Finite Group Databasebeta

← #415171

Subgroups — #415171

Order 512.

1272 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1990C212yesmaximal
2#14714C211yesmaximal, characteristic
3#14692C211yesmaximal, characteristic
4#11357C214yesmaximal
5#11267C214yesmaximal
6#10841C212yesmaximal
7C4×#336C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×C8C2212yes—
10C2×#391—21no—
11C2×#391—21no—
12C2×#384—21no—
13C2×#384—21no—
14C2×#384—21no—
15C2×#384—21no—
16C2×#384C2212yes—
17C2×#384C2211yescharacteristic
18C2×#384C2212yes—
19C2×#384C2212yes—
20C2×C43C2211yescharacteristic
21#1990—28no—
22C2×#340C2218yes—
23C2×#340C2218yes—
24C2×#336C2212yes—
25C2×#336C2212yes—
26C4⋊#95—21no—
27C4⋊#95—21no—
28C4⋊#95C2212yes—
29C4⋊#95C2212yes—
30C23×Q8—21no—
31C23×C8C2314yes—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—24no—
34C2×C4⋊Q8—24no—
35C2×C4⋊Q8D414yes—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2314yes—
47C22×C4⋊C4C2312yes—
48C22×C42D411yescharacteristic
49C22×C42D411yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C2×C4×C8—24no—
56C2×C4×C8—24no—
57C2×C4×C8—24no—
58#391—44no—
59#391—44no—
60#384—24no—
61#384—28no—
62#384—24no—
63#384—24no—
64#384—44no—
65#384—44no—
66#384—44no—
67#384—44no—
68#384—44no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C43—24no—
74#340—216no—
75#340—48no—
76#336—24no—
77#336Q818yescomplemented
78C22×Q8—24no—
79C22×Q8—42no—
80C22×Q8—42no—
81C22×Q8—42no—
82C22×Q8—42no—
83C22×Q8—42no—
84C22×Q8—42no—
85C23×C4—22no—
86C23×C4—24no—
87C23×C4—22no—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2411yescharacteristic
95C22×C8—24no—
96C22×C8—24no—
97C22×C8—24no—
98C22×C8—24no—
99C22×C8—24no—
100C22×C8—24no—
101C22×C8C2×Q818yes—
102C4⋊Q8—216no—
103C4⋊Q8—48no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4C2×D414yes—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42D814yes—
155C2×C42SD1614yes—
156C2×C42SD1614yes—
157C2×C42Q1614yes—
158C2×C42C2×D412yes—
159C2×C42C2×Q814yes—
160C2×C42#4114yes—
161C2×C42#4114yes—
162C2×C42#4114yes—
163C2×C42#4112yes—
164C4×C8—48no—
165#95—44no—
166#95—44no—
167#95—44no—
168#95—44no—
169C24C22×D411yescharacteristic
170C2×Q8—48no—
171C2×Q8—48no—
172C2×Q8—48no—
173C2×Q8—84no—
174C2×Q8—84no—
175C2×Q8—84no—
176C2×Q8—84no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—24no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4#12112yes—
215C22×C4#12112yes—
216C22×C4#12312yes—
217C22×C4#12312yes—
218C22×C4#12412yes—
219C22×C4#12412yes—
220C22×C4#12412yes—
221C22×C4#12412yes—
222C22×C4C4⋊Q812yes—
223C22×C4C4⋊Q812yes—
224C22×C4C2×D812yes—
225C22×C4C2×D812yes—
226C22×C4C2×D812yes—
227C22×C4C2×SD1612yes—
228C22×C4C2×SD1612yes—
229C22×C4C2×SD1612yes—
230C22×C4C2×SD1612yes—
231C22×C4C2×SD1612yes—
232C22×C4C2×SD1612yes—
233C22×C4C2×Q1612yes—
234C22×C4C2×Q1612yes—
235C22×C4C2×Q1612yes—
236C22×C4AGL(1,ℤ/8)12yes—
237C22×C4AGL(1,ℤ/8)12yes—
238C22×C4#13712yes—
239C22×C4#13712yes—
240C22×C4C22×D411yescharacteristic
241C22×C4C22×Q812yes—
242C22×C4C2×#4112yes—
243C22×C4C2×#4112yes—
244C22×C4C2×#4111yescharacteristic
245C22×C4C2×#4112yes—
246C22×C4H(5,𝔽2)12yes—
247C22×C42−1+412yes—
248C2×C8—28no—
249C2×C8—28no—
250C2×C8—28no—
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—44no—
255C4⋊C4—28no—
256C4⋊C4—44no—
257C4⋊C4—48no—
258C4⋊C4—48no—
259C4⋊C4—48no—
260C4⋊C4—48no—
261C4⋊C4—88no—
262C4⋊C4—88no—
263C4⋊C4—88no—
264C4⋊C4—88no—
265C4⋊C4—88no—
266C4⋊C4—88no—
267C4⋊C4—84no—
268C4⋊C4—84no—
269C42—28no—
270C42—28no—
271C42—28no—
272C42—28no—
273C42—28no—
274C42—28no—
275C42#12318yes—
276C42#12618yes—
277C23C2×#12111yescharacteristic
278C23C2×#12311yescharacteristic
279C23C2×#12411yescharacteristic
280C23C2×#12411yescharacteristic
281C23C2×C4⋊Q811yescharacteristic
282C23#53511yescharacteristic
283C23#53611yescharacteristic
284C23C22×D811yescharacteristic
285C23C22×SD1611yescharacteristic
286C23C22×SD1611yescharacteristic
287C23C22×Q1611yescharacteristic
288C23C2×AGL(1,ℤ/8)11yescharacteristic
289C23C2×AGL(1,ℤ/8)11yescharacteristic
290C23C2×#13711yescharacteristic
291C23C2×#13711yescharacteristic
292Q8—816no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—84no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—84no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4#39714yes—
339C2×C4#39714yes—
340C2×C4#45914yes—
341C2×C4#46014yes—
342C2×C4#46114yes—
343C2×C4#46214yes—
344C2×C4#48014yes—
345C2×C4#48114yes—
346C2×C4#48314yes—
347C2×C4#48414yes—
348C2×C4#48614yes—
349C2×C4#48614yes—
350C2×C4#48714yes—
351C2×C4#48714yes—
352C2×C4#48914yes—
353C2×C4#48914yes—
354C2×C4#49814yes—
355C2×C4#50014yes—
356C2×C4C2×#12312yes—
357C2×C4C2×#12612yes—
358C2×C4#53314yes—
359C2×C4#54214yes—
360C2×C4D4×Q812yes—
361C2×C4#55012yes—
362C2×C4#55214yes—
363C2×C4#55414yes—
364C2×C4#55712yes—
365C2×C4#55912yes—
366C8—48no—
367C22C2×#39711yescharacteristic
368C22#250211yescharacteristic
369C22#250311yescharacteristic
370C22#254611yescharacteristic
371C22#254811yescharacteristic
372C22#264011yescharacteristic
373C22#286211yescharacteristic
374C22C2×#45911yescharacteristic
375C22C2×#46111yescharacteristic
376C22C2×#46011yescharacteristic
377C22C2×#46211yescharacteristic
378C22C2×#48011yescharacteristic
379C22C2×#48311yescharacteristic
380C22C2×#48111yescharacteristic
381C22C2×#48411yescharacteristic
382C22C2×#48611yescharacteristic
383C22C2×#48611yescharacteristic
384C22C2×#48711yescharacteristic
385C22C2×#48711yescharacteristic
386C22C2×#48911yescharacteristic
387C22C2×#48911yescharacteristic
388C22#313811yescharacteristic
389C22#313911yescharacteristic
390C22C2×#49811yescharacteristic
391C22C2×#50011yescharacteristic
392C22#316511yescharacteristic
393C22#316511yescharacteristic
394C22#320511yescharacteristic
395C22#320611yescharacteristic
396C22#320811yescharacteristic
397C22#320911yescharacteristic
398C22#322611yescharacteristic
399C22#322711yescharacteristic
400C22#322911yescharacteristic
401C22#323011yescharacteristic
402C4—24no—
403C4—24no—
404C4—24no—
405C4—24no—
406C4—84no—
407C4—164no—
408C4—162no—
409C4#207118yes—
410C4#207218yes—
411C4#242314yes—
412C4#251114yes—
413C2#1518511yescharacteristic
414C2C2×#207111yescharacteristic
415C2C2×#207211yescharacteristic
416C2#2794811yescharacteristic
417C2#2795011yescharacteristic
418C2#2796011yescharacteristic
419C2#2796111yescharacteristic
420C2#2868711yescharacteristic
421C2#2868911yescharacteristic
422C2#2874811yescharacteristic
423C2#2874911yescharacteristic
424C2#3020311yescharacteristic
425C2#3020511yescharacteristic
426C2#3328911yescharacteristic
427C2#3329311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
77. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19902
#147142
#146922
#113572
#112672
#108412
C4×#3362