Finite Group Databasebeta

← #239377

Subgroups — #239377

Order 512.

1127 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
1C2×#3189C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic
3#21759C212yesmaximal
4C2×#1996C212yesmaximal
5#10566C218yesmaximal, complemented
6C2×#1289C211yesmaximal, characteristic
7C22×C22⋊Q8C2212yes—
8#3189—22no—
9#3189—24no—
10#3189C2214yes—
11C2×#484—22no—
12C2×#467—22no—
13C2×#451—22no—
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18#1996—24no—
19#1996C2218yes—
20#1892C2218yes—
21#1872—24no—
22#1850—22no—
23#1850C2214yes—
24#1824—22no—
25#1797—22no—
26#1787—22no—
27C2×#342—22no—
28C2×#342—22no—
29#1289C2214yes—
30C23×Q8D412yes—
31C24×C4C2311yescharacteristic
32C22×Q16—22no—
33C23×C8D412yes—
34C23×C8C2312yes—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8D414yes—
46C2×C22⋊Q8C2314yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D412yes—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4D411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42—21no—
55#484—28no—
56#484—28no—
57#467—28no—
58#467—28no—
59#451—28no—
60#451—28no—
61C2×#107—22no—
62C2×C4⋊C8—24no—
63C2×#103—24no—
64C2×#103—28no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8D412yes—
72C2×C22⋊C8C2312yes—
73#380—22no—
74#380—22no—
75#380—22no—
76#380C2314yes—
77C4×C22⋊C4—21no—
78C4×C22⋊C4—21no—
79C4×C22⋊C4D412yes—
80C4×C22⋊C4C2312yes—
81C2×#95—21no—
82C2×#95—22no—
83#342—28no—
84#342—28no—
85#340—28no—
86#336—24no—
87C25C2×D411yescharacteristic
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—22no—
93C22×Q8—84no—
94C22×Q8C2×D414yes—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—42no—
100C23×C4—82no—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C23×C4#4111yescharacteristic
108C23×C4#4112yes—
109C23×C4#4112yes—
110C23×C4C2411yescharacteristic
111C2×Q16—28no—
112C2×Q16—84no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—24no—
117C22×C8—82no—
118C22×C8—82no—
119C22×C8C2×D414yes—
120C22⋊Q8—24no—
121C22⋊Q8—24no—
122C22⋊Q8—24no—
123C22⋊Q8—24no—
124C22⋊Q8—24no—
125C22⋊Q8—24no—
126C22⋊Q8—24no—
127C22⋊Q8—24no—
128C22⋊Q8—42no—
129C22⋊Q8—42no—
130C22⋊Q8—42no—
131C22⋊Q8—42no—
132C22⋊Q8—42no—
133C22⋊Q8—42no—
134C22⋊Q8—42no—
135C22⋊Q8—42no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—84no—
150C2×C4⋊C4C2×D414yes—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4Q1614yescomplemented
166C2×C22⋊C4C2×D412yes—
167C2×C22⋊C4C2×D412yes—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—82no—
171#107—28no—
172C4⋊C8—84no—
173#103—216no—
174#103—88no—
175C22⋊C8—24no—
176C22⋊C8—24no—
177C22⋊C8—42no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180#95—22no—
181#95—22no—
182#95—24no—
183#95—42no—
184C24—81no—
185C24—81no—
186C24#12112yes—
187C24#12112yes—
188C24#12111yescharacteristic
189C24#12111yescharacteristic
190C24C2×Q1612yes—
191C24#13512yes—
192C24#13714yes—
193C24C22×D411yescharacteristic
194C2×Q8—24no—
195C2×Q8—24no—
196C2×Q8—48no—
197C2×Q8—48no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—84no—
201C2×Q8—84no—
202C2×Q8—84no—
203C2×Q8—84no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—82no—
223C22×C4—81no—
224C22×C4—82no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4—84no—
232C22×C4—84no—
233C22×C4—84no—
234C22×C4#12012yes—
235C22×C4#12011yescharacteristic
236C22×C4#12011yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12112yes—
239C22×C4#12111yescharacteristic
240C22×C4#12112yes—
241C22×C4#12312yes—
242C22×C4#12312yes—
243C22×C4#12312yes—
244C22×C4#12312yes—
245C22×C4#12411yescharacteristic
246C22×C4#12412yes—
247C22×C4#12411yescharacteristic
248C22×C4C22×D412yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C2×#4111yescharacteristic
251C22×C4C2×#4112yes—
252Q16—88no—
253C2×C8—24no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—48no—
257C2×C8—44no—
258C2×C8—42no—
259C2×C8—82no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—48no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—84no—
270C22⋊C4—22no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—42no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—84no—
281C22⋊C4C2×Q1614yescomplemented
282C42—44no—
283C42—82no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—42no—
291C23—42no—
292C23—41no—
293C23—41no—
294C23—82no—
295C23—82no—
296C23—84no—
297C23#39411yescharacteristic
298C23#39412yes—
299C23#39411yescharacteristic
300C23#46214yes—
301C23#46214yes—
302C23#46414yes—
303C23#46414yes—
304C23#46712yes—
305C23#46712yes—
306C23#47012yes—
307C23#47012yes—
308C23C2×#12112yes—
309C23C2×#12111yescharacteristic
310C23C22×Q1611yescharacteristic
311C23C2×#13511yescharacteristic
312C23C2×#13712yes—
313Q8—48no—
314Q8—48no—
315Q8—84no—
316Q8—84no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—168no—
355C2×C4#39411yescharacteristic
356C2×C4#39412yes—
357C2×C4#39411yescharacteristic
358C2×C4C2×#12011yescharacteristic
359C2×C4C2×#12312yes—
360C2×C4C2×#12411yescharacteristic
361C8—44no—
362C8—82no—
363C22—21no—
364C22—22no—
365C22—21no—
366C22—22no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—21no—
371C22—22no—
372C22—21no—
373C22—42no—
374C22—42no—
375C22—44no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—44no—
380C22—44no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22#205612yes—
385C22#205614yes—
386C22#205612yes—
387C22C2×#39411yescharacteristic
388C22C2×#46212yes—
389C22C2×#46412yes—
390C22C2×#46711yescharacteristic
391C22C2×#47011yescharacteristic
392C4—22no—
393C4—21no—
394C4—21no—
395C4—22no—
396C4—21no—
397C4—44no—
398C4—82no—
399C4—84no—
400C4—84no—
401C4—164no—
402C4#302612yes—
403C2—21no—
404C2—21no—
405C2—22no—
406C2—22no—
407C2—24no—
408C2—22no—
409C2—22no—
410C2#2351511yescharacteristic
411C2#2353511yescharacteristic
412C2C2×#205611yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10566C2semidirect
165. C2×C22⋊C4Q16semidirect
281. C22⋊C4C2×Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
#219302
#217592
C2×#19962
#105662
C2×#12892