Finite Group Databasebeta

← #205901

Subgroups — #205901

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic
3#21829C211yesmaximal, characteristic
4#17793C211yesmaximal, characteristic
5#17746C211yesmaximal, characteristic
6#17742C211yesmaximal, characteristic
7#17323C211yesmaximal, characteristic
8#9803C214yesmaximal, complemented
9#9795C214yesmaximal, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#487—22no—
13C2×#487—22no—
14C2×#487—21no—
15C2×#487—21no—
16C2×#487—21no—
17C2×#487—21no—
18C2×#487C414yes—
19C2×#487C2212yes—
20C2×#487C2212yes—
21#2920C2211yescharacteristic
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24#2920C2211yescharacteristic
25C22×#103C414yes—
26C22×C4×C8C2211yescharacteristic
27#1997—22no—
28#1997—22no—
29#1997C2214yes—
30#1997C2214yes—
31#1821—22no—
32#1821—22no—
33#1568—22no—
34#1567C2212yes—
35#1566C2212yes—
36#1564—22no—
37#1558—22no—
38#1556—22no—
39#1549C2214yes—
40#1548—22no—
41#1548C2214yes—
42#1546—22no—
43#1452—21no—
44#1451—21no—
45#1451C2212yes—
46#1450—21no—
47C8×M16—21no—
48C8×M16C2212yes—
49C23×Q8—21no—
50C23×Q8C2×C412yes—
51C23×C8C2×C412yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8C2×C412yes—
73C2×C4⋊Q8C2×C412yes—
74C2×C4⋊Q8C2312yes—
75C2×C4⋊Q8C2312yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2×C412yes—
80C22×C42C2311yescharacteristic
81#487—24no—
82#487—24no—
83#487—24no—
84#487—24no—
85#487—22no—
86#487—22no—
87#487—22no—
88#487—22no—
89#487—24no—
90#487—24no—
91#432—21no—
92#432—21no—
93#432—21no—
94#432—21no—
95#432—21no—
96#432—21no—
97#432—21no—
98#432D412yes—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431C2312yes—
107#423D412yes—
108#423D412yes—
109#423C2312yes—
110#423C2312yes—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—24no—
115C2×#103C2×C418yes—
116C2×#103C2×C418yes—
117#407—21no—
118#407—21no—
119#407—21no—
120#407D412yes—
121C4×M16—21no—
122C4×M16—21no—
123C4×M16—21no—
124C4×M16C2312yes—
125C2×C4×C8C2×C412yes—
126C2×C4×C8D412yes—
127C2×C4×C8D412yes—
128C2×C4×C8D411yescharacteristic
129C2×C4×C8D411yescharacteristic
130C2×C4×C8D411yescharacteristic
131C2×C4×C8D411yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134#343—28no—
135#343—28no—
136#335—24no—
137#334—24no—
138#333—28no—
139#332—28no—
140#322—22no—
141#322—22no—
142#322—22no—
143C82—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—48no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×Q8—84no—
157C22×Q8C22×C414yes—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16C2×D414yes—
167C22×C8—44no—
168C22×C8C22⋊C412yes—
169C22×C8C22⋊C412yes—
170C22×C8C22⋊C412yes—
171C22×C8C22⋊C412yes—
172C22×C8C22×C412yes—
173C22×C8C22×C412yes—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—41no—
191C4⋊Q8—41no—
192C4⋊Q8—41no—
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C4⋊Q8—41no—
196C4⋊Q8—41no—
197C4⋊Q8—41no—
198C4⋊Q8—41no—
199C4⋊Q8—41no—
200C4⋊Q8—41no—
201C4⋊Q8—41no—
202C4⋊Q8—41no—
203C4⋊Q8—41no—
204C4⋊Q8—41no—
205C4⋊Q8—41no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4C22×C414yes—
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42C2411yescharacteristic
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8C2×D414yes—
250#103—28no—
251#103—28no—
252#103—28no—
253#103—28no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—41no—
259C22⋊C8—41no—
260C22⋊C8—41no—
261C22⋊C8—41no—
262#97—22no—
263#97—22no—
264#97—41no—
265#97—41no—
266C4×C8—22no—
267C4×C8—22no—
268C4×C8—41no—
269C4×C8—41no—
270C4×C8C22⋊C412yes—
271C4×C8C22⋊C412yes—
272C4×C8C22⋊C412yes—
273C4×C8C22⋊C412yes—
274C4×C8C2×D411yescharacteristic
275C4×C8C2×D411yescharacteristic
276C4×C8C2×D411yescharacteristic
277C4×C8C2×D411yescharacteristic
278C4×C8C2×D412yes—
279C4×C8C2×D412yes—
280C24C4×D411yescharacteristic
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—48no—
284C2×Q8—48no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—48no—
292C2×Q8—84no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8—84no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12411yescharacteristic
324C22×C4#12411yescharacteristic
325C22×C4#12711yescharacteristic
326C22×C4#12711yescharacteristic
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4111yescharacteristic
330M16—28no—
331M16—44no—
332M16—44no—
333M16—44no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8C2×C22⋊C412yes—
345C2×C8C2×C22⋊C412yes—
346C2×C8C2×C22⋊C412yes—
347C2×C8C2×C22⋊C412yes—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C42C2×C22⋊C411yescharacteristic
365C42C2×C22⋊C411yescharacteristic
366C42C4×D411yescharacteristic
367C42C4×D411yescharacteristic
368C42#12111yescharacteristic
369C42#12111yescharacteristic
370C42#12111yescharacteristic
371C42#12111yescharacteristic
372C42#12411yescharacteristic
373C42#12411yescharacteristic
374C42#12711yescharacteristic
375C42#12711yescharacteristic
376C42C22×D411yescharacteristic
377C42C22×D411yescharacteristic
378C42C22×D411yescharacteristic
379C42C2×#4111yescharacteristic
380C23—22no—
381C23—22no—
382C23—22no—
383C23#39011yescharacteristic
384C23#39011yescharacteristic
385C23C2×C4×D411yescharacteristic
386Q8—48no—
387Q8—84no—
388Q8—84no—
389Q8—84no—
390Q8—84no—
391Q8—84no—
392Q8—84no—
393Q8—84no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4#39011yescharacteristic
418C2×C4#39011yescharacteristic
419C2×C4#39011yescharacteristic
420C2×C4#39011yescharacteristic
421C2×C4#39011yescharacteristic
422C2×C4#39011yescharacteristic
423C2×C4#41314yes—
424C2×C4#47114yes—
425C2×C4C22×C22⋊C411yescharacteristic
426C2×C4C2×C4×D411yescharacteristic
427C2×C4C2×#12111yescharacteristic
428C2×C4C2×#12111yescharacteristic
429C2×C4C2×#12411yescharacteristic
430C2×C4C2×#12711yescharacteristic
431C2×C4#57814yes—
432C8—22no—
433C8—22no—
434C8—22no—
435C8—22no—
436C8—44no—
437C8—44no—
438C8—82no—
439C8—82no—
440C22—28no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22#196712yes—
448C22#197712yes—
449C22C2×#39011yescharacteristic
450C22#295012yes—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—84no—
458C4—84no—
459C4#196712yes—
460C4#197712yes—
461C4C2×#41312yes—
462C4#305912yes—
463C4C2×#47112yes—
464C4#315612yes—
465C2—22no—
466C2—22no—
467C2—22no—
468C2C2×#196711yescharacteristic
469C2#2123211yescharacteristic
470C2C2×#197711yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #9803C2semidirect
9. #9795C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4872
#218292
#177932
#177462
#177422
#173232
#98032
#97952