Finite Group Databasebeta

← #240545

Subgroups — #240545

Order 512.

926 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3249C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21318C211yesmaximal, characteristic
4#21300C211yesmaximal, characteristic
5C22×#334C211yesmaximal, characteristic
6C2×#1550C211yesmaximal, characteristic
7#10482C214yesmaximal, complemented
8#10479C214yesmaximal, complemented
9#8045C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3249—24no—
13#3249—24no—
14C2×#477—21no—
15C2×#477C2212yes—
16C2×#467—22no—
17C2×#462—21no—
18C2×#462C2212yes—
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2285C2211yescharacteristic
24#1991—22no—
25#1990C2212yes—
26#1990C2212yes—
27#1987—21no—
28#1984—22no—
29#1984C2214yes—
30#1981—21no—
31#1892—22no—
32#1803C2214yes—
33#1735—21no—
34#1735—21no—
35#1729—21no—
36#1729—21no—
37C2×#334—24no—
38C2×#334C2214yes—
39#1550—24no—
40C2×#333C2212yes—
41C2×#332C2212yes—
42#1296C2214yes—
43C22×Q16—22no—
44C22×M16—21no—
45C23×C8—21no—
46C23×C8—21no—
47C23×C8D411yescharacteristic
48C23×C8D411yescharacteristic
49#533—22no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533—22no—
55#533—22no—
56#533D414yes—
57C2×C4⋊Q8D411yescharacteristic
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×#125—21no—
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8D412yes—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8C2312yes—
71C2×#117—21no—
72C2×#117—21no—
73C2×#117—21no—
74C22×C42C2311yescharacteristic
75#477—28no—
76#477—24no—
77#477—24no—
78#467—216no—
79#462—28no—
80#462—24no—
81#462—24no—
82C2×#107—22no—
83#423—22no—
84#423—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8D414yes—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—24no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8D412yes—
101C2×C4×C8C2312yes—
102#384C2312yes—
103#382—21no—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—24no—
113#336—24no—
114#336—24no—
115#334—28no—
116#333—24no—
117#332—24no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4#4111yescharacteristic
125C2×Q16—28no—
126C2×Q16—84no—
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—42no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—24no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—42no—
140C22×C8—82no—
141C22×C8—82no—
142C22×C8Q1614yescomplemented
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8C2×D412yes—
156C4⋊Q8C2×D412yes—
157#125—24no—
158#125—42no—
159#125—42no—
160#125—42no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—42no—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×Q8—42no—
179#117—24no—
180#117—42no—
181#117—42no—
182#117—42no—
183#117—42no—
184#117—42no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4C2×D412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C42—22no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42#4112yes—
211C2×C42#4112yes—
212C2×C42#4111yescharacteristic
213C2×C42C2411yescharacteristic
214#107—28no—
215C4⋊C8—24no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—42no—
222C4⋊C8—42no—
223C4⋊C8C2×D414yes—
224#103—28no—
225#103—28no—
226#103—84no—
227C4×C8—24no—
228C4×C8—22no—
229C4×C8—22no—
230#95—41no—
231#95—41no—
232C24#12111yescharacteristic
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—44no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—82no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4#12012yes—
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12112yes—
258C22×C4#12112yes—
259C22×C4#12111yescharacteristic
260C22×C4#12312yes—
261C22×C4#12312yes—
262C22×C4#12412yes—
263C22×C4C2×Q1612yes—
264C22×C4#13512yes—
265C22×C4#13712yes—
266C22×C4#13712yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4111yescharacteristic
270Q16—88no—
271M16—84no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—28no—
275C2×C8—24no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—44no—
286C2×C8—84no—
287C2×C8—84no—
288C2×C8C2×Q1614yes—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—82no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—82no—
315C42—42no—
316C42—42no—
317C42—44no—
318C42—82no—
319C42#12012yes—
320C42#12112yes—
321C42#12112yes—
322C42#12312yes—
323C42#12312yes—
324C42#12412yes—
325C42C22×D411yescharacteristic
326C42C22×D411yescharacteristic
327C42C2×#4111yescharacteristic
328C42C2×#4111yescharacteristic
329C23—22no—
330C23—22no—
331C23#39412yes—
332C23#46212yes—
333C23#46412yes—
334C23C2×#12111yescharacteristic
335Q8—44no—
336Q8—42no—
337Q8—42no—
338Q8—42no—
339Q8—42no—
340Q8—82no—
341Q8—84no—
342Q8—82no—
343Q8—82no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4#39412yes—
364C2×C4#39412yes—
365C2×C4#39412yes—
366C2×C4#46212yes—
367C2×C4#46214yes—
368C2×C4#46412yes—
369C2×C4#46414yes—
370C2×C4#46714yes—
371C2×C4#47014yes—
372C2×C4C2×#12011yescharacteristic
373C2×C4C2×#12111yescharacteristic
374C2×C4C2×#12111yescharacteristic
375C2×C4C2×#12311yescharacteristic
376C2×C4C2×#12311yescharacteristic
377C2×C4C2×#12411yescharacteristic
378C2×C4C22×Q1611yescharacteristic
379C2×C4C2×#13511yescharacteristic
380C2×C4C2×#13711yescharacteristic
381C2×C4C2×#13711yescharacteristic
382C2×C4#57712yes—
383C2×C4#57712yes—
384C8—24no—
385C8—44no—
386C8—42no—
387C8—82no—
388C8—82no—
389C22—24no—
390C22—24no—
391C22—22no—
392C22—22no—
393C22#205614yes—
394C22C2×#39411yescharacteristic
395C22C2×#46211yescharacteristic
396C22C2×#46411yescharacteristic
397C22#306012yes—
398C22#306712yes—
399C4—22no—
400C4—21no—
401C4—21no—
402C4—22no—
403C4—82no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—162no—
408C4#205614yes—
409C4#300911yescharacteristic
410C4#301011yescharacteristic
411C4C2×#46211yescharacteristic
412C4C2×#46411yescharacteristic
413C4C2×#46711yescharacteristic
414C4C2×#47011yescharacteristic
415C4#309912yes—
416C2—24no—
417C2#1921012yes—
418C2#1921612yes—
419C2#2348911yescharacteristic
420C2#2353011yescharacteristic
421C2C2×#205611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×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
7. #10482C2semidirect
8. #10479C2semidirect
142. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32492
#218852
#213182
#213002
C22×#3342
C2×#15502
#104822
#104792
#80452