Finite Group Databasebeta

← #214107

Subgroups — #214107

Order 512.

947 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2637C211yesmaximal, characteristic
2#21822C211yesmaximal, characteristic, complemented
3#21818C211yesmaximal, characteristic, complemented
4#9086C212yesmaximal, complemented
5#9084C212yesmaximal
6#9082C212yesmaximal, complemented
7#9080C212yesmaximal
8#8703C212yesmaximal, complemented
9#8309C212yesmaximal
10C22×#126C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2212yes—
13C2×#431C2212yes—
14#2637—24no—
15#2637—24no—
16C2×#398C411yescharacteristic
17C2×#398C411yescharacteristic
18C2×#398C411yescharacteristic
19C2×#397C411yescharacteristic
20C2×#396C411yescharacteristic
21C2×#396C411yescharacteristic
22C2×#395C411yescharacteristic
23C2×#393C411yescharacteristic
24C2×#388C2211yescharacteristic
25C2×#388C2211yescharacteristic
26C2×#387C2211yescharacteristic
27C2×#387C2211yescharacteristic
28C2×#382C2211yescharacteristic
29#1985—24no—
30#1984—24no—
31#1817C2212yes—
32#1817C2212yes—
33#1817C2212yes—
34#1817C2212yes—
35C2×#324C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×Q8C2×C411yescharacteristic
40C2×#126—22no—
41C2×#126—22no—
42C2×#126—44no—
43C2×C4×Q8—44no—
44C2×C4×Q8D414yes—
45C2×C4×Q8D414yes—
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⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#432—44no—
69#431—44no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8C2314yes—
73C2×C22⋊C8C2314yes—
74C2×#97—22no—
75C2×C4×C8—22no—
76#398—24no—
77#398—24no—
78#398—24no—
79#397—24no—
80#396—24no—
81#396—24no—
82#395—24no—
83#393—24no—
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#382—22no—
93#382—22no—
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×#95C2311yescharacteristic
105C2×#95C2311yescharacteristic
106#336—24no—
107#336—24no—
108#336—24no—
109#336—24no—
110#324—44no—
111#323—44no—
112C25C22×C411yescharacteristic
113C22×Q8—24no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8C22⋊C414yes—
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×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—24no—
132C22×C8—24no—
133#126—48no—
134#126—48no—
135C4×Q8—44no—
136C4×Q8—44no—
137C4×Q8—44no—
138C4×Q8—44no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—81no—
158C2×C4⋊C4—81no—
159C2×C4⋊C4—81no—
160C2×C4⋊C4—81no—
161C2×C4⋊C4C22⋊C414yes—
162C2×C4⋊C4C22⋊C414yes—
163C2×C4⋊C4C22⋊C414yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—81no—
178C2×C22⋊C4—81no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C4⋊C8—44no—
189C4⋊C8—44no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192C22⋊C8—24no—
193C22⋊C8—24no—
194#97—44no—
195C4×C8—44no—
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—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210C24—21no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24—41no—
215C24—41no—
216C24—41no—
217C24—41no—
218C24C23×C411yescharacteristic
219C2×Q8—28no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—416no—
223C22×C4—22no—
224C22×C4—22no—
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—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
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×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4#13114yes—
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4H(5,𝔽2)14yes—
280C2×C8—44no—
281C2×C8—44no—
282C2×C8—44no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—44no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—24no—
312C42—24no—
313C42—24no—
314C42—24no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—44no—
327C23—82no—
328C23—82no—
329C23—82no—
330C23—82no—
331C23—82no—
332C23—82no—
333C23C22×C22⋊C411yescharacteristic
334C23#51812yes—
335C23C2×#13112yes—
336C23#56812yes—
337Q8—416no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—28no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
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—44no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#40814yes—
395C2×C4#53412yes—
396C2×C4#53412yes—
397C8—84no—
398C8—84no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—164no—
413C22#230811yescharacteristic
414C22C2×#40811yescharacteristic
415C22#287911yescharacteristic
416C22#288411yescharacteristic
417C22#288411yescharacteristic
418C22#288811yescharacteristic
419C22#288911yescharacteristic
420C22#297614yes—
421C4—24no—
422C4—22no—
423C4—22no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—42no—
428C4—42no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4—81no—
433C4—81no—
434C4—81no—
435C4—81no—
436C2—24no—
437C2—161no—
438C2#1739011yescharacteristic
439C2#1739111yescharacteristic
440C2#1740911yescharacteristic
441C2#1741011yescharacteristic
442C2#2174111yescharacteristic
443C2#2175111yescharacteristic
444C2#2175711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21822C2semidirect
3. #21818C2semidirect
4. #9086C2semidirect
6. #9082C2semidirect
8. #8703C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26372
#218222
#218182
#90862
#90842
#90822
#90802
#87032
#83092