Finite Group Databasebeta

← #302380

Subgroups — #302380

Order 512.

946 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic
2#33312C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic
4#29208C211yesmaximal, characteristic
5#10573C214yesmaximal
6#9896C211yesmaximal, characteristic
7#9870C211yesmaximal, characteristic
8#8884C214yesmaximal
9C4×Q8⋊C8C211yesmaximal, characteristic
10C2×Q82—21no—
11C2×#500—22no—
12C2×#477—21no—
13#2868—21no—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2567—22no—
17#2567—21no—
18#2551—22no—
19#2551—21no—
20#2551—22no—
21#2551C2212yes—
22Q8×C4⋊C4—21no—
23Q8×C4⋊C4C2212yes—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27#2057—22no—
28#1990—21no—
29#1987—21no—
30#1947—21no—
31#1947C2214yes—
32#1947C2212yes—
33#1845—21no—
34#1845—22no—
35#1845C2214yes—
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#500—48no—
66#477—44no—
67C2×#107—22no—
68C2×#107—28no—
69C2×#107—24no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×#103—22no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8C2312yes—
79#395—24no—
80#395—24no—
81#395—24no—
82#391—22no—
83#391—21no—
84#389—22no—
85#389—24no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—24no—
90#389—21no—
91#389D412yes—
92#384—22no—
93#384—22no—
94#384—22no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384D411yescharacteristic
101#384D412yes—
102#384C2312yes—
103#384C2312yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4C2312yes—
113C4×C4⋊C4C2311yescharacteristic
114C43—21no—
115C43C2311yescharacteristic
116#340—24no—
117#340—22no—
118#340—22no—
119#336—21no—
120Q8⋊C8Q818yescomplemented
121C22×Q8—24no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—41no—
125C22×Q8C2×D411yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—48no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—416no—
134C4⋊Q8—44no—
135C4⋊Q8—44no—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—82no—
141C4⋊Q8—82no—
142C4⋊Q8—84no—
143C4⋊Q8—84no—
144C4⋊Q8Q1614yes—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—48no—
149C4×Q8—44no—
150C4×Q8—44no—
151C4×Q8—42no—
152C4×Q8—42no—
153C4×Q8D814yes—
154C4×Q8Q1614yes—
155C4×Q8C2×Q814yes—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D414yes—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—24no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2411yescharacteristic
199#107—416no—
200#107—44no—
201#107—84no—
202C4⋊C8—22no—
203C4⋊C8—24no—
204C4⋊C8—42no—
205C4⋊C8—42no—
206C4⋊C8C2×Q814yes—
207#103—44no—
208C4×C8—24no—
209C4×C8—24no—
210C4×C8—22no—
211C4×C8—42no—
212C4×C8C2×Q814yes—
213#95—24no—
214#95—24no—
215#95—21no—
216#95—21no—
217#95—22no—
218#95—21no—
219#95—22no—
220C2×Q8—21no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—24no—
224C2×Q8—48no—
225C2×Q8—42no—
226C2×Q8—44no—
227C2×Q8—42no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—84no—
233C2×Q8—84no—
234C2×Q8—82no—
235C2×Q8C4⋊Q812yes—
236C2×Q8C2×D812yes—
237C2×Q8C2×Q1612yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4#12011yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C4⋊C4—21no—
263C4⋊C4—24no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—44no—
272C4⋊C4—48no—
273C4⋊C4—48no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—84no—
289C4⋊C4—84no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4C4⋊Q812yes—
293C4⋊C4C2×D812yes—
294C4⋊C4C2×Q1612yes—
295C4⋊C4C2×Q1614yes—
296C42—24no—
297C42—22no—
298C42—24no—
299C42—24no—
300C42—22no—
301C42—22no—
302C42—48no—
303C42—42no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42C2×D812yes—
309C42C2×Q1612yes—
310C42C2×Q1612yes—
311C42AGL(1,ℤ/8)12yes—
312C42#13712yes—
313C42#13712yes—
314C42C22×Q812yes—
315C422−1+412yes—
316C23C2×#12011yescharacteristic
317Q8—22no—
318Q8—22no—
319Q8—22no—
320Q8—48no—
321Q8—164no—
322Q8—164no—
323Q8—168no—
324Q8#50014yescomplemented
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4#44712yes—
356C2×C4#44712yes—
357C2×C4#45112yes—
358C2×C4#45112yes—
359C2×C4#45112yes—
360C2×C4#45112yes—
361C2×C4#45312yes—
362C2×C4#45612yes—
363C2×C4C2×C4⋊Q811yescharacteristic
364C2×C4#53611yescharacteristic
365C2×C4D4×Q812yes—
366C2×C4D4×Q812yes—
367C2×C4C22×D811yescharacteristic
368C2×C4C22×Q1611yescharacteristic
369C2×C4C22×Q1611yescharacteristic
370C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
371C2×C4C2×#13711yescharacteristic
372C2×C4C2×#13711yescharacteristic
373C2×C4#57612yes—
374C8—28no—
375C8—44no—
376C8—82no—
377C22#243411yescharacteristic
378C22C2×#44711yescharacteristic
379C22C2×#45111yescharacteristic
380C22C2×#45111yescharacteristic
381C22#300811yescharacteristic
382C22C2×#45311yescharacteristic
383C22C2×#45611yescharacteristic
384C4—22no—
385C4—24no—
386C4—22no—
387C4—22no—
388C4—44no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—82no—
393C4—84no—
394C4—162no—
395C4#163314yes—
396C4#164914yes—
397C4C2×#50011yescharacteristic
398C4#316511yescharacteristic
399C4#322611yescharacteristic
400C4#323011yescharacteristic
401C4Q8×Q1612yes—
402C4#338512yes—
403C2C2×#164911yescharacteristic
404C2C2×#163311yescharacteristic
405C2#1862011yescharacteristic
406C2#1865311yescharacteristic
407C2#2581311yescharacteristic
408C2#2583011yescharacteristic
409C2#2584111yescharacteristic
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
120. Q8⋊C8Q8semidirect
324. Q8#500semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#333122
#331982
#292082
#105732
#98962
#98702
#88842
C4×Q8⋊C82