Finite Group Databasebeta

← #216384

Subgroups — #216384

Order 512.

877 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#541C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic
3#21350C211yesmaximal, characteristic
4#9866C212yesmaximal, complemented
5#9673C212yesmaximal, complemented
6#9077C212yesmaximal
7#9075C212yesmaximal
8#8566C212yesmaximal, complemented
9#8564C212yesmaximal, complemented
10C2×#541C411yescharacteristic
11C2×#431C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2336—22no—
14#2334—22no—
15#2330C411yescharacteristic
16#2327C411yescharacteristic
17#2323C412yes—
18#2320C412yes—
19#2314C411yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C4×#125—22no—
22C4×#125C2211yescharacteristic
23C4×#126—22no—
24C4×C22⋊Q8C2212yes—
25C42×Q8C2212yes—
26C4×#117C2211yescharacteristic
27#1940—22no—
28#1923—22no—
29#1916—22no—
30#1851—22no—
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33#1797C2212yes—
34C4×C4⋊C8C2212yes—
35C4×C4⋊C8C2212yes—
36C4×C4⋊C8C2212yes—
37C2×Q8⋊C8C2212yes—
38C2×Q8⋊C8C2212yes—
39C2×Q8⋊C8C2212yes—
40C2×Q8⋊C8C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45C22×M16—21no—
46#541—24no—
47#541—24no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#126—22no—
50C2×#125—22no—
51C2×#125C2×C411yescharacteristic
52C2×C22⋊Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2312yes—
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C22×C42C2311yescharacteristic
59#431—41no—
60#431—41no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—22no—
69C4×M16—41no—
70C4×M16—41no—
71C2×#97—21no—
72C2×C4×C8—22no—
73C2×C4×C8C2312yes—
74C2×C4×C8C2312yes—
75#391C2×C412yes—
76#391C2×C412yes—
77#389—22no—
78#389—22no—
79#389—22no—
80#389C2×C412yes—
81#389C2×C412yes—
82#388—22no—
83#388—22no—
84#388—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#385—22no—
95#384C2×C411yescharacteristic
96#383—22no—
97#382—22no—
98#382C2×C411yescharacteristic
99#380C2×C412yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4C2312yes—
111C4×C22⋊C4—22no—
112C4×C22⋊C4D412yes—
113C4×C22⋊C4D412yes—
114#376C2×C411yescharacteristic
115#376C2×C411yescharacteristic
116C43—22no—
117C43C2311yescharacteristic
118C43C2311yescharacteristic
119#336—22no—
120#336—22no—
121#336—22no—
122#336—22no—
123#336—22no—
124Q8⋊C8—42no—
125Q8⋊C8—42no—
126Q8⋊C8—42no—
127Q8⋊C8—42no—
128C22×Q8C22×C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—44no—
133C2×M16—42no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C4⋊Q8—22no—
139C4⋊Q8M1614yescomplemented
140#126—216no—
141#125—216no—
142#125—22no—
143#125M1614yescomplemented
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154#117—22no—
155#117—22no—
156#117—22no—
157#117—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4C22⋊C412yes—
175C2×C22⋊C4C22⋊C412yes—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2411yescharacteristic
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—42no—
203C4⋊C8—44no—
204C4⋊C8—44no—
205C4⋊C8—82no—
206C22⋊C8—42no—
207C22⋊C8—82no—
208#97—42no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—82no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95C22⋊C412yes—
226#95C22⋊C412yes—
227#95C22⋊C412yes—
228#95C22⋊C412yes—
229C24C2×C22⋊C411yescharacteristic
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8C2×M1614yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263M16—48no—
264C2×C8—24no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—22no—
279C4⋊C4—28no—
280C4⋊C4—28no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4C2×M1614yes—
294C22⋊C4—28no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—82no—
298C42—28no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—44no—
320C42—82no—
321C42—82no—
322C42C2×M1612yes—
323C42C2×M1612yes—
324C42#13112yes—
325C42#13112yes—
326C42H(5,𝔽2)11yescharacteristic
327C42H(5,𝔽2)11yescharacteristic
328C422−1+411yescharacteristic
329C422−1+411yescharacteristic
330C23—21no—
331C23—21no—
332C23—21no—
333C23—22no—
334C23—22no—
335C23C22×C22⋊C411yescharacteristic
336Q8—24no—
337Q8—24no—
338Q8—42no—
339Q8—42no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4#40714yes—
380C2×C4#40814yes—
381C2×C4#52011yescharacteristic
382C2×C4#52011yescharacteristic
383C2×C4#53511yescharacteristic
384C2×C4#53611yescharacteristic
385C2×C4#53711yescharacteristic
386C2×C4#53711yescharacteristic
387C2×C4C22×M1611yescharacteristic
388C2×C4C22×M1611yescharacteristic
389C2×C4C2×#13111yescharacteristic
390C2×C4C2×#13111yescharacteristic
391C2×C4#57712yes—
392C2×C4#57812yes—
393C8—44no—
394C8—82no—
395C8—82no—
396C8—82no—
397C22—42no—
398C22—42no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22#231111yescharacteristic
403C22C2×#40711yescharacteristic
404C22C2×#40811yescharacteristic
405C22#289812yes—
406C22#290112yes—
407C4—24no—
408C4—24no—
409C4—21no—
410C4—21no—
411C4—42no—
412C4—44no—
413C4—42no—
414C4—41no—
415C4—44no—
416C4—41no—
417C4—41no—
418C4—82no—
419C4—82no—
420C4#299011yescharacteristic
421C4#299011yescharacteristic
422C4#299111yescharacteristic
423C4#299111yescharacteristic
424C4#325812yes—
425C4#326011yescharacteristic
426C4#326111yescharacteristic
427C2—81no—
428C2#1745412yes—
429C2#1746912yes—
430C2#2184911yescharacteristic
431C2#2189811yescharacteristic
432C2#2190111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
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
4. #9866C2semidirect
5. #9673C2semidirect
8. #8566C2semidirect
9. #8564C2semidirect
139. C4⋊Q8M16semidirect
143. #125M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5412
#219222
#213502
#98662
#96732
#90772
#90752
#85662
#85642