Finite Group Databasebeta

← #235958

Subgroups — #235958

Order 512.

1342 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2C22×#477C211yesmaximal, characteristic
3#35179C211yesmaximal, characteristic
4#21834C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic
6C2×#1479C211yesmaximal, characteristic
7#10061C214yesmaximal, complemented
8#10057C214yesmaximal, complemented
9#7893C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#3209—24no—
14#3209—24no—
15C2×#494C2212yes—
16C2×#487C2212yes—
17C2×#477—24no—
18C2×#477—24no—
19C2×#477—22no—
20C2×#477C2214yes—
21C2×#462—22no—
22C2×#451—22no—
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C22×#107—21no—
26C22×C4⋊C8C2211yescharacteristic
27C22×#103—21no—
28#2855—22no—
29#2627—22no—
30#2582—22no—
31#2581—22no—
32#2568C2212yes—
33#2567C2212yes—
34#2565—22no—
35#2317—21no—
36#2310—21no—
37#2285C2211yescharacteristic
38#1990—22no—
39#1987C2214yes—
40#1824—22no—
41#1542—24no—
42C2×#333C2212yes—
43C2×#332C2212yes—
44#1479—24no—
45C2×Q8⋊C8C2214yes—
46#1279C2214yes—
47C23×Q8—21no—
48C23×Q8D411yescharacteristic
49C22×Q16—24no—
50C23×C8—21no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8D414yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2311yescharacteristic
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8D412yes—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8D412yes—
76C2×C4×Q8D412yes—
77C2×C4×Q8C2312yes—
78C2×#117—21no—
79C2×#117—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#494—24no—
87#494—24no—
88#487—24no—
89#487—24no—
90#477—216no—
91#477—28no—
92#477—28no—
93#462—216no—
94#451—216no—
95#431—22no—
96#431—22no—
97#431—22no—
98#431D414yes—
99C2×#107—28no—
100C2×#107—24no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8D414yes—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107C2×#103—28no—
108C2×#103—24no—
109C2×#103—24no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2312yes—
114#400—24no—
115#398—28no—
116#395—24no—
117#391—24no—
118#391—22no—
119#391—22no—
120#391—24no—
121#389—24no—
122#389—24no—
123#387—28no—
124#385—24no—
125#384—22no—
126#384D412yes—
127#382—22no—
128#382—21no—
129#381—24no—
130#380—21no—
131#380D412yes—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2312yes—
134#340—28no—
135#333—24no—
136#332—24no—
137Q8⋊C8—28no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—41no—
148C22×Q8—42no—
149C22×Q8—84no—
150C22×Q8Q1614yescomplemented
151C22×Q8C2×D412yes—
152C22×Q8C2×D412yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—41no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C2×Q16—216no—
160C2×Q16—88no—
161C22×C8—24no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—42no—
166C22×C8—42no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—22no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—22no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—22no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184C4⋊Q8C2×D414yes—
185#125—42no—
186#125—42no—
187C22⋊Q8—24no—
188C22⋊Q8—24no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205#117—42no—
206#117—42no—
207#117—42no—
208#117—42no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42#4112yes—
250C2×C42C2411yescharacteristic
251#107—216no—
252C4⋊C8—24no—
253C4⋊C8—24no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8C2×D414yes—
258#103—216no—
259#103—216no—
260C22⋊C8—24no—
261C22⋊C8—42no—
262C4×C8—22no—
263C4×C8—42no—
264C4×C8C2×D414yes—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—22no—
269#95—22no—
270C24#12011yescharacteristic
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—22no—
274C2×Q8—24no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—28no—
281C2×Q8—28no—
282C2×Q8—44no—
283C2×Q8—48no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—41no—
287C2×Q8—44no—
288C2×Q8—42no—
289C2×Q8—44no—
290C2×Q8—42no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—84no—
294C2×Q8—88no—
295C2×Q8—88no—
296C2×Q8C2×Q1614yes—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—82no—
311C22×C4—84no—
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12012yes—
315C22×C4#12012yes—
316C22×C4#12011yescharacteristic
317C22×C4#12112yes—
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4#12712yes—
321C22×C4C2×Q1612yes—
322C22×C4#13512yes—
323C22×C4#13712yes—
324C22×C4#13712yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328Q16—816no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—44no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—48no—
335C2×C8—42no—
336C2×C8—44no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—28no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—82no—
362C4⋊C4—88no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C42—42no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—44no—
370C42—44no—
371C42#12012yes—
372C42#12012yes—
373C42#12112yes—
374C42#12112yes—
375C42#12112yes—
376C42#12712yes—
377C42C22×D411yescharacteristic
378C42C22×D411yescharacteristic
379C42C22×D411yescharacteristic
380C42C2×#4111yescharacteristic
381C23—22no—
382C23—22no—
383C23#39212yes—
384C23#45112yes—
385C23#45212yes—
386C23C2×#12011yescharacteristic
387Q8—24no—
388Q8—24no—
389Q8—44no—
390Q8—48no—
391Q8—42no—
392Q8—48no—
393Q8—42no—
394Q8—82no—
395Q8—84no—
396Q8—88no—
397Q8—82no—
398Q8—82no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—48no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—84no—
421C2×C4—88no—
422C2×C4—84no—
423C2×C4#39212yes—
424C2×C4#39212yes—
425C2×C4#39212yes—
426C2×C4#45114yes—
427C2×C4#45112yes—
428C2×C4#45214yes—
429C2×C4#45212yes—
430C2×C4#46714yes—
431C2×C4#47014yes—
432C2×C4C2×#12011yescharacteristic
433C2×C4C2×#12011yescharacteristic
434C2×C4C2×#12111yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C2×C4C2×#12111yescharacteristic
437C2×C4C2×#12711yescharacteristic
438C2×C4C22×Q1611yescharacteristic
439C2×C4C2×#13511yescharacteristic
440C2×C4C2×#13711yescharacteristic
441C2×C4C2×#13711yescharacteristic
442C2×C4#57814yes—
443C8—44no—
444C8—44no—
445C8—82no—
446C22—22no—
447C22—22no—
448C22—24no—
449C22—24no—
450C22#200114yes—
451C22C2×#39211yescharacteristic
452C22C2×#45111yescharacteristic
453C22C2×#45211yescharacteristic
454C22#301112yes—
455C22#302112yes—
456C4—21no—
457C4—21no—
458C4—22no—
459C4—22no—
460C4—44no—
461C4—82no—
462C4—82no—
463C4—84no—
464C4—82no—
465C4—84no—
466C4#200114yes—
467C4C2×#45111yescharacteristic
468C4C2×#45211yescharacteristic
469C4#304412yes—
470C4C2×#46711yescharacteristic
471C4C2×#47011yescharacteristic
472C4#315612yes—
473C2—24no—
474C2#1889912yes—
475C2#1891512yes—
476C2C2×#200111yescharacteristic
477C2#2340011yescharacteristic
478C2#2342511yescharacteristic
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. #10061C2semidirect
8. #10057C2semidirect
150. C22×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C22×#4772
#351792
#218342
C2×#15422
C2×#14792
#100612
#100572
#78932