Finite Group Databasebeta

← #193239

Subgroups — #193239

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4C2×#1948C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic
6#8927C214yesmaximal, complemented
7#8926C214yesmaximal, complemented
8#7893C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×#106—21no—
14C22×C4⋊C8C2211yescharacteristic
15#2370—22no—
16#2369C2212yes—
17#2365C2212yes—
18#2355—21no—
19#2355—22no—
20Q8×C4⋊C4—21no—
21#2329—22no—
22#2317—21no—
23C4×C22⋊Q8—22no—
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#2274—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990C2212yes—
29#1956C2212yes—
30#1948—22no—
31#1948C2214yes—
32#1947C2214yes—
33#1872—22no—
34#1852—22no—
35#1845—21no—
36#1834—21no—
37#1803—22no—
38C2×#340—21no—
39#1542C418yescomplemented
40C2×#333C2212yes—
41C2×#332C2212yes—
42#1279C2214yes—
43#1279C2214yes—
44C23×C8—21no—
45#533—22no—
46#533—22no—
47#533C2×C414yes—
48#533Q814yes—
49C2×C4⋊Q8C2311yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8D412yes—
54C2×C4×Q8—22no—
55C2×C4×Q8—21no—
56C2×C4×Q8D412yes—
57C2×#117—21no—
58C2×#117—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66#431—22no—
67#431—22no—
68#431—22no—
69#431C2×C414yes—
70C2×#107—24no—
71C2×#106—24no—
72C2×#106—44no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2×C414yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#103—24no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8C2312yes—
84#391—24no—
85#391—22no—
86#389—24no—
87#389—22no—
88#389—22no—
89#389—22no—
90#387—24no—
91#387—24no—
92#387—24no—
93#385—24no—
94#384—22no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384D411yescharacteristic
99#384D411yescharacteristic
100#384C2312yes—
101#382—22no—
102#382—21no—
103#380—21no—
104#380D412yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4C2312yes—
116C4×C4⋊C4C2312yes—
117C4×C22⋊C4—24no—
118C2×#95—21no—
119#340—24no—
120#340—24no—
121#340—24no—
122#333C2×C418yescomplemented
123#332C2×C418yescomplemented
124C22×Q8—21no—
125C22×Q8C2×D412yes—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—41no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C22×C8—24no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—42no—
137C22×C8—42no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8C22×C412yes—
143C4⋊Q8C22×C412yes—
144C4⋊Q8C2×Q812yes—
145C4⋊Q8C2×Q812yes—
146#125—24no—
147#125—24no—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C22⋊Q8—24no—
151C22⋊Q8C4⋊C418yes—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—84no—
156C4×Q8C4⋊C418yes—
157#117—24no—
158#117—24no—
159#117—42no—
160#117—42no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4Q1614yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—22no—
201C2×C42—82no—
202C2×C42C22×C412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×Q812yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42C2411yescharacteristic
210#107—84no—
211#106—48no—
212C4⋊C8—24no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—42no—
216C4⋊C8C22×C414yes—
217C4⋊C8C22×C414yes—
218#103—84no—
219C22⋊C8—24no—
220C22⋊C8—42no—
221C4×C8—24no—
222C4×C8—22no—
223C4×C8—22no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232C24#12011yescharacteristic
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8C2×C4⋊C414yes—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4C2×C4⋊C412yes—
262C22×C4C4×D412yes—
263C22×C4C4×D412yes—
264C22×C4#12011yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4C22⋊Q812yes—
268C22×C4C22⋊Q812yes—
269C22×C4#12312yes—
270C22×C4C2×Q1612yes—
271C22×C4#13512yes—
272C22×C4AGL(1,ℤ/8)12yes—
273C22×C4#13712yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—48no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—82no—
316C4⋊C4C2×C4⋊C414yes—
317C4⋊C4C2×Q1614yes—
318C22⋊C4—28no—
319C22⋊C4—44no—
320C42—28no—
321C42—42no—
322C42—42no—
323C42—44no—
324C42—82no—
325C42—84no—
326C42—84no—
327C42C2×C4⋊C412yes—
328C42C4×D412yes—
329C42C4×D412yes—
330C42C22⋊Q812yes—
331C42C22⋊Q812yes—
332C42#12312yes—
333C42C23×C411yescharacteristic
334C42C22×Q811yescharacteristic
335C42C2×#4111yescharacteristic
336C42C2×#4111yescharacteristic
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23#38512yes—
342C23#44912yes—
343C23#45112yes—
344C23C2×#12011yescharacteristic
345Q8—44no—
346Q8—44no—
347Q8—44no—
348Q8—82no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—164no—
383C2×C4#38512yes—
384C2×C4#38512yes—
385C2×C4#38512yes—
386C2×C4C4×Q1614yes—
387C2×C4#44014yes—
388C2×C4#44912yes—
389C2×C4#45112yes—
390C2×C4#47714yes—
391C2×C4#47914yes—
392C2×C4C22×C4⋊C411yescharacteristic
393C2×C4C2×C4×D411yescharacteristic
394C2×C4C2×C4×D411yescharacteristic
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4C2×C22⋊Q811yescharacteristic
397C2×C4C2×#12311yescharacteristic
398C2×C4C22×Q1611yescharacteristic
399C2×C4C2×#13511yescharacteristic
400C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
401C2×C4C2×#13711yescharacteristic
402C2×C4#57712yes—
403C2×C4#57712yes—
404C8—44no—
405C8—44no—
406C8—82no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22#186314yes—
414C22C2×#38511yescharacteristic
415C22C2×#45111yescharacteristic
416C22C2×#44911yescharacteristic
417C22#300911yescharacteristic
418C22#301011yescharacteristic
419C22#302111yescharacteristic
420C22#302211yescharacteristic
421C4—21no—
422C4—21no—
423C4—22no—
424C4—22no—
425C4—44no—
426C4—44no—
427C4—44no—
428C4—82no—
429C4—82no—
430C4—82no—
431C4—162no—
432C4#186314yes—
433C4C2×C4×Q1611yescharacteristic
434C4C2×#44011yescharacteristic
435C4#295212yes—
436C4C2×#47711yescharacteristic
437C4C2×#47911yescharacteristic
438C4#308412yes—
439C2—24no—
440C2#1889511yescharacteristic
441C2#1889811yescharacteristic
442C2#1892511yescharacteristic
443C2#1892811yescharacteristic
444C2#2085811yescharacteristic
445C2C2×#186311yescharacteristic
446C2#2088211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8927C2semidirect
7. #8926C2semidirect
39. #1542C4semidirect
122. #333C2×C4semidirect
123. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#219822
#218602
C2×#19482
C2×#15422
#89272
#89262
#78932
#78922