28no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4C22⋊C8116yes—
305C42—22no—
306C42—22no—
307C42—24no—
308C42—28no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—44no—
314C42—82no—
315C42—82no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23C22×C22⋊C411yescharacteristic
321C23#52012yes—
322C23C23×C811yescharacteristic
323C23C22×M1611yescharacteristic
324C23#56812yes—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4C2×C22⋊C814yes—
392C2×C4C2×C22⋊C814yes—
393C2×C4C2×C22⋊C814yes—
394C2×C4#53511yescharacteristic
395C2×C4#53611yescharacteristic
396C2×C4#53712yes—
397C8—84no—
398C8—84no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—22no—
407C22#231111yescharacteristic
408C22C22×C22⋊C811yescharacteristic
409C22#287911yescharacteristic
410C22#288311yescharacteristic
411C22#288411yescharacteristic
412C22#288911yescharacteristic
413C22#288911yescharacteristic
414C22#298812yes—
415C22#298912yes—
416C4—24no—
417C4—21no—
418C4—22no—
419C4—21no—
420C4—24no—
421C4—24no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—42no—
428C4—42no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4—81no—
433C2—22no—
434C2—22no—
435C2#1735111yescharacteristic
436C2#1735211yescharacteristic
437C2#1740611yescharacteristic
438C2#1741011yescharacteristic
439C2#2184711yescharacteristic
440C2#2187711yescharacteristic
441C2#2188311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
41. C2×#125C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23362
#219512
#219122
#98802
#96502
#90822
#85602
#85582