Finite Group Databasebeta

← #216134

Subgroups — #216134

Order 512.

878 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#541C211yesmaximal, characteristic
2#21972C211yesmaximal, characteristic
3#21431C211yesmaximal, characteristic
4#9876C211yesmaximal, characteristic, complemented
5#9875C211yesmaximal, characteristic, complemented
6#9647C211yesmaximal, characteristic, complemented
7#9643C211yesmaximal, characteristic, complemented
8#8938C212yesmaximal
9#8937C212yesmaximal
10#8566C212yesmaximal, complemented
11#8564C212yesmaximal, complemented
12C2×#541C411yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
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#1939—22no—
30#1924—22no—
31#1917—22no—
32#1850—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37C4×C4⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39#1744C2211yescharacteristic
40C4×#97C2211yescharacteristic
41C2×Q8⋊C8C2212yes—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49C23×C8—21no—
50#541—28no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#126—22no—
53C2×#125—22no—
54C2×#125C2×C411yescharacteristic
55C2×C22⋊Q8C2×C412yes—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2312yes—
59C2×#117C2×C411yescharacteristic
60C2×#117C2×C411yescharacteristic
61C22×C42C2311yescharacteristic
62#431—42no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—42no—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×C22⋊C8—21no—
73C2×C22⋊C8—22no—
74C2×#97—22no—
75C2×C4×C8—21no—
76C2×C4×C8C2312yes—
77C2×C4×C8C2312yes—
78#391C2×C412yes—
79#391C2×C412yes—
80#389—22no—
81#389—22no—
82#389—22no—
83#389C2×C412yes—
84#389C2×C412yes—
85#388—22no—
86#388—22no—
87#388—22no—
88#387—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#385—22no—
98#384C2×C411yescharacteristic
99#383—22no—
100#382—22no—
101#382C2×C411yescharacteristic
102#380C2×C412yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4C2312yes—
114C4×C22⋊C4—22no—
115C4×C22⋊C4D412yes—
116C4×C22⋊C4D412yes—
117#376C2×C411yescharacteristic
118#376C2×C411yescharacteristic
119C43—22no—
120C43C2311yescharacteristic
121C43C2311yescharacteristic
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127Q8⋊C8—42no—
128Q8⋊C8—42no—
129Q8⋊C8—42no—
130Q8⋊C8—42no—
131C22×Q8C22×C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—42no—
142C22×C8—82no—
143C4⋊Q8—24no—
144#126—216no—
145#125—216no—
146#125—24no—
147C22⋊Q8—24no—
148C22⋊Q8M1618yes—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157#117—24no—
158#117—22no—
159#117—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⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
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×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D411yescharacteristic
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—82no—
205C4⋊C8—82no—
206C22⋊C8—44no—
207C22⋊C8—42no—
208#97—44no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—42no—
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—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×M1614yes—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C2×C8—24no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—84no—
275C4⋊C4—28no—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—22no—
279C4⋊C4—28no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4C2×M1614yes—
293C22⋊C4—28no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—82no—
297C42—28no—
298C42—22no—
299C42—21no—
300C42—21no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—21no—
305C42—21no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—44no—
317C42—82no—
318C42—82no—
319C42#13112yes—
320C42#13112yes—
321C42H(5,𝔽2)12yes—
322C422−1+412yes—
323C23—21no—
324C23—21no—
325C23—22no—
326C23—22no—
327C23#40712yes—
328C23C22×C22⋊C411yescharacteristic
329Q8—24no—
330Q8—24no—
331Q8—42no—
332Q8—42no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4#40712yes—
370C2×C4#40814yes—
371C2×C4#52012yes—
372C2×C4#53511yescharacteristic
373C2×C4#53611yescharacteristic
374C2×C4#53712yes—
375C2×C4C22×M1612yes—
376C2×C4C2×#13111yescharacteristic
377C2×C4C2×#13111yescharacteristic
378C2×C4#57712yes—
379C2×C4#57812yes—
380C8—44no—
381C8—82no—
382C8—81no—
383C8—81no—
384C8—81no—
385C8—81no—
386C22—24no—
387C22—22no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22#231111yescharacteristic
392C22C2×#40711yescharacteristic
393C22C2×#40811yescharacteristic
394C22#289812yes—
395C22#290112yes—
396C4—24no—
397C4—22no—
398C4—22no—
399C4—42no—
400C4—42no—
401C4—44no—
402C4—42no—
403C4—44no—
404C4—41no—
405C4—41no—
406C4—82no—
407C4—82no—
408C4#299012yes—
409C4#299012yes—
410C4#322211yescharacteristic
411C4#322311yescharacteristic
412C4#324212yes—
413C2—42no—
414C2#1745412yes—
415C2#1746912yes—
416C2#2185111yescharacteristic
417C2#2185811yescharacteristic
418C2#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. #9876C2semidirect
5. #9875C2semidirect
6. #9647C2semidirect
7. #9643C2semidirect
10. #8566C2semidirect
11. #8564C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5412
#219722
#214312
#98762
#98752
#96472
#96432
#89382
#89372
#85662
#85642