Finite Group Databasebeta

← #119579

Subgroups — #119579

Order 512.

715 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21765C211yesmaximal, characteristic
2#21765C211yesmaximal, characteristic
3C2×#1472C211yesmaximal, characteristic
4#8107C211yesmaximal, characteristic
5#8107C211yesmaximal, characteristic
6#8057C211yesmaximal, characteristic
7C2×#1289C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1894—22no—
16#1894C414yescomplemented
17#1872—22no—
18#1872—22no—
19#1823—21no—
20#1823—21no—
21#1787—21no—
22#1787—21no—
23#1472—22no—
24#1472—22no—
25C2×#324—22no—
26#1305—22no—
27#1305—22no—
28#1298—22no—
29#1289—22no—
30C23×Q8D411yescharacteristic
31C24×C4C2311yescharacteristic
32C22×M16—21no—
33C22×M16—21no—
34C22×M16C2×C412yes—
35C23×C8—21no—
36C2×C22⋊Q8—21no—
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8C2×C412yes—
47C2×C22⋊Q8D412yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D411yescharacteristic
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C2×#103—24no—
56C2×#103—24no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407—21no—
62#407—21no—
63#407—21no—
64#407—21no—
65#407—21no—
66#407—21no—
67#407—21no—
68#407D412yes—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8D412yes—
80C2×C22⋊C8D412yes—
81#380—21no—
82#380—21no—
83#380—21no—
84#380—21no—
85#380—21no—
86#380—21no—
87#380C2×C412yes—
88#380D412yes—
89C2×#95—21no—
90C2×#95—21no—
91#340—24no—
92#340—24no—
93#324—28no—
94C25#4111yescharacteristic
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—82no—
101C22×Q8C22⋊C412yes—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—82no—
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4#4111yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—82no—
128C2×M16—82no—
129C2×M16—82no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—82no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C22⋊C412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—81no—
182C2×C22⋊C4—81no—
183C2×C22⋊C4—81no—
184C2×C22⋊C4—81no—
185#103—84no—
186#103—84no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203#95—22no—
204#95—22no—
205#95—41no—
206#95—41no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—42no—
212C24—81no—
213C24C4×D411yescharacteristic
214C24#12111yescharacteristic
215C24#12311yescharacteristic
216C24#12311yescharacteristic
217C24#12411yescharacteristic
218C24#12611yescharacteristic
219C24#12611yescharacteristic
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—82no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4#11711yescharacteristic
265C22×C4#11711yescharacteristic
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4#12311yescharacteristic
279C22×C4#12311yescharacteristic
280C22×C4#12311yescharacteristic
281C22×C4#12311yescharacteristic
282C22×C4#12311yescharacteristic
283C22×C4#12311yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4#12311yescharacteristic
286C22×C4#12411yescharacteristic
287C22×C4#12411yescharacteristic
288C22×C4#12411yescharacteristic
289C22×C4#12611yescharacteristic
290C22×C4#12611yescharacteristic
291M16—82no—
292M16—82no—
293M16—82no—
294M16—82no—
295M16—82no—
296M16—82no—
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—82no—
304C2×C8—82no—
305C2×C8—82no—
306C2×C8—82no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—81no—
324C22⋊C4—81no—
325C22⋊C4—81no—
326C22⋊C4—81no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—82no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23#38611yescharacteristic
360C23#38811yescharacteristic
361C23#38811yescharacteristic
362C23#39411yescharacteristic
363C23#39711yescharacteristic
364C23#39711yescharacteristic
365C23#39711yescharacteristic
366C23#47111yescharacteristic
367C23#47111yescharacteristic
368C23#47211yescharacteristic
369C23#47311yescharacteristic
370Q8—44no—
371Q8—44no—
372Q8—82no—
373Q8—82no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—164no—
420C2×C4—164no—
421C2×C4#38111yescharacteristic
422C2×C4#38611yescharacteristic
423C2×C4#38811yescharacteristic
424C2×C4#38811yescharacteristic
425C2×C4#39411yescharacteristic
426C2×C4#39411yescharacteristic
427C2×C4#39411yescharacteristic
428C2×C4#39711yescharacteristic
429C8—82no—
430C8—82no—
431C8—82no—
432C8—82no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22#144011yescharacteristic
467C22#190711yescharacteristic
468C22#190811yescharacteristic
469C22#205111yescharacteristic
470C22#205211yescharacteristic
471C22#207811yescharacteristic
472C22#207911yescharacteristic
473C22#208212yes—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—82no—
482C4—82no—
483C4—162no—
484C4—162no—
485C4#191312yes—
486C2—21no—
487C2—21no—
488C2—22no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—41no—
494C2—41no—
495C2—41no—
496C2#1117211yescharacteristic
497C2#1118211yescharacteristic
498C2#1119211yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1894C4semidirect

Maximal subgroups

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

subgroupindex
#217652
C2×#14722
#81072
#80572
C2×#12892