Finite Group Databasebeta

← #303426

Subgroups — #303426

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic
2#33225C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#28717C211yesmaximal, characteristic
5#10415C214yesmaximal
6#9691C211yesmaximal, characteristic
7#9637C211yesmaximal, characteristic
8#8451C214yesmaximal
9C4×Q8⋊C8C211yesmaximal, characteristic
10C2×Q82C2212yes—
11C2×#487—21no—
12C2×#487—22no—
13C2×#475—21no—
14#2868—21no—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17#2567—22no—
18#2567C2212yes—
19#2551—21no—
20#2551—21no—
21#2551—22no—
22#2551—22no—
23Q8×C4⋊C4—21no—
24Q8×C4⋊C4—21no—
25C4×C4⋊Q8—21no—
26C4×C4⋊Q8C2211yescharacteristic
27C42×Q8C2211yescharacteristic
28#2072—22no—
29#1990—21no—
30#1988—21no—
31#1956—22no—
32#1956C2214yes—
33#1948—21no—
34#1948C2212yes—
35#1842C2211yescharacteristic
36#1803—21no—
37#1803C2212yes—
38#1803C2214yes—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×#333C2214yes—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2214yes—
44#1279C2212yes—
45Q82—28no—
46Q82—44no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8D411yescharacteristic
56C2×C4⋊Q8D412yes—
57C2×C4⋊Q8D412yes—
58C2×C4⋊Q8C2312yes—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8C2312yes—
64C2×C4×Q8C2311yescharacteristic
65#487—48no—
66#487—44no—
67#475—44no—
68C2×#106—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×#103—22no—
74C2×#103—24no—
75C2×#103—28no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8C2312yes—
80#395—24no—
81#395—24no—
82#395—24no—
83#391—22no—
84#391D412yes—
85#389—22no—
86#389—22no—
87#389—24no—
88#389—24no—
89#389—22no—
90#389—22no—
91#389—21no—
92#389—21no—
93#384—22no—
94#384—22no—
95#384—22no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384D411yescharacteristic
104#384C2312yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D411yescharacteristic
113C4×C4⋊C4D411yescharacteristic
114C4×C4⋊C4C2311yescharacteristic
115C43—21no—
116C43C2311yescharacteristic
117#340—22no—
118#340—24no—
119#340—22no—
120#336—21no—
121#333—44no—
122Q8⋊C8—24no—
123Q8⋊C8—44no—
124C22×Q8—24no—
125C22×Q8—41no—
126C22×Q8C2×D412yes—
127C22×Q8C2×D411yescharacteristic
128C22×Q8C2×D414yes—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—24no—
134C4⋊Q8—24no—
135C4⋊Q8—48no—
136C4⋊Q8—44no—
137C4⋊Q8—44no—
138C4⋊Q8—44no—
139C4⋊Q8—416no—
140C4⋊Q8—44no—
141C4⋊Q8—44no—
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—84no—
147C4⋊Q8—84no—
148C4⋊Q8SD1614yes—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—48no—
155C4×Q8—44no—
156C4×Q8—44no—
157C4×Q8SD1614yes—
158C4×Q8Q1614yes—
159C4×Q8#4114yes—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—24no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2411yescharacteristic
203#106—44no—
204C4⋊C8—22no—
205C4⋊C8—24no—
206C4⋊C8—22no—
207C4⋊C8—42no—
208C4⋊C8—42no—
209#103—416no—
210#103—44no—
211#103—48no—
212C4×C8—24no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—42no—
216C4×C8—42no—
217#95—24no—
218#95—24no—
219#95—21no—
220#95—21no—
221#95—22no—
222#95—22no—
223#95—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—22no—
228C2×Q8—24no—
229C2×Q8—28no—
230C2×Q8—24no—
231C2×Q8—48no—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—84no—
235C2×Q8—84no—
236C2×Q8—82no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8#12412yes—
240C2×Q8C2×SD1614yes—
241C2×Q8C2×SD1612yes—
242C2×Q8C2×Q1612yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C2×C8—24no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C4⋊C4—21no—
268C4⋊C4—24no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—48no—
277C4⋊C4—48no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4#12412yes—
299C4⋊C4C2×SD1612yes—
300C4⋊C4C2×Q1612yes—
301C42—24no—
302C42—22no—
303C42—24no—
304C42—24no—
305C42—22no—
306C42—22no—
307C42—48no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42C2×SD1612yes—
314C42C2×SD1612yes—
315C42C2×Q1612yes—
316C42#13712yes—
317C42#13712yes—
318C42#13712yes—
319C42C2×#4112yes—
320C42H(5,𝔽2)12yes—
321C23C2×#12011yescharacteristic
322Q8—22no—
323Q8—22no—
324Q8—22no—
325Q8—28no—
326Q8—44no—
327Q8—88no—
328Q8—168no—
329Q8—164no—
330Q8#48714yescomplemented
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4#44812yes—
362C2×C4#44812yes—
363C2×C4#44812yes—
364C2×C4#44812yes—
365C2×C4#45112yes—
366C2×C4#45112yes—
367C2×C4#45312yes—
368C2×C4#45512yes—
369C2×C4C2×#12411yescharacteristic
370C2×C4#53511yescharacteristic
371C2×C4#54612yes—
372C2×C4#54612yes—
373C2×C4C22×SD1611yescharacteristic
374C2×C4C22×SD1611yescharacteristic
375C2×C4C22×Q1611yescharacteristic
376C2×C4C2×#13711yescharacteristic
377C2×C4C2×#13711yescharacteristic
378C2×C4C2×#13711yescharacteristic
379C2×C4#57812yes—
380C8—44no—
381C8—82no—
382C8—82no—
383C22#243511yescharacteristic
384C22C2×#44811yescharacteristic
385C22C2×#44811yescharacteristic
386C22C2×#45111yescharacteristic
387C22#301111yescharacteristic
388C22C2×#45311yescharacteristic
389C22C2×#45511yescharacteristic
390C4—24no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—42no—
395C4—44no—
396C4—42no—
397C4—42no—
398C4—84no—
399C4—82no—
400C4—162no—
401C4#162314yes—
402C4#164514yes—
403C4C2×#48711yescharacteristic
404C4#313911yescharacteristic
405C4#320611yescharacteristic
406C4#320911yescharacteristic
407C4#329812yes—
408C4#332812yes—
409C2C2×#162311yescharacteristic
410C2C2×#164511yescharacteristic
411C2#1862311yescharacteristic
412C2#1863211yescharacteristic
413C2#2585611yescharacteristic
414C2#2587511yescharacteristic
415C2#2588411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
330. Q8#487semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#332252
#331962
#287172
#104152
#96912
#96372
#84512
C4×Q8⋊C82