Finite Group Databasebeta

← #204262

Subgroups — #204262

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#135C211yesmaximal, characteristic
2#21847C212yesmaximal, complemented
3C22×#334C211yesmaximal, characteristic, complemented
4C2×#1477C212yesmaximal, complemented
5C2×#1450C211yesmaximal, characteristic, complemented
6#9659C212yesmaximal, complemented
7#9658C212yesmaximal, complemented
8#9655C214yesmaximal, complemented
9C22×#135C411yescharacteristic
10C2×C4×#41C2212yes—
11C4×#135—24no—
12C4×#135—24no—
13C2×C4×Q16C411yescharacteristic
14C2×C4×Q16C2211yescharacteristic
15C2×C4×SD16C412yes—
16C2×C4×SD16C2212yes—
17C2×C4×D8C411yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#427C411yescharacteristic
22C22×C4⋊C8C2212yes—
23C2×#416C412yes—
24C22×C4×C8C2211yescharacteristic
25#1984C2214yes—
26#1954C2214yes—
27#1869—24no—
28#1797—24no—
29C2×#334—24no—
30C2×#334C2212yes—
31C2×#334C2212yes—
32#1477—28no—
33C2×Q8⋊C8C2214yes—
34C2×D4⋊C8C2214yes—
35#1450—24no—
36C2×#322C2212yes—
37C2×C82C2212yes—
38C22×#41C2×C412yes—
39C2×#135—24no—
40C2×#135—24no—
41C2×#135C818yescomplemented
42C22×Q16C2×C411yescharacteristic
43C22×SD16C2×C412yes—
44C22×D8C2×C411yescharacteristic
45C23×C8—22no—
46C23×C8C2×C411yescharacteristic
47C23×C8C2×C411yescharacteristic
48C4×#41—24no—
49C4×#41—24no—
50C4×#41—24no—
51C4×#41—24no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2312yes—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4C2×C412yes—
57C2×C4×D4C2312yes—
58C2×#117—22no—
59C2×#117C2×C412yes—
60C22×C42—22no—
61C22×C42C2311yescharacteristic
62C4×Q16—24no—
63C4×Q16—22no—
64C4×Q16—22no—
65C4×SD16—28no—
66C4×SD16—24no—
67C4×SD16—24no—
68C4×D8—24no—
69C4×D8—22no—
70C4×D8—22no—
71#431—22no—
72#431—22no—
73#431—22no—
74#431D414yes—
75#427—24no—
76C2×#107C2×C411yescharacteristic
77C2×#107C2×C411yescharacteristic
78C2×#106C2×C412yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8D412yes—
84C2×C4⋊C8C2314yes—
85#416—28no—
86C2×#103C2×C412yes—
87C2×#103C2×C412yes—
88C2×#102C2×C412yes—
89C2×#102C2×C412yes—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—28no—
92C2×C22⋊C8D412yes—
93C2×C4×C8C2×C411yescharacteristic
94C2×C4×C8C2×C411yescharacteristic
95C2×C4×C8D414yes—
96C2×C4×C8D414yes—
97C2×C4×C8C2312yes—
98C2×C4×C8C2312yes—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#336—28no—
102#334—24no—
103#334—24no—
104Q8⋊C8—28no—
105D4⋊C8—28no—
106#322—24no—
107C82—24no—
108C2×#41—24no—
109C2×#41—24no—
110C2×#41—24no—
111C2×#41—24no—
112C2×#41—24no—
113C2×#41C2×C818yes—
114C22×Q8C22×C412yes—
115C22×D4—22no—
116C22×D4C22×C412yes—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4#4111yescharacteristic
123#135—232no—
124C2×Q16—22no—
125C2×Q16—22no—
126C2×Q16C2×C814yescomplemented
127C2×SD16—24no—
128C2×SD16—24no—
129C2×SD16C2×C818yescomplemented
130C2×D8—22no—
131C2×D8—22no—
132C2×D8C2×C814yescomplemented
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—24no—
137C22×C8—44no—
138C22×C8—84no—
139C22×C8C22⋊C414yes—
140C22×C8C22⋊C414yes—
141C22×C8C2×C814yes—
142C22×C8M1614yes—
143C22×C8C22×C411yescharacteristic
144C22×C8C22×C411yescharacteristic
145C22×C8C22×C411yescharacteristic
146C22×C8C22×C411yescharacteristic
147C22×C8C2×D414yes—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—44no—
161C4×D4—44no—
162C4×D4—44no—
163C4×D4—44no—
164#117—24no—
165#117—24no—
166#117—44no—
167#117—44no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—44no—
179C2×C42—44no—
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D411yescharacteristic
185C2×C42#4112yes—
186C2×C42#4111yescharacteristic
187C2×C42C2411yescharacteristic
188#107—22no—
189#107—22no—
190#106—24no—
191C4⋊C8—24no—
192C4⋊C8—24no—
193C4⋊C8—24no—
194C4⋊C8—24no—
195C4⋊C8—44no—
196#103—24no—
197#103—24no—
198#102—24no—
199#102—24no—
200C22⋊C8—24no—
201C22⋊C8—24no—
202C22⋊C8—88no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8C22⋊C414yes—
207C4×C8C22⋊C414yes—
208C4×C8C2×D414yes—
209C4×C8C2×D412yes—
210C4×C8C2×D412yes—
211C4×C8C2×D412yes—
212C4×C8C2×D412yes—
213C24—22no—
214C24C4×D411yescharacteristic
215#41—216no—
216#41—216no—
217#41—48no—
218#41—48no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8C22×C814yes—
225C2×D4—28no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4C22×C814yes—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—28no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—84no—
248C22×C4C2×C22⋊C412yes—
249C22×C4C4×D411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D412yes—
252C22×C4C4×D411yescharacteristic
253C22×C4#12112yes—
254C22×C4#12112yes—
255C22×C4#12412yes—
256C22×C4#12712yes—
257C22×C4C22×C812yes—
258C22×C4C2×M1612yes—
259C22×C4#13112yes—
260C22×C4#13112yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C2×#4111yescharacteristic
264Q16—28no—
265SD16—216no—
266D8—28no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—28no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—28no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—48no—
278C2×C8—88no—
279C2×C8C22⋊C8116yescomplemented
280C2×C8C2×C22⋊C412yes—
281C2×C8C2×C22⋊C412yes—
282C2×C8C2×C22⋊C412yes—
283C2×C8C2×C22⋊C412yes—
284C2×C8C22×C812yes—
285C2×C8C22×C812yes—
286C2×C8C2×M1612yes—
287C2×C8C2×M1612yes—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—44no—
299C42—42no—
300C42—42no—
301C42—44no—
302C42—44no—
303C42C2×C22⋊C412yes—
304C42C4×D412yes—
305C42#12112yes—
306C42#12112yes—
307C42#12412yes—
308C42#12712yes—
309C42C22×D411yescharacteristic
310C42C22×D411yescharacteristic
311C42C22×D411yescharacteristic
312C42C2×#4111yescharacteristic
313C23—22no—
314C23—22no—
315C23—24no—
316C23—42no—
317C23—44no—
318C23—44no—
319C23—42no—
320C23#39012yes—
321C23C8×D412yes—
322C23#43612yes—
323C23C2×C4×D411yescharacteristic
324Q8—24no—
325Q8—24no—
326Q8—42no—
327Q8—42no—
328D4—24no—
329D4—24no—
330D4—416no—
331D4—42no—
332D4—42no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—416no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—88no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—84no—
352C2×C4#39012yes—
353C2×C4#39012yes—
354C2×C4#39012yes—
355C2×C4C2×C22⋊C814yes—
356C2×C4#40814yes—
357C2×C4C8×D414yes—
358C2×C4C8×D412yes—
359C2×C4#43614yes—
360C2×C4#43612yes—
361C2×C4C22×C22⋊C411yescharacteristic
362C2×C4C2×C4×D411yescharacteristic
363C2×C4C2×#12111yescharacteristic
364C2×C4C2×#12111yescharacteristic
365C2×C4C2×#12411yescharacteristic
366C2×C4C2×#12711yescharacteristic
367C2×C4C23×C811yescharacteristic
368C2×C4C22×M1611yescharacteristic
369C2×C4C2×#13111yescharacteristic
370C2×C4C2×#13111yescharacteristic
371C2×C4#57714yes—
372C8—22no—
373C8—22no—
374C8—44no—
375C8—44no—
376C8—84no—
377C8C2×C22⋊C814yes—
378C8C2×C22⋊C814yes—
379C22—24no—
380C22—24no—
381C22—22no—
382C22—22no—
383C22—44no—
384C22—48no—
385C22—84no—
386C22—82no—
387C22—82no—
388C22#195414yes—
389C22C2×#39011yescharacteristic
390C22C2×C8×D411yescharacteristic
391C22C2×#43611yescharacteristic
392C22#295212yes—
393C22#295712yes—
394C4—22no—
395C4—21no—
396C4—22no—
397C4—21no—
398C4—44no—
399C4—82no—
400C4—82no—
401C4#195414yes—
402C4C22×C22⋊C811yescharacteristic
403C4C2×#40811yescharacteristic
404C4C2×C8×D411yescharacteristic
405C4C2×#43611yescharacteristic
406C4#306012yes—
407C4#315412yes—
408C2—24no—
409C2—84no—
410C2#1785812yes—
411C2#1789912yes—
412C2C2×#195411yescharacteristic
413C2#2122911yescharacteristic
414C2#2124011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#135C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21847C2semidirect
3. C22×#334C2semidirect
4. C2×#1477C2semidirect
5. C2×#1450C2semidirect
6. #9659C2semidirect
7. #9658C2semidirect
8. #9655C2semidirect
41. C2×#135C8semidirect
126. C2×Q16C2×C8semidirect
129. C2×SD16C2×C8semidirect
132. C2×D8C2×C8semidirect
279. C2×C8C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1352
#218472
C22×#3342
C2×#14772
C2×#14502
#96592
#96582
#96552