Finite Group Databasebeta

← #213593

Subgroups — #213593

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2545C211yesmaximal, characteristic
2#21820C212yesmaximal, complemented
3#9084C214yesmaximal, complemented
4#9080C214yesmaximal, complemented
5#8701C212yesmaximal, complemented
6#8305C212yesmaximal
7C22×C4⋊Q8C411yescharacteristic
8C22×#124C411yescharacteristic
9C22×#124C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#432C2212yes—
12C2×#431C2212yes—
13#2545—22no—
14#2545—22no—
15#2545—22no—
16#2545—22no—
17C2×#396C414yes—
18C2×#391C411yescharacteristic
19C2×#390C411yescharacteristic
20C2×#388C2214yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22#1985—22no—
23#1985—22no—
24#1984—22no—
25#1984—22no—
26#1817C2214yes—
27#1817C2214yes—
28C2×#323C2212yes—
29C2×#323C2214yes—
30C2×#323C2212yes—
31#1280C2212yes—
32#1280C2214yes—
33#1280C2212yes—
34C23×Q8C2×C411yescharacteristic
35C23×Q8C2×C411yescharacteristic
36C23×D4C2×C411yescharacteristic
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—44no—
41C2×#124—22no—
42C2×#124—21no—
43C2×#124—21no—
44C2×#124—22no—
45C2×#124—22no—
46C2×#124—41no—
47C2×#124—41no—
48C2×#124—42no—
49C2×#124—44no—
50C2×C4×Q8—41no—
51C2×C4×Q8—42no—
52C2×C4×Q8—41no—
53C2×C4×Q8D412yes—
54C2×C4×Q8D412yes—
55C2×C4×Q8D412yes—
56C2×C4×Q8D412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C414yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C414yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C414yes—
64C22×C22⋊C4C2312yes—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2312yes—
69#432—42no—
70#432—42no—
71#431—42no—
72#431—42no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C22⋊C8C2314yes—
76C2×C22⋊C8C2314yes—
77C2×#97—22no—
78C2×C4×C8—22no—
79#396—24no—
80#396—24no—
81#396—24no—
82#396—24no—
83#391—21no—
84#391—21no—
85#391—22no—
86#390—21no—
87#390—21no—
88#390—22no—
89#388—24no—
90#388—24no—
91#388—24no—
92#388—24no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C2×#95C2×C414yes—
96C2×#95C2312yes—
97#336—24no—
98#336—24no—
99#336—24no—
100#336—24no—
101#323—42no—
102#323—44no—
103#323—42no—
104C25C22×C412yes—
105C22×Q8—24no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—41no—
109C22×Q8—41no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×Q8—42no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8C22⋊C412yes—
116C22×Q8C22⋊C411yescharacteristic
117C22×Q8C22⋊C411yescharacteristic
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—41no—
121C22×D4—41no—
122C22×D4—42no—
123C22×D4—42no—
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C412yes—
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C412yes—
128C23×C4C22×C412yes—
129C23×C4C22×C412yes—
130C23×C4C22×C412yes—
131C23×C4C2411yescharacteristic
132C22×C8—24no—
133C22×C8—24no—
134C4⋊Q8—48no—
135C4⋊Q8—44no—
136C4⋊Q8—44no—
137#124—44no—
138#124—44no—
139#124—44no—
140#124—44no—
141#124—48no—
142#124—44no—
143#124—44no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×Q8—42no—
151C4×Q8—42no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—48no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C4⋊C8—44no—
203C4⋊C8—44no—
204C22⋊C8—24no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208#97—44no—
209C4×C8—44no—
210#95—24no—
211#95—24no—
212#95—24no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—24no—
217C24—24no—
218C24—42no—
219C24—42no—
220C24—44no—
221C24C23×C411yescharacteristic
222C2×Q8—24no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—41no—
226C2×Q8—42no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—48no—
231C2×Q8—48no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—84no—
239C2×Q8—84no—
240C2×Q8#10014yes—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—84no—
248C2×D4—84no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4#13114yes—
288C22×C4C22×D412yes—
289C22×C4H(5,𝔽2)11yescharacteristic
290C22×C4H(5,𝔽2)11yescharacteristic
291C22×C4H(5,𝔽2)12yes—
292C2×C8—48no—
293C2×C8—48no—
294C2×C8—44no—
295C2×C8—44no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C22⋊C4—48no—
307C22⋊C4—48no—
308C22⋊C4—48no—
309C22⋊C4—48no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—48no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C23—22no—
330C23—21no—
331C23—21no—
332C23—42no—
333C23—44no—
334C23—44no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—44no—
340C23—82no—
341C23—82no—
342C23—84no—
343C23—84no—
344C23—84no—
345C23—84no—
346C23—82no—
347C23—82no—
348C23C22×C22⋊C411yescharacteristic
349C23#51811yescharacteristic
350C23#51811yescharacteristic
351C23C2×#13112yes—
352C23#56812yes—
353Q8—44no—
354Q8—48no—
355Q8—44no—
356Q8—84no—
357Q8—84no—
358D4—84no—
359D4—84no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4C2×#10012yes—
405C2×C4C2×#10014yes—
406C2×C4#41312yes—
407C2×C4#53412yes—
408C2×C4#53412yes—
409C8—84no—
410C8—84no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—84no—
420C22—84no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—162no—
425C22—162no—
426C22—164no—
427C22#230811yescharacteristic
428C22#288012yes—
429C22#288412yes—
430C22C22×#10011yescharacteristic
431C22C2×#41311yescharacteristic
432C22#297911yescharacteristic
433C22#298011yescharacteristic
434C22#298112yes—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—24no—
440C4—44no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—82no—
446C4—82no—
447C2—21no—
448C2—21no—
449C2—22no—
450C2—162no—
451C2#1742512yes—
452C2#1742812yes—
453C2#2174511yescharacteristic
454C2#2175111yescharacteristic
455C2#2175211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×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. #21820C2semidirect
3. #9084C2semidirect
4. #9080C2semidirect
5. #8701C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25452
#218202
#90842
#90802
#87012
#83052