Finite Group Databasebeta

← #239890

Subgroups — #239890

Order 512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic
6C2×#1481C211yesmaximal, characteristic
7#10419C214yesmaximal, complemented
8#10414C214yesmaximal, complemented
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×#107—21no—
15C22×C4⋊C8C2211yescharacteristic
16#2857—22no—
17#2659—22no—
18#2584—22no—
19#2584—22no—
20#2567—21no—
21#2567C2212yes—
22#2546—21no—
23#2546C2212yes—
24#2316—21no—
25#2316—21no—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990—21no—
29#1990C2212yes—
30#1956—21no—
31#1956C2212yes—
32#1947—22no—
33#1947C2214yes—
34#1872—22no—
35#1872—22no—
36#1803—22no—
37#1803C2214yes—
38C2×#340—21no—
39#1542—24no—
40C2×#333C2212yes—
41C2×#332C2212yes—
42#1481—24no—
43C2×Q8⋊C8C2214yes—
44#1279C2214yes—
45C23×C8—21no—
46#533—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8D411yescharacteristic
56C2×C4⋊Q8D411yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2311yescharacteristic
59C2×#125—21no—
60C2×#125—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8D412yes—
64C2×C22⋊Q8D412yes—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8D412yes—
68C2×C4×Q8C2312yes—
69C2×#117—21no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#431—22no—
79#431—22no—
80#431—22no—
81#431—22no—
82C2×#107—24no—
83C2×#107—44no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8D414yes—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×#103—24no—
91C2×#103—24no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8C2312yes—
96#398—24no—
97#398—24no—
98#395—24no—
99#395—24no—
100#391—22no—
101#391—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—24no—
107#389—24no—
108#389—22no—
109#389—22no—
110#389—22no—
111#389—22no—
112#384—22no—
113#384—21no—
114#384—21no—
115#384—21no—
116#384C2312yes—
117#380—24no—
118#380—24no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4C2312yes—
123C2×#95—21no—
124#340—24no—
125#340—24no—
126#340—24no—
127#333—24no—
128#332—24no—
129Q8⋊C8—28no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8C2×D412yes—
133C22×Q8C2×D412yes—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—41no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C22×C8—24no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—42no—
145C22×C8—42no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—84no—
159#125—42no—
160#125—42no—
161#125—42no—
162#125—42no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C22⋊Q8—24no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C4×Q8—24no—
172C4×Q8—24no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181#117—42no—
182#117—42no—
183#117—42no—
184#117—42no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
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—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4SD1614yes—
214C2×C4⋊C4Q1614yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—82no—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4112yes—
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42C2411yescharacteristic
234#107—48no—
235C4⋊C8—24no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—42no—
240C4⋊C8C2×D414yes—
241#103—84no—
242#103—84no—
243C22⋊C8—42no—
244C22⋊C8—42no—
245C4×C8—22no—
246C4×C8—22no—
247C4×C8—42no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95—22no—
254C24#12011yescharacteristic
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—81no—
269C2×Q8—81no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4#12112yes—
296C22×C4#12312yes—
297C22×C4#12312yes—
298C22×C4#12412yes—
299C22×C4C2×SD1612yes—
300C22×C4C2×Q1612yes—
301C22×C4#13712yes—
302C22×C4#13712yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C2×C8—24no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—48no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—44no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C4⋊C4—82no—
349C4⋊C4C2×SD1614yes—
350C4⋊C4C2×Q1614yes—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C42—42no—
354C42—42no—
355C42—44no—
356C42—44no—
357C42—82no—
358C42#12112yes—
359C42#12112yes—
360C42#12112yes—
361C42#12312yes—
362C42#12312yes—
363C42#12412yes—
364C42C22×D411yescharacteristic
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C42C2×#4111yescharacteristic
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23#39412yes—
373C23#44812yes—
374C23#45112yes—
375C23C2×#12011yescharacteristic
376Q8—44no—
377Q8—44no—
378Q8—44no—
379Q8—82no—
380Q8—84no—
381Q8—82no—
382Q8—82no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—164no—
417C2×C4#39412yes—
418C2×C4#39412yes—
419C2×C4#39412yes—
420C2×C4#44812yes—
421C2×C4#45112yes—
422C2×C4#46114yes—
423C2×C4#46214yes—
424C2×C4#48714yes—
425C2×C4#49114yes—
426C2×C4C2×#12111yescharacteristic
427C2×C4C2×#12111yescharacteristic
428C2×C4C2×#12111yescharacteristic
429C2×C4C2×#12311yescharacteristic
430C2×C4C2×#12311yescharacteristic
431C2×C4C2×#12411yescharacteristic
432C2×C4C22×SD1611yescharacteristic
433C2×C4C22×Q1611yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4C2×#13711yescharacteristic
436C2×C4#57612yes—
437C2×C4#57812yes—
438C8—44no—
439C8—82no—
440C8—82no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22#204314yes—
448C22C2×#39411yescharacteristic
449C22C2×#44811yescharacteristic
450C22C2×#45111yescharacteristic
451C22#300811yescharacteristic
452C22#301111yescharacteristic
453C22#302111yescharacteristic
454C22#302211yescharacteristic
455C4—22no—
456C4—21no—
457C4—21no—
458C4—22no—
459C4—44no—
460C4—44no—
461C4—82no—
462C4—82no—
463C4—82no—
464C4—82no—
465C4—162no—
466C4#204314yes—
467C4C2×#46111yescharacteristic
468C4C2×#46211yescharacteristic
469C4#306112yes—
470C4#309812yes—
471C4C2×#48711yescharacteristic
472C4C2×#49111yescharacteristic
473C2—24no—
474C2#1889411yescharacteristic
475C2#1890511yescharacteristic
476C2#1891511yescharacteristic
477C2#1892411yescharacteristic
478C2#2348811yescharacteristic
479C2#2349511yescharacteristic
480C2C2×#204311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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. #10419C2semidirect
8. #10414C2semidirect

Maximal subgroups

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

subgroupindex
#351732
#218602
C2×#19472
C2×#15422
C2×#14812
#104192
#104142
#78922