Finite Group Databasebeta

← #204521

Subgroups — #204521

Order 512.

1019 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34631C211yesmaximal, characteristic
2#21350C211yesmaximal, characteristic, complemented
3#20547C211yesmaximal, characteristic, complemented
4C2×#1483C211yesmaximal, characteristic
5C2×#1485C211yesmaximal, characteristic, complemented
6C2×#1482C211yesmaximal, characteristic
7C2×#1481C211yesmaximal, characteristic, complemented
8#9677C214yesmaximal, complemented
9#9676C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#529C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2664—22no—
17#2663—22no—
18#2660C411yescharacteristic
19#2658C411yescharacteristic
20#2642C411yescharacteristic
21#2640C411yescharacteristic
22#2621C412yes—
23#2620C412yes—
24#2396—22no—
25#2395—22no—
26C4×C4⋊Q8C2212yes—
27C4×#124C2212yes—
28C4×#117C2211yescharacteristic
29#1940—22no—
30#1851—22no—
31#1770—22no—
32#1767C2214yes—
33C4×C4⋊C8C2214yes—
34#1752—22no—
35#1485—24no—
36#1483—24no—
37#1482—24no—
38#1481—24no—
39C2×Q8⋊C8C2214yes—
40C2×Q8⋊C8C2212yes—
41C2×Q8⋊C8C2214yes—
42C2×Q8⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45C22×M16—21no—
46C23×C8—21no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C412yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54#529—22no—
55#529—22no—
56#529—22no—
57#529—22no—
58C2×#126—21no—
59C2×#125—21no—
60C2×#125C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#124C2×C412yes—
63C2×#123—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8C2×C412yes—
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×C4×Q8—21no—
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2312yes—
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C2×#117—21no—
75C2×#117C2×C412yes—
76C22×C42C2311yescharacteristic
77#431—22no—
78#431—22no—
79#431—22no—
80#431D414yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8D414yes—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C4×M16—22no—
91C4×M16—22no—
92C2×#97—21no—
93C2×#97—24no—
94C2×C4×C8—21no—
95C2×C4×C8C2312yes—
96C2×C4×C8C2312yes—
97#400—22no—
98#400—22no—
99#399—22no—
100#399—22no—
101#398—24no—
102#398—22no—
103#397—24no—
104#397—22no—
105#396—24no—
106#395—22no—
107#394—24no—
108#393—22no—
109#391C2×C412yes—
110#391C2×C412yes—
111#391C2×C412yes—
112#390C2×C412yes—
113#389—24no—
114#388—24no—
115#384C2×C412yes—
116#382C2×C412yes—
117C4×C4⋊C4—24no—
118C4×C4⋊C4D412yes—
119C4×C4⋊C4D412yes—
120C4×C22⋊C4—24no—
121C4×C22⋊C4D412yes—
122C4×C22⋊C4D412yes—
123#376C2×C411yescharacteristic
124#376C2×C411yescharacteristic
125C43C2312yes—
126#336—24no—
127#336—24no—
128Q8⋊C8—28no—
129Q8⋊C8—24no—
130Q8⋊C8—24no—
131Q8⋊C8—28no—
132D4⋊C8—24no—
133D4⋊C8—24no—
134C22×Q8—21no—
135C22×Q8C22×C411yescharacteristic
136C22×Q8C22×C412yes—
137C22×D4—21no—
138C22×D4C22×C411yescharacteristic
139C23×C4—21no—
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C2×M16—44no—
144C2×M16—42no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—24no—
149C22×C8—22no—
150C22×C8—82no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—44no—
156#126—42no—
157#126—42no—
158#125—22no—
159#125—22no—
160#125—42no—
161#125—42no—
162#124—22no—
163#124—22no—
164#124—44no—
165#123—42no—
166#123—42no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173#121—22no—
174#121—22no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—42no—
190C4×D4—42no—
191#117—44no—
192#117—42no—
193#117—42no—
194#117—42no—
195#117—42no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4C22⋊C414yes—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C412yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4C22⋊C414yes—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42C22×C412yes—
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D414yes—
226C2×C42C2×D414yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42C2411yescharacteristic
234C4⋊C8—24no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—48no—
238C4⋊C8—42no—
239C4⋊C8C2×D414yes—
240C4⋊C8C2×D414yes—
241C22⋊C8—24no—
242C22⋊C8—42no—
243#97—42no—
244#97—84no—
245C4×C8—24no—
246C4×C8—22no—
247C4×C8—22no—
248C4×C8—22no—
249C4×C8—22no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258C24—21no—
259C24C2×C22⋊C411yescharacteristic
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—42no—
268C2×Q8—82no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—24no—
272C2×D4—42no—
273C2×D4—82no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4C2×C22⋊C414yes—
288C22×C4C2×C22⋊C414yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C4×D412yes—
293C22×C4C4×D412yes—
294C22×C4#12112yes—
295C22×C4#12112yes—
296C22×C4#12412yes—
297C22×C4#12712yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301M16—48no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—44no—
309C2×C8—42no—
310C2×C8—44no—
311C2×C8—42no—
312C2×C8—84no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—44no—
320C4⋊C4—48no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—84no—
333C22⋊C4—44no—
334C22⋊C4—48no—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—84no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—24no—
347C42—44no—
348C42—41no—
349C42—41no—
350C42—42no—
351C42—42no—
352C42C4×D412yes—
353C42C4×D412yes—
354C42#12112yes—
355C42#12112yes—
356C42#12412yes—
357C42#12712yes—
358C42C22×D411yescharacteristic
359C42C22×D411yescharacteristic
360C42C2×#4111yescharacteristic
361C42C2×#4111yescharacteristic
362C23—22no—
363C23—22no—
364C23—22no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—41no—
369C23#39012yes—
370C23C22×C22⋊C411yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—44no—
374Q8—44no—
375Q8—84no—
376D4—42no—
377D4—42no—
378D4—84no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—82no—
410C2×C4#39012yes—
411C2×C4#39012yes—
412C2×C4#39012yes—
413C2×C4C2×C4×D411yescharacteristic
414C2×C4C2×C4×D411yescharacteristic
415C2×C4C2×#12111yescharacteristic
416C2×C4C2×#12111yescharacteristic
417C2×C4C2×#12411yescharacteristic
418C2×C4C2×#12711yescharacteristic
419C2×C4#56814yes—
420C2×C4#57712yes—
421C2×C4#57812yes—
422C8—44no—
423C8—44no—
424C8—44no—
425C8—82no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—44no—
430C22—81no—
431C22—81no—
432C22—84no—
433C22—81no—
434C22C2×#39011yescharacteristic
435C22#287912yes—
436C22#289812yes—
437C22#290112yes—
438C4—21no—
439C4—21no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—44no—
444C4—44no—
445C4—81no—
446C4—82no—
447C4—81no—
448C4—82no—
449C4—81no—
450C4#293212yes—
451C4#293212yes—
452C4#304211yescharacteristic
453C4#304411yescharacteristic
454C4#304612yes—
455C2—42no—
456C2—161no—
457C2#1750614yes—
458C2#2118811yescharacteristic
459C2#2125311yescharacteristic
460C2#2126211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21350C2semidirect
3. #20547C2semidirect
5. C2×#1485C2semidirect
7. C2×#1481C2semidirect
8. #9677C2semidirect
9. #9676C2semidirect

Maximal subgroups

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

subgroupindex
#346312
#213502
#205472
C2×#14832
C2×#14852
C2×#14822
C2×#14812
#96772
#96762