Finite Group Databasebeta

← #215319

Subgroups — #215319

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3#20777C211yesmaximal, characteristic
4#9677C214yesmaximal, complemented
5#9641C214yesmaximal, complemented
6#8311C214yesmaximal
7C22×C4⋊Q8C411yescharacteristic
8C22×C4×Q8C2212yes—
9C4×C4⋊Q8—21no—
10C4×C4⋊Q8—21no—
11C4×C4⋊Q8—21no—
12C4×C4⋊Q8—21no—
13C4×C4⋊Q8C412yes—
14C4×C4⋊Q8C412yes—
15C4×C4⋊Q8C2212yes—
16C4×C4⋊Q8C2212yes—
17C2×#391C412yes—
18C2×#389—21no—
19C2×#389—21no—
20C2×#384C411yescharacteristic
21C2×C4×C4⋊C4—21no—
22C2×C4×C4⋊C4—21no—
23C2×C43C2211yescharacteristic
24#1839—21no—
25#1839—21no—
26#1831—21no—
27#1831C2212yes—
28#1817—21no—
29#1817—21no—
30#1797—21no—
31#1797C2212yes—
32#1767C2212yes—
33C4×C4⋊C8C2212yes—
34#1749—21no—
35C4×C22⋊C8C2212yes—
36#1745—21no—
37#1744C2212yes—
38C2×Q8⋊C8C22116yes—
39C23×Q8C2×C412yes—
40C2×C4⋊Q8—28no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—48no—
48C2×C4×Q8C2×C414yes—
49C2×C4×Q8C2314yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2×C412yes—
56C22×C42—22no—
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C4⋊C8—22no—
65C2×C4⋊C8C2314yes—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×#97—22no—
69C2×C4×C8C2314yes—
70#391—22no—
71#391—22no—
72#391C2×C414yes—
73#391C2×C414yes—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#384—21no—
83#384—21no—
84#384C2×C412yes—
85#384C2×C412yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C2×#95—22no—
95C2×#95—22no—
96C43C2×C411yescharacteristic
97C43C2×C411yescharacteristic
98C43D411yescharacteristic
99C43D411yescharacteristic
100C43D411yescharacteristic
101C43D411yescharacteristic
102C43C2311yescharacteristic
103C43C2311yescharacteristic
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112Q8⋊C8—416no—
113C22×Q8—28no—
114C22×Q8—22no—
115C22×Q8—28no—
116C22×Q8—44no—
117C22×Q8C22×C414yes—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22⋊C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C22×C8—24no—
128C22×C8—24no—
129C4⋊Q8—28no—
130C4⋊Q8—28no—
131C4⋊Q8—28no—
132C4⋊Q8—24no—
133C4⋊Q8M1618yes—
134C4×Q8—216no—
135C4×Q8—48no—
136C4×Q8—48no—
137C4×Q8—48no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—48no—
155C2×C4⋊C4C22×C414yes—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—24no—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42—48no—
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2411yescharacteristic
194C4⋊C8—28no—
195C4⋊C8—44no—
196C22⋊C8—44no—
197C22⋊C8—44no—
198#97—44no—
199C4×C8—28no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204C24C2×C22⋊C411yescharacteristic
205C2×Q8—28no—
206C2×Q8—24no—
207C2×Q8—28no—
208C2×Q8—28no—
209C2×Q8—432no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8C2×M1618yes—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—28no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4#13114yes—
252C22×C4#13714yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—44no—
259C2×C8—44no—
260C2×C8—44no—
261C2×C8—44no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—28no—
265C4⋊C4—28no—
266C4⋊C4—28no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—48no—
273C4⋊C4—44no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42—41no—
297C42—41no—
298C42—41no—
299C42—48no—
300C42—48no—
301C42C2×M1614yes—
302C42#13714yes—
303C42H(5,𝔽2)14yes—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23#40812yes—
309C23#41912yes—
310C23#42112yes—
311C23C22×C22⋊C411yescharacteristic
312Q8—216no—
313Q8—416no—
314Q8—48no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—28no—
334C2×C4—28no—
335C2×C4—21no—
336C2×C4—28no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—416no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—88no—
358C2×C4#40712yes—
359C2×C4#40712yes—
360C2×C4#40812yes—
361C2×C4#41912yes—
362C2×C4#41912yes—
363C2×C4#41912yes—
364C2×C4#42112yes—
365C2×C4#51812yes—
366C2×C4#53512yes—
367C2×C4#53512yes—
368C2×C4C22×M1612yes—
369C2×C4C2×#13112yes—
370C2×C4C2×#13712yes—
371C2×C4C2×#13712yes—
372C8—84no—
373C8—84no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22#149714yes—
381C22#230911yescharacteristic
382C22C2×#40711yescharacteristic
383C22C2×#40811yescharacteristic
384C22C2×#41911yescharacteristic
385C22C2×#41911yescharacteristic
386C22C2×#42111yescharacteristic
387C22C2×#42111yescharacteristic
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—48no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—84no—
402C4#297414yes—
403C4#321814yes—
404C2—24no—
405C2#1744911yescharacteristic
406C2#1744911yescharacteristic
407C2C2×#149711yescharacteristic
408C2#1747911yescharacteristic
409C2#2178811yescharacteristic
410C2#2181111yescharacteristic
411C2#2181611yescharacteristic
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 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9677C2semidirect
5. #9641C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
#207782
#207772
#96772
#96412
#83112