Finite Group Databasebeta

← #276975

Subgroups — #276975

Order 512.

645 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
1#33198C211yesmaximal, characteristic
2#28717C211yesmaximal, characteristic
3Q8×C4⋊C8C211yesmaximal, characteristic
4C4×#475C211yesmaximal, characteristic
5C2×#1684C212yesmaximal
6#10601C212yesmaximal
7#10541C212yesmaximal
8#9691C211yesmaximal, characteristic
9#9601C211yesmaximal, characteristic
10#9216C212yesmaximal
11#8586C211yesmaximal, characteristic
12C2×#487C2212yes—
13C2×#477C2212yes—
14C2×#475C2211yescharacteristic
15C2×C8×Q8C2212yes—
16#2868C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C42×Q8C2211yescharacteristic
19#2072—22no—
20#1990C2212yes—
21#1988C2211yescharacteristic
22#1987C2212yes—
23#1948C2211yescharacteristic
24#1947—21no—
25#1947—21no—
26#1939—22no—
27#1939C2212yes—
28#1919—21no—
29#1865—21no—
30#1842—21no—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1803—21no—
34#1803—21no—
35#1803—21no—
36#1803—21no—
37#1803C2211yescharacteristic
38C4×#106—21no—
39C4×C4⋊C8—21no—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#103—21no—
43#1684—28no—
44C2×#335C2212yes—
45C2×#333C2212yes—
46C2×#333C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1277C2212yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2312yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4×Q8—21no—
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58#487—24no—
59#487—42no—
60#477—24no—
61#477—24no—
62#475—22no—
63#475D414yes—
64C8×Q8—24no—
65C8×Q8—42no—
66C2×#107—22no—
67C2×#106—21no—
68C2×#106—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8D411yescharacteristic
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×#103—21no—
82C2×#103—22no—
83C2×#103—21no—
84C2×#103—22no—
85C2×#103—22no—
86C2×C4×C8—22no—
87C2×C4×C8—21no—
88C2×C4×C8C2312yes—
89#391—21no—
90#389—21no—
91#389—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384C2311yescharacteristic
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4D411yescharacteristic
103C4×C4⋊C4D411yescharacteristic
104C4×C4⋊C4C2311yescharacteristic
105C43—21no—
106C43C2311yescharacteristic
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—21no—
112#340—21no—
113#340—21no—
114#340—21no—
115#336—22no—
116#336—21no—
117#336D412yes—
118#335—24no—
119#333—24no—
120#333—42no—
121#332—24no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8C2×D411yescharacteristic
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8C2×D412yes—
130C22×C8C2×D412yes—
131C4⋊Q8—22no—
132C4⋊Q8—21no—
133C4⋊Q8—22no—
134C4⋊Q8—21no—
135C4⋊Q8—22no—
136C4⋊Q8—21no—
137C4⋊Q8—22no—
138C4⋊Q8—82no—
139C4⋊Q8—82no—
140C4⋊Q8C2×D412yes—
141C4×Q8—22no—
142C4×Q8—21no—
143C4×Q8—21no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—42no—
147C4×Q8—42no—
148C4×Q8Q1614yes—
149C4×Q8C2×D412yes—
150C4×Q8#4112yes—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—41no—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42C2411yescharacteristic
183#107—44no—
184#106—24no—
185#106—42no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—21no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—44no—
196C4⋊C8—42no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C4⋊C8—81no—
200C4⋊C8—82no—
201C4⋊C8C2×D412yes—
202#103—24no—
203#103—44no—
204#103—44no—
205#103—44no—
206#103—42no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—44no—
210C4×C8—42no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—41no—
215#95—41no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—21no—
219C2×Q8—22no—
220C2×Q8—41no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8#12112yes—
230C2×Q8C2×Q1612yes—
231C2×Q8#13712yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C2×#4111yescharacteristic
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—42no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—81no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4#12112yes—
291C4⋊C4C2×Q1612yes—
292C42—22no—
293C42—22no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—24no—
299C42—22no—
300C42—41no—
301C42—42no—
302C42—41no—
303C42—41no—
304C42—42no—
305C42—81no—
306C42—82no—
307C42C2×Q1612yes—
308C42#13512yes—
309C42#13512yes—
310C42#13712yes—
311C42C22×D411yescharacteristic
312C42C2×#4111yescharacteristic
313C42H(5,𝔽2)11yescharacteristic
314C422−1+411yescharacteristic
315C23C2×#12111yescharacteristic
316Q8—22no—
317Q8—22no—
318Q8—22no—
319Q8—42no—
320Q8—42no—
321Q8—81no—
322Q8—81no—
323Q8—162no—
324Q8—162no—
325Q8#46214yes—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4#46412yes—
357C2×C4#46412yes—
358C2×C4#46712yes—
359C2×C4#46712yes—
360C2×C4#47112yes—
361C2×C4C2×#12111yescharacteristic
362C2×C4#53711yescharacteristic
363C2×C4D4211yescharacteristic
364C2×C4#54711yescharacteristic
365C2×C4#54811yescharacteristic
366C2×C4#55311yescharacteristic
367C2×C4C22×Q1611yescharacteristic
368C2×C4C2×#13511yescharacteristic
369C2×C4C2×#13511yescharacteristic
370C2×C4C2×#13711yescharacteristic
371C2×C4C2×#13711yescharacteristic
372C2×C4#57511yescharacteristic
373C2×C4#57711yescharacteristic
374C2×C4#57811yescharacteristic
375C8—42no—
376C8—42no—
377C8—42no—
378C8—81no—
379C8—82no—
380C8—82no—
381C22#241611yescharacteristic
382C22C2×#46411yescharacteristic
383C22#304611yescharacteristic
384C22C2×#46711yescharacteristic
385C22#305611yescharacteristic
386C22C2×#47111yescharacteristic
387C22#306811yescharacteristic
388C4—24no—
389C4—21no—
390C4—21no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—42no—
395C4—42no—
396C4—81no—
397C4—82no—
398C4—162no—
399C4—161no—
400C4#168414yescomplemented
401C4C2×#46211yescharacteristic
402C4#303511yescharacteristic
403C4#324811yescharacteristic
404C4#324911yescharacteristic
405C4#327611yescharacteristic
406C4#328411yescharacteristic
407C4#336411yescharacteristic
408C4#336911yescharacteristic
409C2C2×#168411yescharacteristic
410C2#1917611yescharacteristic
411C2#1941711yescharacteristic
412C2#1950011yescharacteristic
413C2#2472911yescharacteristic
414C2#2475211yescharacteristic
415C2#2477311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
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
400. C4#1684semidirect

Maximal subgroups

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

subgroupindex
#331982
#287172
Q8×C4⋊C82
C4×#4752
C2×#16842
#106012
#105412
#96912
#96012
#92162
#85862