Finite Group Databasebeta

← #6386234

Subgroups — #6386234

Order 512.

2731 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3502C211yesmaximal, characteristic
2#60249C212yesmaximal
3#22517C214yesmaximal, complemented
4#22517C218yesmaximal, complemented
5Q8×#101C214yesmaximal, complemented
6#21832C214yesmaximal
7#21831C212yesmaximal
8#20730C214yesmaximal
9#20727C211yesmaximal, characteristic
10#20727C211yesmaximal, characteristic
11#3502—24no—
12#3502—24no—
13#3502—24no—
14#3502—24no—
15C2×Q82C412yes—
16C2×Q82C2212yes—
17C2×#557C414yes—
18C2×#557C412yes—
19C2×#557C2214yes—
20C2×#557C2212yes—
21C2×#544C414yes—
22C2×#544C2212yes—
23C2×#533C414yes—
24C2×#533C2211yescharacteristic
25C2×#533C2211yescharacteristic
26C2×#519C2212yes—
27C22×C4×Q8C2211yescharacteristic
28#2987—24no—
29#2986—24no—
30#2985—24no—
31#2985—24no—
32Q8×M16C2218yes—
33Q8×M16C2218yes—
34#2921C2218yes—
35#2920C2212yes—
36#2920C2212yes—
37#1998C2218yes—
38#1998C22116yes—
39#1997C2214yes—
40#1997C2214yes—
41#1997C22116yes—
42#1822C2218yes—
43#1821C2212yes—
44#1821C2212yes—
45#1801C2214yes—
46#1801C2218yes—
47#1801C2214yes—
48#1801C2218yes—
49#1784C2212yes—
50#1784C2212yes—
51C4×#101C2214yes—
52C4×#101C2218yes—
53C4×#101C2214yes—
54C4×#101C2218yes—
55C23×Q8C2311yescharacteristic
56Q82—28no—
57Q82—28no—
58Q82—28no—
59Q82—28no—
60#557—28no—
61#557—28no—
62#557—216no—
63#557—216no—
64#557—216no—
65#557—216no—
66#557—28no—
67#557—28no—
68#544—216no—
69#544—216no—
70#544—28no—
71#544—28no—
72#533—28no—
73#533—28no—
74#533—28no—
75#533—28no—
76#533—22no—
77#533—22no—
78#533—22no—
79#533—22no—
80#533D414yes—
81#533D414yes—
82#533D414yes—
83#533D414yes—
84C2×C4⋊Q8C2×C414yes—
85C2×C4⋊Q8C2×C414yes—
86C2×C4⋊Q8C2×C418yes—
87C2×C4⋊Q8C2×C418yes—
88C2×C4⋊Q8C2×C414yes—
89C2×C4⋊Q8C2×C411yescharacteristic
90C2×C4⋊Q8C2×C411yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2314yes—
94C2×#125C2×C418yes—
95C2×#125C2×C418yes—
96C2×#125C2×C418yes—
97C2×#125C2×C414yes—
98C2×#125C2×C412yes—
99C2×#125C2312yes—
100C2×#125C2314yes—
101C2×C22⋊Q8C2×C412yes—
102C2×C22⋊Q8C2×C418yes—
103C2×C22⋊Q8C2×C412yes—
104C2×C22⋊Q8C2×C412yes—
105C2×C22⋊Q8C2×C412yes—
106#519—28no—
107#519—28no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8C2×C412yes—
113C2×C4×Q8C2×C412yes—
114C2×C4×Q8C2×C414yes—
115C2×C4×Q8C2×C414yes—
116C2×C4×Q8C2×C418yes—
117C2×C4×Q8C2×C414yes—
118C2×C4×Q8C2×C414yes—
119C2×C4×Q8C2×C412yes—
120C2×C4×Q8C2×C412yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2314yes—
124C2×C4×Q8C2314yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×#117C2×C418yes—
128C2×#117C2312yes—
129C2×#117C2312yes—
130C22×C4⋊C4C2×C412yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C42C2×C412yes—
133C22×C42C2311yescharacteristic
134#446—28no—
135#446—28no—
136#446—216no—
137#446—216no—
138C8×Q8—28no—
139C8×Q8—28no—
140#433—216no—
141#433—216no—
142#432—22no—
143#432—22no—
144#432—22no—
145#432—22no—
146#431—22no—
147#431—22no—
148#431—22no—
149#431—22no—
150#423C2314yes—
151#423C2314yes—
152#423C23116yes—
153C2×#101C2318yes—
154C2×#101C2314yes—
155C2×#101C2314yes—
156#407C2314yes—
157#407C2314yes—
158C4×M16C2314yes—
159C4×M16C2314yes—
160C4×M16C23116yes—
161#343C2318yes—
162#343C2318yes—
163#343C23116yes—
164#343C23116yes—
165C22×Q8—28no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8C22×C412yes—
169C22×Q8C22×C412yes—
170C22×Q8C22×C412yes—
171C22×Q8C22×C418yes—
172C22×Q8C22×C412yes—
173C22×Q8C22×C412yes—
174C22×Q8C2412yes—
175C23×C4C22×C412yes—
176C23×C4C22×C412yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C2411yescharacteristic
180C2×M16C2418yes—
181C2×M16C2418yes—
182C4⋊Q8—28no—
183C4⋊Q8—28no—
184C4⋊Q8—232no—
185C4⋊Q8—28no—
186C4⋊Q8—28no—
187C4⋊Q8—232no—
188C4⋊Q8—216no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—216no—
194C4⋊Q8C22⋊C414yes—
195C4⋊Q8C22⋊C414yes—
196C4⋊Q8C22⋊C414yes—
197C4⋊Q8C22⋊C414yes—
198#125—216no—
199#125—216no—
200#125—232no—
201#125—232no—
202#125—216no—
203#125—24no—
204#125—24no—
205#125—216no—
206#125C22⋊C418yes—
207#125C22⋊C418yes—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C22⋊Q8—216no—
215C22⋊Q8—216no—
216C22⋊Q8C22⋊C414yes—
217C22⋊Q8C22⋊C414yes—
218C22⋊Q8C22⋊C414yes—
219C22⋊Q8C22⋊C414yes—
220C22⋊Q8C22⋊C418yes—
221C22⋊Q8C22⋊C418yes—
222C4×Q8—28no—
223C4×Q8—28no—
224C4×Q8—216no—
225C4×Q8—216no—
226C4×Q8—216no—
227C4×Q8—216no—
228C4×Q8—28no—
229C4×Q8—28no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—216no—
243C4×Q8—216no—
244C4×Q8C2×D418yes—
245C4×Q8C2×D418yes—
246C4×Q8C2×D418yes—
247C4×Q8C2×D418yes—
248#117—24no—
249#117—24no—
250#117—216no—
251#117—216no—
252#117C2×D418yes—
253#117C2×D418yes—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4C22×C414yes—
259C2×C4⋊C4C22×C414yes—
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C22×C418yes—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C418yes—
270C2×C4⋊C4C22×C412yes—
271C2×C4⋊C4C22×C412yes—
272C2×C4⋊C4C22×C414yes—
273C2×C4⋊C4C22×C412yes—
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C22×C414yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C22⋊C4C22×C414yes—
280C2×C22⋊C4C22×C414yes—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42C22×C418yes—
286C2×C42C22×C418yes—
287C2×C42C22×C414yes—
288C2×C42C22×C414yes—
289C2×C42C22×C414yes—
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2414yes—
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299C4⋊C8—24no—
300C4⋊C8—24no—
301C4⋊C8—24no—
302C4⋊C8—24no—
303C4⋊C8—216no—
304C4⋊C8—216no—
305#101—216no—
306#101—28no—
307#101—28no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311C22⋊C8—24no—
312#97—24no—
313#97—24no—
314#97—216no—
315C4×C8—24no—
316C4×C8—24no—
317C4×C8—216no—
318C24C23×C411yescharacteristic
319C2×Q8—216no—
320C2×Q8—216no—
321C2×Q8—24no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—216no—
326C2×Q8—28no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×Q8—88no—
334C2×Q8C2×C22⋊C414yes—
335C2×Q8C2×C22⋊C414yes—
336C2×Q8C2×C22⋊C418yes—
337C2×Q82−1+418yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—28no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—81no—
346C22×C4—82no—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C23×C411yescharacteristic
355C22×C4C23×C414yes—
356C22×C4C23×C414yes—
357C22×C4C23×C414yes—
358C22×C4C23×C414yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C23×C414yes—
361C22×C4C23×C414yes—
362C22×C4C23×C411yescharacteristic
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C2511yescharacteristic
367M16—28no—
368M16—28no—
369M16—216no—
370C2×C8—28no—
371C2×C8—28no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—24no—
375C4⋊C4—216no—
376C4⋊C4—216no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—216no—
383C4⋊C4—28no—
384C4⋊C4—216no—
385C4⋊C4—216no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4C2×C22⋊C418yes—
390C4⋊C4C2×C22⋊C418yes—
391C4⋊C4C2×C22⋊C418yes—
392C4⋊C4C2×C22⋊C418yes—
393C4⋊C4C2×C22⋊C414yes—
394C4⋊C4C2×C22⋊C414yes—
395C4⋊C4C2×C22⋊C418yes—
396C4⋊C4C2×C22⋊C4116yes—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4C2×C22⋊C418yes—
400C22⋊C4C2×C22⋊C418yes—
401C42—24no—
402C42—24no—
403C42—24no—
404C42—24no—
405C42—216no—
406C42—216no—
407C42—28no—
408C42C2×C22⋊C412yes—
409C42C2×C22⋊C412yes—
410C42C2×C22⋊C412yes—
411C42C2×C22⋊C412yes—
412C42C22×D412yes—
413C42C22×D412yes—
414C42C22×D4116yes—
415C42C22×D412yes—
416C42C22×D412yes—
417C23—22no—
418C23—41no—
419C23—41no—
420C23C22×C22⋊C411yescharacteristic
421C23C22×C22⋊C411yescharacteristic
422C23C24×C411yescharacteristic
423Q8—216no—
424Q8—28no—
425Q8—216no—
426Q8—44no—
427Q8—48no—
428Q8—44no—
429Q8—44no—
430Q8—84no—
431Q8—816no—
432C2×C4—28no—
433C2×C4—28no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4C22×C22⋊C412yes—
460C2×C4C22×C22⋊C411yescharacteristic
461C2×C4C22×C22⋊C411yescharacteristic
462C2×C4C22×C22⋊C411yescharacteristic
463C2×C4C22×C22⋊C418yes—
464C2×C4C22×C22⋊C418yes—
465C2×C4C22×C22⋊C412yes—
466C2×C4C22×C22⋊C412yes—
467C2×C4C22×C22⋊C411yescharacteristic
468C2×C4#51918yes—
469C2×C4C23×D412yes—
470C2×C4C2×2−1+414yes—
471C8—48no—
472C8—48no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—44no—
482C22C23×C22⋊C411yescharacteristic
483C22C2×#51912yes—
484C4—22no—
485C4—28no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—28no—
490C4—22no—
491C4—22no—
492C4—28no—
493C4—44no—
494C4—44no—
495C4—44no—
496C4#288912yes—
497C4#348614yes—
498C2—22no—
499C2—41no—
500C2—41no—
501C2#3417311yescharacteristic
502C2C2×#288911yescharacteristic
503C2#6007611yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22517C2semidirect
4. #22517C2semidirect
5. Q8×#101C2semidirect

Maximal subgroups

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

subgroupindex
C2×#35022
#602492
#225172
Q8×#1012
#218322
#218312
#207302
#207272