Finite Group Databasebeta

← #214872

Subgroups — #214872

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35739C211yesmaximal, characteristic
2#20640C211yesmaximal, characteristic
3#17372C211yesmaximal, characteristic
4#9488C212yesmaximal, complemented
5#9400C212yesmaximal, complemented
6#9191C212yesmaximal, complemented
7#9033C212yesmaximal, complemented
8#8764C212yesmaximal, complemented
9#8391C212yesmaximal, complemented
10C2×D4×Q8C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2983C2212yes—
13#2905C2212yes—
14C2×#413—21no—
15#2628C2211yescharacteristic
16#2627C412yes—
17#2581C2211yescharacteristic
18#2567C2211yescharacteristic
19#2564C412yes—
20#2451—21no—
21#2449—22no—
22#2440—21no—
23#2438—22no—
24#2409C412yes—
25#2396C2211yescharacteristic
26C2×#395C412yes—
27#2349C2211yescharacteristic
28D4×C4⋊C4—21no—
29#2339—21no—
30#1801—21no—
31#1801C2212yes—
32#1800—21no—
33#1800C2212yes—
34C2×#343—21no—
35#1548C2214yes—
36#1487C2214yes—
37#1466—22no—
38#1464—22no—
39#1460C2212yes—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C22×#41D411yescharacteristic
44C23×Q8C2×C412yes—
45C22×M16—22no—
46D4×Q8—44no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C412yes—
52#520—44no—
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8C2311yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D412yes—
61C2×C4×D4C2311yescharacteristic
62C2×#117D412yes—
63C2×#117C2311yescharacteristic
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4D411yescharacteristic
67#446—44no—
68#432—24no—
69#431C2314yes—
70#424—44no—
71#423—22no—
72#423C2314yes—
73#413—44no—
74C2×#101—22no—
75C2×#100—22no—
76#400—21no—
77#400C2×C412yes—
78#400C2×C411yescharacteristic
79#399—21no—
80#399C2×C411yescharacteristic
81#398—22no—
82#398—22no—
83#398—21no—
84#396—21no—
85#395—22no—
86#395—22no—
87#395C2×C412yes—
88#395C2×C412yes—
89#395C2×C412yes—
90#395C2×C412yes—
91#395C2×C411yescharacteristic
92#393C2×C412yes—
93#393C2×C411yescharacteristic
94#391—21no—
95#391—21no—
96#391—22no—
97#389—21no—
98#389—21no—
99#389—22no—
100#389—21no—
101#387—22no—
102#387—22no—
103#386—22no—
104#386—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#384—21no—
110#382C2311yescharacteristic
111#381—22no—
112#380—21no—
113#380—22no—
114#380C2×C412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—21no—
118C2×#95—22no—
119C2×#95C2×C412yes—
120#343—22no—
121#343—22no—
122#333—44no—
123Q8⋊C8—44no—
124#324—22no—
125#323—22no—
126#323—24no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—24no—
130C2×#41—42no—
131C22×Q8—22no—
132C22×Q8—44no—
133C22×Q8—84no—
134C22×Q8C22×C411yescharacteristic
135C22×Q8C22×C412yes—
136C22×Q8C2×D411yescharacteristic
137C22×D4C22⋊C412yes—
138C22×D4C2×D411yescharacteristic
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22×C412yes—
145C23×C4C2×D411yescharacteristic
146C2×M16—24no—
147C2×M16—42no—
148C2×M16—44no—
149C2×M16—44no—
150C22×C8—42no—
151C4⋊Q8—24no—
152C4⋊Q8—44no—
153C22⋊Q8—48no—
154C22⋊Q8—44no—
155C22⋊Q8—84no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—42no—
159C4×D4—24no—
160C4×D4—44no—
161C4×D4—42no—
162C4×D4—44no—
163C4×D4—44no—
164C4×D4—42no—
165C4×D4—42no—
166#117—24no—
167#117—44no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—84no—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22×C411yescharacteristic
199C2×C22⋊C4C2×D412yes—
200C2×C22⋊C4C2×D411yescharacteristic
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22⋊C412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2411yescharacteristic
207C4⋊C8—24no—
208C4⋊C8—24no—
209C4⋊C8—24no—
210C4⋊C8—44no—
211C4⋊C8—44no—
212#101—44no—
213#100—44no—
214C22⋊C8—24no—
215#97—44no—
216C4×C8—24no—
217#95—21no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95C22×C411yescharacteristic
231#95C22×C412yes—
232#95C22×C411yescharacteristic
233C24C2×C22⋊C412yes—
234C24C2×C22⋊C411yescharacteristic
235#41—48no—
236#41—48no—
237C2×Q8—22no—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—42no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—82no—
247C2×Q8—82no—
248C2×Q8—88no—
249C2×Q8—88no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—42no—
255C2×D4AGL(1,ℤ/8)12yes—
256C2×D4#13712yes—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4#13714yes—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291M16—44no—
292M16—44no—
293M16—84no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—84no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—24no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—82no—
328C42—24no—
329C42—22no—
330C42—42no—
331C42—42no—
332C42—41no—
333C42—41no—
334C42H(5,𝔽2)14yes—
335C23—21no—
336C23—22no—
337C23—22no—
338C23—21no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23#41714yes—
344C23#41912yes—
345C23C22×C22⋊C411yescharacteristic
346Q8—24no—
347Q8—42no—
348Q8—42no—
349Q8—42no—
350Q8—84no—
351Q8—88no—
352D4—24no—
353D4—24no—
354D4—48no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—48no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—84no—
392C2×C4—88no—
393C2×C4—88no—
394C2×C4—162no—
395C2×C4—162no—
396C2×C4#41912yes—
397C2×C4#51812yes—
398C2×C4#53412yes—
399C2×C4#53412yes—
400C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
401C2×C4C2×#13712yes—
402C2×C4C2×#13711yescharacteristic
403C8—84no—
404C8—84no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—24no—
409C22—44no—
410C22—42no—
411C22—44no—
412C22—42no—
413C22—41no—
414C22—42no—
415C22—41no—
416C22—81no—
417C22#230811yescharacteristic
418C22C2×#41911yescharacteristic
419C22C2×#41711yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—41no—
428C4—42no—
429C4—41no—
430C4—81no—
431C4—82no—
432C4—82no—
433C4—81no—
434C4#319312yes—
435C4#319512yes—
436C2—21no—
437C2—21no—
438C2—44no—
439C2—42no—
440C2#2176511yescharacteristic
441C2#2177811yescharacteristic
442C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9488C2semidirect
5. #9400C2semidirect
6. #9191C2semidirect
7. #9033C2semidirect
8. #8764C2semidirect
9. #8391C2semidirect

Maximal subgroups

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

subgroupindex
#357392
#206402
#173722
#94882
#94002
#91912
#90332
#87642
#83912