Finite Group Databasebeta

← #330181

Subgroups — #330181

Order 512.

604 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33242C211yesmaximal, characteristic
2#33202C212yesmaximal
3#21339C211yesmaximal, characteristic
4#10543C211yesmaximal, characteristic
5#10541C212yesmaximal
6#10538C211yesmaximal, characteristic
7#10279C211yesmaximal, characteristic
8#10277C212yesmaximal
9#10276C211yesmaximal, characteristic
10#9697C212yesmaximal
11#8784C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#494C2211yescharacteristic
14C2×#487C2212yes—
15C2×#477C2212yes—
16C2×#475C2212yes—
17#2862C2212yes—
18C42×Q8C2211yescharacteristic
19#1988C2212yes—
20#1987C2212yes—
21#1984C2211yescharacteristic
22#1984C2211yescharacteristic
23#1969—21no—
24#1969—21no—
25#1969—21no—
26#1969—21no—
27#1949—22no—
28#1949—22no—
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1916—21no—
32#1916—21no—
33#1842C2212yes—
34#1805—22no—
35#1805—22no—
36C4×C4⋊C8—21no—
37C4×C4⋊C8C2211yescharacteristic
38#1744—21no—
39C42×C8C2211yescharacteristic
40C2×#333C2212yes—
41C2×#333C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44#1277C2211yescharacteristic
45#1277C2212yes—
46#1277C2211yescharacteristic
47C22×Q16—21no—
48C22×Q16—21no—
49C2×C4⋊Q8C2312yes—
50C2×C4⋊Q8C2312yes—
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8C2311yescharacteristic
54#500—42no—
55#494—42no—
56#487—42no—
57#487—42no—
58#477—42no—
59#477—42no—
60#475—42no—
61#475—42no—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107—22no—
65C2×#106—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—22no—
78C2×C4×C8—21no—
79C2×C4×C8D411yescharacteristic
80C2×C4×C8D411yescharacteristic
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#391—22no—
86#391—22no—
87#391—22no—
88#391—22no—
89#382—22no—
90#382—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4D411yescharacteristic
95C4×C4⋊C4D411yescharacteristic
96C4×C4⋊C4C2311yescharacteristic
97C43—21no—
98C43C2311yescharacteristic
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#336—21no—
104#336—21no—
105#336—21no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336—21no—
110#333—42no—
111#333—42no—
112#332—42no—
113#332—42no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8C2×D411yescharacteristic
117C2×Q16—42no—
118C2×Q16—42no—
119C2×Q16—42no—
120C2×Q16—42no—
121C2×Q16—42no—
122C2×Q16—42no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4×Q8—22no—
140C4×Q8—24no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8Q1614yescomplemented
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42C2411yescharacteristic
167#107—44no—
168#107—42no—
169#107—42no—
170#106—44no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—42no—
178C4⋊C8—42no—
179C4⋊C8—44no—
180C4⋊C8—44no—
181#103—44no—
182#103—44no—
183#103—44no—
184#103—44no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—41no—
190C4×C8—41no—
191C4×C8—42no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—42no—
196#95—42no—
197#95—42no—
198#95—42no—
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—82no—
211C2×Q8—82no—
212C2×Q8C2×Q1612yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4C22×D411yescharacteristic
225C22×C4C2×#4111yescharacteristic
226Q16—84no—
227Q16—84no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8#13512yes—
251C2×C8#13712yes—
252C4⋊C4—21no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4C2×Q1612yes—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—24no—
272C42—41no—
273C42—41no—
274C42—42no—
275C42—42no—
276C42—41no—
277C42—41no—
278C42—41no—
279C42—41no—
280C42—42no—
281C42C2×Q1612yes—
282C42#13512yes—
283C42#13714yes—
284C42H(5,𝔽2)12yes—
285C42H(5,𝔽2)12yes—
286C23C2×#12111yescharacteristic
287Q8—24no—
288Q8—42no—
289Q8—82no—
290Q8—82no—
291Q8—82no—
292Q8—82no—
293Q8—162no—
294Q8—162no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4#46712yes—
324C2×C4#46712yes—
325C2×C4#47012yes—
326C2×C4#47012yes—
327C2×C4#53512yes—
328C2×C4#53511yescharacteristic
329C2×C4#53511yescharacteristic
330C2×C4#54012yes—
331C2×C4C22×Q1611yescharacteristic
332C2×C4C2×#13511yescharacteristic
333C2×C4C2×#13511yescharacteristic
334C2×C4C2×#13712yes—
335C2×C4C2×#13711yescharacteristic
336C2×C4#57511yescharacteristic
337C2×C4#57511yescharacteristic
338C8—22no—
339C8—22no—
340C8—41no—
341C8—41no—
342C8—82no—
343C8—82no—
344C8—81no—
345C8—81no—
346C22#261911yescharacteristic
347C22#303511yescharacteristic
348C22#303711yescharacteristic
349C22C2×#46711yescharacteristic
350C22C2×#47011yescharacteristic
351C22#306711yescharacteristic
352C22#306711yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—22no—
357C4—22no—
358C4—42no—
359C4—42no—
360C4—81no—
361C4—81no—
362C4—162no—
363C4—162no—
364C4#320912yes—
365C4#321011yescharacteristic
366C4#321012yes—
367C4#321111yescharacteristic
368C4#321511yescharacteristic
369C4#321911yescharacteristic
370C2#1912611yescharacteristic
371C2#1915811yescharacteristic
372C2#1926211yescharacteristic
373C2#1928711yescharacteristic
374C2#2980111yescharacteristic
375C2#2980811yescharacteristic
376C2#2982111yescharacteristic
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
143. C4×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#332422
#332022
#213392
#105432
#105412
#105382
#102792
#102772
#102762
#96972
#87842