Finite Group Databasebeta

← #214454

Subgroups — #214454

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2#21822C211yesmaximal, characteristic, complemented
3#13961C211yesmaximal, characteristic, complemented
4#9157C212yesmaximal, complemented
5#9149C212yesmaximal, complemented
6#9118C214yesmaximal, complemented
7#8716C212yesmaximal, complemented
8#8332C212yesmaximal
9C2×#536—21no—
10C2×#532C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2212yes—
13#2854C2212yes—
14#2854C2212yes—
15#2618—21no—
16#2618—21no—
17#2548C412yescomplemented
18#2546C412yescomplemented
19#2545—22no—
20#2538C414yescomplemented
21#2536—22no—
22#2311—21no—
23#2285C2211yescharacteristic
24C2×#383C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#1985—22no—
27#1985—22no—
28#1817C2212yes—
29#1817C2212yes—
30C2×#327C2214yes—
31C2×#327C2214yes—
32#1446—21no—
33#1445—21no—
34#1436—21no—
35#1436—21no—
36#1434C2212yes—
37#1434C2212yes—
38#1294C2214yes—
39#1294C2214yes—
40C22×#41—21no—
41C23×Q8D411yescharacteristic
42#536—48no—
43#532—42no—
44#532—42no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#127C2×C411yescharacteristic
48C2×#124—22no—
49C2×#124—21no—
50C2×#123—22no—
51C2×#123—22no—
52C2×C22⋊Q8—24no—
53C2×#121—21no—
54C2×#121C2312yes—
55C2×C4×Q8—42no—
56C2×C4×Q8—42no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8D412yes—
60C2×C4×Q8D412yes—
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C2×#117—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#432—42no—
72#432—42no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8C2314yes—
75C2×#97—22no—
76#398—22no—
77#398—22no—
78#396—24no—
79#396—24no—
80#394—22no—
81#394—22no—
82#391—22no—
83#390—22no—
84#390C2×C414yes—
85#390C2×C414yes—
86#389C2×C414yes—
87#389C2×C414yes—
88#388—24no—
89#388—24no—
90#386—22no—
91#386—22no—
92#384C2×C412yes—
93#383—42no—
94#382—21no—
95#380—21no—
96#380—24no—
97#380C2312yes—
98C4×C4⋊C4—21no—
99C4×C4⋊C4C2×C412yes—
100#376—42no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95C2312yes—
109C2×#95C2312yes—
110#336—22no—
111#336—22no—
112#336—24no—
113#327—44no—
114#327—44no—
115C2×#41—42no—
116C2×#41—42no—
117C2×#41—44no—
118C2×#41—44no—
119C22×Q8—21no—
120C22×Q8—24no—
121C22×Q8—41no—
122C22×Q8—41no—
123C22×Q8—42no—
124C22×Q8C22⋊C412yes—
125C22×Q8C22⋊C412yes—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4C22×C412yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—24no—
141C4⋊Q8—42no—
142C4⋊Q8—48no—
143#127—42no—
144#124—44no—
145#124—48no—
146#123—48no—
147#123—48no—
148C22⋊Q8—416no—
149#121—24no—
150#121—24no—
151#121—42no—
152#121—42no—
153C4×Q8—44no—
154C4×Q8—44no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×D4—44no—
160C4×D4—44no—
161#117—44no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C414yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—42no—
196C2×C42—44no—
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C412yes—
200C2×C42C22×C412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C4⋊C8—44no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208#97—44no—
209#95—22no—
210#95—24no—
211#95—24no—
212#95—22no—
213#95—24no—
214#95—24no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—42no—
219#95—42no—
220#95—42no—
221#95—44no—
222C24—22no—
223C24C22×D411yescharacteristic
224#41—88no—
225#41—88no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—48no—
233C2×Q8—48no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×D4—24no—
238C2×D4—44no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—42no—
247C2×D4—82no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4AGL(1,ℤ/8)14yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4H(5,𝔽2)12yes—
300C22×C4H(5,𝔽2)12yes—
301C2×C8—48no—
302C2×C8—44no—
303C4⋊C4—24no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—48no—
313C4⋊C4—48no—
314C4⋊C4—44no—
315C4⋊C4—82no—
316C22⋊C4—44no—
317C22⋊C4—48no—
318C22⋊C4—48no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C42—22no—
324C42—24no—
325C42—24no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—48no—
334C42—44no—
335C23—22no—
336C23—22no—
337C23—42no—
338C23—44no—
339C23—42no—
340C23—42no—
341C23—44no—
342C23C22×C22⋊C411yescharacteristic
343C23#53411yescharacteristic
344C23#53411yescharacteristic
345C23C2×AGL(1,ℤ/8)12yes—
346C23#57512yes—
347Q8—48no—
348Q8—48no—
349Q8—84no—
350D4—84no—
351D4—84no—
352D4—84no—
353D4—84no—
354D4—84no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—88no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4#41812yes—
411C2×C4#41812yes—
412C2×C4#51812yes—
413C2×C4#53412yes—
414C8—84no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—21no—
422C22—21no—
423C22—82no—
424C22—82no—
425C22—84no—
426C22—84no—
427C22—82no—
428C22#230811yescharacteristic
429C22#288412yes—
430C22C2×#41811yescharacteristic
431C22#289611yescharacteristic
432C22#290112yes—
433C22#319412yes—
434C22#319412yes—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4—41no—
443C4—42no—
444C4—41no—
445C4—84no—
446C4—82no—
447C4—84no—
448C4—162no—
449C2—22no—
450C2—22no—
451C2—162no—
452C2#1762412yes—
453C2#1763112yes—
454C2#2175111yescharacteristic
455C2#2176711yescharacteristic
456C2#2177511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #13961C2semidirect
4. #9157C2semidirect
5. #9149C2semidirect
6. #9118C2semidirect
7. #8716C2semidirect
17. #2548C4semidirect
18. #2546C4semidirect
20. #2538C4semidirect

Maximal subgroups

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

subgroupindex
#351782
#218222
#139612
#91572
#91492
#91182
#87162
#83322