Finite Group Databasebeta

← #216162

Subgroups — #216162

Order 512.

967 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#540C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic, complemented
3#21431C211yesmaximal, characteristic, complemented
4#9876C211yesmaximal, characteristic, complemented
5#9875C211yesmaximal, characteristic, complemented
6#9648C211yesmaximal, characteristic, complemented
7#9644C211yesmaximal, characteristic, complemented
8#8944C212yesmaximal, complemented
9#8943C212yesmaximal, complemented
10#8565C212yesmaximal, complemented
11#8563C212yesmaximal, complemented
12C2×#540C411yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#2335—22no—
16#2333—22no—
17#2330C411yescharacteristic
18#2328C411yescharacteristic
19#2324C412yes—
20#2319C412yes—
21#2314C411yescharacteristic
22C4×#127C2211yescharacteristic
23C4×#125C2211yescharacteristic
24C4×#123—22no—
25C4×#121—22no—
26C4×#121C2212yes—
27C42×D4C2212yes—
28C4×#117C2211yescharacteristic
29#1940—22no—
30#1924—22no—
31#1917—22no—
32#1851—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37C4×C4⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39#1744C2211yescharacteristic
40C4×#97C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49C22×M16—21no—
50#540—28no—
51C2×#127C2×C411yescharacteristic
52C2×#125C2×C411yescharacteristic
53C2×#123—22no—
54C2×#121—22no—
55C2×#121C2×C412yes—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2312yes—
61C2×#117C2×C411yescharacteristic
62C2×#117C2×C411yescharacteristic
63C22×C42—22no—
64C22×C42C2311yescharacteristic
65#432—42no—
66#423—42no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2311yescharacteristic
75C2×C22⋊C8—21no—
76C2×C22⋊C8—22no—
77C2×#97—21no—
78C2×#97—22no—
79C2×C4×C8C2312yes—
80C2×C4×C8C2312yes—
81#390—22no—
82#390—22no—
83#390C2×C412yes—
84#390C2×C412yes—
85#389—22no—
86#389C2×C412yes—
87#389C2×C412yes—
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#386—22no—
97#386—22no—
98#386—22no—
99#386—22no—
100#385—22no—
101#384C2×C411yescharacteristic
102#382C2×C411yescharacteristic
103#381—22no—
104#380—22no—
105#380C2×C412yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4C2312yes—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4D412yes—
117C4×C22⋊C4D412yes—
118#376C2×C411yescharacteristic
119#376C2×C411yescharacteristic
120C43C2311yescharacteristic
121C43C2311yescharacteristic
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127D4⋊C8—42no—
128D4⋊C8—42no—
129D4⋊C8—42no—
130D4⋊C8—42no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4C22×C412yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C2×M16—42no—
141C2×M16—82no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148#127—24no—
149#125—24no—
150#123—216no—
151#121—216no—
152#121—24no—
153#121—24no—
154C4×D4—24no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—42no—
160C4×D4—42no—
161C4×D4—82no—
162C4×D4—82no—
163C4×D4—82no—
164C4×D4—82no—
165C4×D4M1618yescomplemented
166#117—24no—
167#117—41no—
168#117—41no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—82no—
195C2×C42—82no—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2411yescharacteristic
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—42no—
213C4⋊C8—44no—
214C4⋊C8—82no—
215C4⋊C8—82no—
216C22⋊C8—44no—
217C22⋊C8—42no—
218#97—42no—
219#97—44no—
220C4×C8—22no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235C24—22no—
236C24C2×C22⋊C411yescharacteristic
237C2×D4—28no—
238C2×D4—28no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4C2×M1614yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—28no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284M16—84no—
285C2×C8—24no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C4⋊C4—28no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4C2×M1614yes—
311C22⋊C4—28no—
312C22⋊C4—28no—
313C22⋊C4—24no—
314C22⋊C4—42no—
315C22⋊C4—44no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C42—22no—
324C42—21no—
325C42—21no—
326C42—22no—
327C42—22no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—82no—
341C42—82no—
342C42C2×M1614yes—
343C42H(5,𝔽2)12yes—
344C422−1+412yes—
345C23—21no—
346C23—21no—
347C23—24no—
348C23—21no—
349C23—22no—
350C23—22no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23C22×C22⋊C411yescharacteristic
358D4—24no—
359D4—24no—
360D4—42no—
361D4—42no—
362D4—88no—
363D4—88no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—82no—
398C2×C4—88no—
399C2×C4—88no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4#40714yes—
405C2×C4#52012yes—
406C2×C4#53511yescharacteristic
407C2×C4#53611yescharacteristic
408C2×C4#53712yes—
409C2×C4C22×M1612yes—
410C2×C4#56811yescharacteristic
411C2×C4#56811yescharacteristic
412C2×C4#57612yes—
413C2×C4#57712yes—
414C8—44no—
415C8—82no—
416C8—81no—
417C8—81no—
418C8—81no—
419C8—81no—
420C22—44no—
421C22—42no—
422C22—44no—
423C22—41no—
424C22—44no—
425C22—42no—
426C22—41no—
427C22—41no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22#231111yescharacteristic
433C22C2×#40711yescharacteristic
434C22#287911yescharacteristic
435C22#289712yes—
436C22#290112yes—
437C4—24no—
438C4—22no—
439C4—42no—
440C4—44no—
441C4—41no—
442C4—42no—
443C4—42no—
444C4—41no—
445C4—41no—
446C4—82no—
447C4—82no—
448C4#298912yes—
449C4#298912yes—
450C4#322011yescharacteristic
451C4#322111yescharacteristic
452C4#324112yes—
453C2—81no—
454C2—84no—
455C2#1745312yes—
456C2#1750512yes—
457C2#2185311yescharacteristic
458C2#2185811yescharacteristic
459C2#2187211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#540C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21978C2semidirect
3. #21431C2semidirect
4. #9876C2semidirect
5. #9875C2semidirect
6. #9648C2semidirect
7. #9644C2semidirect
8. #8944C2semidirect
9. #8943C2semidirect
10. #8565C2semidirect
11. #8563C2semidirect
165. C4×D4M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5402
#219782
#214312
#98762
#98752
#96482
#96442
#89442
#89432
#85652
#85632