Finite Group Databasebeta

← #207447

Subgroups — #207447

Order 512.

903 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#541C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic
3#20549C211yesmaximal, characteristic
4#9867C212yesmaximal, complemented
5#9677C212yesmaximal, complemented
6#8942C212yesmaximal
7#8941C212yesmaximal
8#8171C212yesmaximal, complemented
9C4×Q8⋊C8C212yesmaximal, complemented
10C2×#541C411yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#423C2211yescharacteristic
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#1923—22no—
28#1916—22no—
29#1771—22no—
30#1767C2212yes—
31C4×C4⋊C8C2212yes—
32C4×C4⋊C8C2212yes—
33#1753—22no—
34C4×C22⋊C8C2212yes—
35#1744C2211yescharacteristic
36C42×C8C2211yescharacteristic
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—28no—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#126—22no—
49C2×#125—22no—
50C2×#125C2×C411yescharacteristic
51C2×C22⋊Q8C2×C412yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2312yes—
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C22×C42C2311yescharacteristic
58#432—24no—
59#423—24no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—22no—
66C2×#97—21no—
67C2×#97—22no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8C2312yes—
71C2×C4×C8C2312yes—
72#391C2×C412yes—
73#391C2×C412yes—
74#389—22no—
75#389—22no—
76#389—22no—
77#389C2×C412yes—
78#389C2×C412yes—
79#388—22no—
80#388—22no—
81#388—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#385—22no—
92#384C2×C411yescharacteristic
93#383—22no—
94#382—22no—
95#382C2×C411yescharacteristic
96#380C2×C412yes—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4C2312yes—
108C4×C22⋊C4—22no—
109C4×C22⋊C4D412yes—
110C4×C22⋊C4D412yes—
111#376C2×C411yescharacteristic
112#376C2×C411yescharacteristic
113C43—22no—
114C43C2311yescharacteristic
115C43C2311yescharacteristic
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121Q8⋊C8—24no—
122Q8⋊C8—24no—
123Q8⋊C8—24no—
124Q8⋊C8—24no—
125C22×Q8C22×C412yes—
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C2×M16—24no—
130C2×M16—82no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—24no—
136#126—216no—
137#125—216no—
138#125—24no—
139C22⋊Q8—24no—
140C22⋊Q8—24no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8M1618yescomplemented
149#117—24no—
150#117—22no—
151#117—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4C22⋊C412yes—
169C2×C22⋊C4C22⋊C412yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2411yescharacteristic
190C4⋊C8—24no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—44no—
194C4⋊C8#4114yes—
195C4⋊C8#4114yes—
196C22⋊C8—24no—
197C22⋊C8—44no—
198#97—24no—
199#97—82no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—44no—
205C4×C8—82no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95C22⋊C412yes—
218#95C22⋊C412yes—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221C24C2×C22⋊C411yescharacteristic
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8C2×M1614yes—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C23×C411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252M16—84no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C4⋊C4—28no—
266C4⋊C4—28no—
267C4⋊C4—28no—
268C4⋊C4—22no—
269C4⋊C4—28no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4C2×M1614yes—
283C22⋊C4—28no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—82no—
287C42—22no—
288C42—21no—
289C42—21no—
290C42—22no—
291C42—22no—
292C42—21no—
293C42—21no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—28no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—44no—
306C42—82no—
307C42—82no—
308C42C2×M1614yes—
309C42#13112yes—
310C42#13112yes—
311C42C2×#4112yes—
312C42C2×#4112yes—
313C23—21no—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—21no—
318C23C22×C22⋊C411yescharacteristic
319Q8—24no—
320Q8—24no—
321Q8—42no—
322Q8—42no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4#40714yes—
360C2×C4#40814yes—
361C2×C4C4×#4112yes—
362C2×C4#52511yescharacteristic
363C2×C4#52511yescharacteristic
364C2×C4#53212yes—
365C2×C4C22×M1612yes—
366C2×C4C2×#13111yescharacteristic
367C2×C4C2×#13111yescharacteristic
368C2×C4#57712yes—
369C2×C4#57812yes—
370C8—44no—
371C8—44no—
372C8—44no—
373C8—82no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22#229711yescharacteristic
380C22C2×#40711yescharacteristic
381C22C2×#40811yescharacteristic
382C22#289812yes—
383C22#290112yes—
384C4—24no—
385C4—24no—
386C4—22no—
387C4—42no—
388C4—44no—
389C4—42no—
390C4—41no—
391C4—44no—
392C4—41no—
393C4—41no—
394C4—82no—
395C4—82no—
396C4#296512yes—
397C4#296512yes—
398C4#312012yes—
399C4#312311yescharacteristic
400C4#312711yescharacteristic
401C2—81no—
402C2#1745412yes—
403C2#1748412yes—
404C2#2137411yescharacteristic
405C2#2138411yescharacteristic
406C2#2139811yescharacteristic
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. #9867C2semidirect
5. #9677C2semidirect
8. #8171C2semidirect
9. C4×Q8⋊C8C2semidirect
148. C4×Q8M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5412
#214302
#205492
#98672
#96772
#89422
#89412
#81712
C4×Q8⋊C82