Finite Group Databasebeta

← #302376

Subgroups — #302376

Order 512.

956 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic
2#33310C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#28687C211yesmaximal, characteristic
5#10574C214yesmaximal
6#9896C211yesmaximal, characteristic
7#9870C211yesmaximal, characteristic
8#8888C214yesmaximal
9C4×Q8⋊C8C211yesmaximal, characteristic
10C2×Q82—21no—
11C2×#498—21no—
12C2×#498—22no—
13C2×#475—21no—
14#2868—21no—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17#2567—22no—
18#2567—21no—
19#2551—22no—
20#2551—21no—
21#2551—22no—
22#2551C2212yes—
23Q8×C4⋊C4—21no—
24Q8×C4⋊C4C2212yes—
25C4×C4⋊Q8—21no—
26C4×C4⋊Q8C2211yescharacteristic
27C42×Q8C2211yescharacteristic
28#2084—22no—
29#1990—21no—
30#1988—21no—
31#1948—21no—
32#1948C2214yes—
33#1948C2212yes—
34#1844—22no—
35#1844C2214yes—
36#1842C2211yescharacteristic
37#1803—21no—
38#1803C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×Q8⋊C8C2212yes—
42#1296C2214yes—
43#1279C2212yes—
44#1279C2214yes—
45Q82—44no—
46Q82—44no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—24no—
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⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8D411yescharacteristic
58C2×C4⋊Q8D412yes—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—21no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65#498—48no—
66#498—44no—
67#475—44no—
68C2×#106—22no—
69C2×#106—24no—
70C2×#106—28no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×#103—22no—
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#391—21no—
85#389—22no—
86#389—24no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—24no—
91#389—21no—
92#389D412yes—
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#384D411yescharacteristic
102#384D412yes—
103#384C2312yes—
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⋊C4D411yescharacteristic
112C4×C4⋊C4D411yescharacteristic
113C4×C4⋊C4C2312yes—
114C4×C4⋊C4C2311yescharacteristic
115C43—21no—
116C43C2311yescharacteristic
117#340—22no—
118#340—22no—
119#340—24no—
120#336—21no—
121Q8⋊C8Q818yescomplemented
122C22×Q8—24no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—41no—
126C22×Q8C2×D411yescharacteristic
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—48no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—416no—
135C4⋊Q8—44no—
136C4⋊Q8—44no—
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—42no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8—82no—
144C4⋊Q8—84no—
145C4⋊Q8—84no—
146C4⋊Q8SD1614yes—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—48no—
151C4×Q8—44no—
152C4×Q8—44no—
153C4×Q8—42no—
154C4×Q8—42no—
155C4×Q8SD1614yes—
156C4×Q8SD1614yes—
157C4×Q8C2×Q814yes—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D414yes—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—24no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2411yescharacteristic
201#106—416no—
202#106—44no—
203#106—48no—
204C4⋊C8—22no—
205C4⋊C8—24no—
206C4⋊C8—42no—
207C4⋊C8—42no—
208C4⋊C8C2×Q814yes—
209#103—44no—
210C4×C8—24no—
211C4×C8—24no—
212C4×C8—22no—
213C4×C8—42no—
214C4×C8C2×Q814yes—
215#95—24no—
216#95—24no—
217#95—21no—
218#95—21no—
219#95—22no—
220#95—21no—
221#95—22no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—24no—
226C2×Q8—48no—
227C2×Q8—42no—
228C2×Q8—44no—
229C2×Q8—42no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—84no—
233C2×Q8—84no—
234C2×Q8—82no—
235C2×Q8—82no—
236C2×Q8—82no—
237C2×Q8C4⋊Q812yes—
238C2×Q8C2×SD1612yes—
239C2×Q8C2×SD1612yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C4⋊C4—21no—
265C4⋊C4—24no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4—48no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4C4⋊Q812yes—
296C4⋊C4C2×SD1612yes—
297C4⋊C4C2×SD1614yes—
298C4⋊C4C2×SD1612yes—
299C42—24no—
300C42—22no—
301C42—24no—
302C42—24no—
303C42—22no—
304C42—22no—
305C42—48no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42C2×SD1612yes—
312C42C2×SD1612yes—
313C42C2×SD1612yes—
314C42AGL(1,ℤ/8)12yes—
315C42#13712yes—
316C42#13712yes—
317C42C22×Q812yes—
318C422−1+412yes—
319C23C2×#12011yescharacteristic
320Q8—22no—
321Q8—22no—
322Q8—22no—
323Q8—48no—
324Q8—164no—
325Q8—168no—
326Q8—164no—
327Q8#49814yescomplemented
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4#44812yes—
359C2×C4#44812yes—
360C2×C4#44812yes—
361C2×C4#44812yes—
362C2×C4#45012yes—
363C2×C4#45012yes—
364C2×C4#45312yes—
365C2×C4#45612yes—
366C2×C4C2×C4⋊Q811yescharacteristic
367C2×C4#53611yescharacteristic
368C2×C4D4×Q812yes—
369C2×C4D4×Q812yes—
370C2×C4C22×SD1611yescharacteristic
371C2×C4C22×SD1611yescharacteristic
372C2×C4C22×SD1611yescharacteristic
373C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
374C2×C4C2×#13711yescharacteristic
375C2×C4C2×#13711yescharacteristic
376C2×C4#57712yes—
377C8—28no—
378C8—44no—
379C8—82no—
380C22#243411yescharacteristic
381C22C2×#45011yescharacteristic
382C22C2×#44811yescharacteristic
383C22C2×#44811yescharacteristic
384C22#300911yescharacteristic
385C22C2×#45311yescharacteristic
386C22C2×#45611yescharacteristic
387C4—22no—
388C4—24no—
389C4—22no—
390C4—22no—
391C4—44no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—82no—
396C4—84no—
397C4—162no—
398C4#162314yes—
399C4#165114yes—
400C4C2×#49811yescharacteristic
401C4#316511yescharacteristic
402C4#322711yescharacteristic
403C4#322911yescharacteristic
404C4Q8×SD1612yes—
405C4#338712yes—
406C2C2×#165111yescharacteristic
407C2C2×#162311yescharacteristic
408C2#1862111yescharacteristic
409C2#1865211yescharacteristic
410C2#2581111yescharacteristic
411C2#2583011yescharacteristic
412C2#2583711yescharacteristic
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
121. Q8⋊C8Q8semidirect
327. Q8#498semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#333102
#331962
#286872
#105742
#98962
#98702
#88882
C4×Q8⋊C82