Finite Group Databasebeta

← #216103

Subgroups — #216103

Order 512.

871 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#541C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic
3#21430C211yesmaximal, characteristic
4#9872C211yesmaximal, characteristic, complemented
5#9871C211yesmaximal, characteristic, complemented
6#9641C211yesmaximal, characteristic, complemented
7#9637C211yesmaximal, characteristic, complemented
8#8942C212yesmaximal
9#8941C212yesmaximal
10#8566C212yesmaximal, complemented
11#8564C212yesmaximal, complemented
12C2×#541C411yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#2336—22no—
16#2334—22no—
17#2330C411yescharacteristic
18#2327C411yescharacteristic
19#2323C412yes—
20#2320C412yes—
21#2314C411yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×#125—22no—
24C4×#125C2211yescharacteristic
25C4×#126—22no—
26C4×C22⋊Q8C2212yes—
27C42×Q8C2212yes—
28C4×#117C2211yescharacteristic
29#1940—22no—
30#1923—22no—
31#1916—22no—
32#1851—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37C4×C4⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39#1744C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×Q8⋊C8C2212yes—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49C22×M16—21no—
50#541—24no—
51#541—24no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#126—22no—
54C2×#125—22no—
55C2×#125C2×C411yescharacteristic
56C2×C22⋊Q8C2×C412yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2312yes—
60C2×#117C2×C411yescharacteristic
61C2×#117C2×C411yescharacteristic
62C22×C42C2311yescharacteristic
63#432—42no—
64#423—42no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2311yescharacteristic
73C2×C22⋊C8—21no—
74C2×C22⋊C8—22no—
75C2×#97—21no—
76C2×C4×C8—22no—
77C2×C4×C8C2312yes—
78C2×C4×C8C2312yes—
79#391C2×C412yes—
80#391C2×C412yes—
81#389—22no—
82#389—22no—
83#389—22no—
84#389C2×C412yes—
85#389C2×C412yes—
86#388—22no—
87#388—22no—
88#388—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#385—22no—
99#384C2×C411yescharacteristic
100#383—22no—
101#382—22no—
102#382C2×C411yescharacteristic
103#380C2×C412yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C4×C22⋊C4—22no—
116C4×C22⋊C4D412yes—
117C4×C22⋊C4D412yes—
118#376C2×C411yescharacteristic
119#376C2×C411yescharacteristic
120C43—22no—
121C43C2311yescharacteristic
122C43C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128Q8⋊C8—42no—
129Q8⋊C8—42no—
130Q8⋊C8—42no—
131Q8⋊C8—42no—
132C22×Q8C22×C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—42no—
137C2×M16—82no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C4⋊Q8—22no—
145C4⋊Q8M1614yescomplemented
146#126—216no—
147#125—216no—
148#125—22no—
149#125M1614yescomplemented
150C22⋊Q8—24no—
151C22⋊Q8—24no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160#117—22no—
161#117—22no—
162#117—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C22⋊C412yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2411yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—42no—
206C4⋊C8—44no—
207C4⋊C8—82no—
208C4⋊C8—82no—
209C22⋊C8—44no—
210C22⋊C8—42no—
211#97—42no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—44no—
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#95—22no—
226#95—22no—
227#95—22no—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232C24C2×C22⋊C411yescharacteristic
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8C2×M1614yes—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
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—84no—
264C2×C8—24no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C4⋊C4—22no—
281C4⋊C4—28no—
282C4⋊C4—28no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4C2×M1614yes—
296C22⋊C4—28no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—82no—
300C42—28no—
301C42—22no—
302C42—21no—
303C42—21no—
304C42—22no—
305C42—21no—
306C42—21no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—44no—
319C42—82no—
320C42—82no—
321C42C2×M1612yes—
322C42C2×M1612yes—
323C42#13114yes—
324C42H(5,𝔽2)12yes—
325C422−1+412yes—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23C22×C22⋊C411yescharacteristic
332Q8—24no—
333Q8—24no—
334Q8—42no—
335Q8—42no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—82no—
372C2×C4#40714yes—
373C2×C4#40814yes—
374C2×C4#52012yes—
375C2×C4#53511yescharacteristic
376C2×C4#53611yescharacteristic
377C2×C4#53712yes—
378C2×C4C22×M1611yescharacteristic
379C2×C4C22×M1611yescharacteristic
380C2×C4C2×#13112yes—
381C2×C4#57712yes—
382C2×C4#57812yes—
383C8—44no—
384C8—82no—
385C8—81no—
386C8—81no—
387C8—81no—
388C8—81no—
389C22—42no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22#231111yescharacteristic
395C22C2×#40711yescharacteristic
396C22C2×#40811yescharacteristic
397C22#289812yes—
398C22#290112yes—
399C4—24no—
400C4—24no—
401C4—22no—
402C4—42no—
403C4—44no—
404C4—42no—
405C4—41no—
406C4—44no—
407C4—41no—
408C4—41no—
409C4—82no—
410C4—82no—
411C4#299012yes—
412C4#299112yes—
413C4#322211yescharacteristic
414C4#322311yescharacteristic
415C4#324212yes—
416C2—81no—
417C2#1745412yes—
418C2#1748412yes—
419C2#2184911yescharacteristic
420C2#2185811yescharacteristic
421C2#2187311yescharacteristic
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. #9872C2semidirect
5. #9871C2semidirect
6. #9641C2semidirect
7. #9637C2semidirect
10. #8566C2semidirect
11. #8564C2semidirect
145. C4⋊Q8M16semidirect
149. #125M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5412
#219782
#214302
#98722
#98712
#96412
#96372
#89422
#89412
#85662
#85642