Finite Group Databasebeta

← #97021

Subgroups — #97021

Order 512.

1608 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1608 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2311C211yesmaximal, characteristic
2#8300C212yesmaximal, complemented
3#7595C212yesmaximal, complemented
4#7592C212yesmaximal, complemented
5C23×#41C411yescharacteristic
6C22×#117C2211yescharacteristic
7C23×C4⋊C4C411yescharacteristic
8#2311—24no—
9#2311—24no—
10C2×#391C411yescharacteristic
11C2×#391C2211yescharacteristic
12C2×#390C411yescharacteristic
13C2×#390C2211yescharacteristic
14C2×#386—22no—
15C2×#380—22no—
16#1817C2214yes—
17C2×#324—22no—
18C2×#323—22no—
19#1272—24no—
20#1270—24no—
21C22×#41—22no—
22C22×#41—22no—
23C22×#41—22no—
24C22×#41—22no—
25C22×#41—22no—
26C22×#41—42no—
27C22×#41—42no—
28C22×#41—42no—
29C22×#41—42no—
30C22×#41C814yescomplemented
31C23×Q8C2×C411yescharacteristic
32C23×D4—21no—
33C23×D4C2×C411yescharacteristic
34C24×C4—21no—
35C24×C4—21no—
36C24×C4C2×C411yescharacteristic
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—42no—
40C2×#117—42no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—42no—
45C22×C4⋊C4—42no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4D412yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D412yes—
52C22×C42C2×C411yescharacteristic
53C22×C42C2311yescharacteristic
54C2×C22⋊C8—24no—
55#391—22no—
56#391—22no—
57#391—22no—
58#391D414yes—
59#390—22no—
60#390—22no—
61#390—22no—
62#390D414yes—
63#386—216no—
64#380—216no—
65C2×#95—22no—
66C2×#95—22no—
67#336—24no—
68#336—24no—
69#324—44no—
70#323—44no—
71C25—21no—
72C25C22⋊C411yescharacteristic
73C2×#41—28no—
74C2×#41—28no—
75C2×#41—28no—
76C2×#41—28no—
77C2×#41—48no—
78C2×#41—48no—
79C2×#41—48no—
80C2×#41—48no—
81C2×#41—48no—
82C2×#41—48no—
83C2×#41—48no—
84C2×#41—48no—
85C2×#41—48no—
86C2×#41—48no—
87C2×#41—48no—
88C2×#41—48no—
89C22×Q8—21no—
90C22×Q8—21no—
91C22×Q8—21no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8—41no—
95C22×Q8—41no—
96C22×Q8C22⋊C412yes—
97C22×Q8C2×C812yes—
98C22×Q8M1612yes—
99C22×D4—24no—
100C22×D4—24no—
101C22×D4—24no—
102C22×D4—21no—
103C22×D4—21no—
104C22×D4—21no—
105C22×D4—41no—
106C22×D4—41no—
107C22×D4—41no—
108C22×D4—41no—
109C22×D4—42no—
110C22×D4—42no—
111C22×D4—42no—
112C22×D4—42no—
113C22×D4C22⋊C412yes—
114C22×D4C2×C812yes—
115C22×D4M1612yes—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—21no—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—84no—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C412yes—
140C23×C4C2×C812yes—
141C23×C4M1612yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D412yes—
144C22×C8—44no—
145#117—48no—
146#117—48no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—84no—
169C2×C22⋊C4—84no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C22⋊C8—44no—
179C22⋊C8—44no—
180#95—28no—
181#95—28no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—22no—
187C24—22no—
188C24—22no—
189C24—42no—
190C24—42no—
191C24—44no—
192C24—44no—
193C24—41no—
194C24—41no—
195C24—41no—
196C24—41no—
197C24C22⋊C812yes—
198C24#9912yes—
199C24#10012yes—
200C24C2×C22⋊C411yescharacteristic
201#41—432no—
202#41—432no—
203#41—432no—
204#41—432no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8C22⋊C814yes—
221C2×D4—216no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—48no—
238C2×D4—48no—
239C2×D4—48no—
240C2×D4—48no—
241C2×D4—48no—
242C2×D4—48no—
243C2×D4C22⋊C814yes—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—216no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—48no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—48no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—88no—
301C22×C4—88no—
302C22×C4—88no—
303C22×C4C22⋊C812yes—
304C22×C4#9912yes—
305C22×C4#10012yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4#12014yes—
309C22×C4C22×C811yescharacteristic
310C22×C4C2×M1611yescharacteristic
311C22×C4C2×M1611yescharacteristic
312C22×C4C2×M1611yescharacteristic
313C2×C8—84no—
314C2×C8—84no—
315C2×C8—84no—
316C4⋊C4—48no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—48no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C22⋊C4—48no—
323C22⋊C4—88no—
324C22⋊C4—88no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—24no—
338C23—42no—
339C23—44no—
340C23—44no—
341C23—42no—
342C23—42no—
343C23—44no—
344C23—42no—
345C23—44no—
346C23—48no—
347C23—48no—
348C23—42no—
349C23—42no—
350C23—44no—
351C23—44no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—84no—
359C23—84no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23#32314yes—
365C23#37911yescharacteristic
366C23C2×C22⋊C811yescharacteristic
367C23#40711yescharacteristic
368C23C2×#9911yescharacteristic
369C23C2×#9911yescharacteristic
370C23C2×#10011yescharacteristic
371C23C2×#10011yescharacteristic
372Q8—44no—
373Q8—44no—
374Q8—44no—
375Q8—44no—
376D4—44no—
377D4—44no—
378D4—44no—
379D4—44no—
380D4—432no—
381C2×C4—28no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—432no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—816no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4#32314yes—
431C2×C4C2×C22⋊C811yescharacteristic
432C2×C4C2×C22⋊C811yescharacteristic
433C2×C4#40711yescharacteristic
434C2×C4#40711yescharacteristic
435C8—164no—
436C22—22no—
437C22—22no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—22no—
443C22—22no—
444C22—24no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—42no—
451C22—42no—
452C22—44no—
453C22—44no—
454C22—48no—
455C22—84no—
456C22—84no—
457C22—84no—
458C22—84no—
459C22—84no—
460C22—82no—
461C22—82no—
462C22C2×#32311yescharacteristic
463C22C2×#32311yescharacteristic
464C22#146211yescharacteristic
465C22#146211yescharacteristic
466C22#177911yescharacteristic
467C22#178111yescharacteristic
468C22#178311yescharacteristic
469C22#179512yes—
470C22#179612yes—
471C4—24no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—44no—
477C4—44no—
478C4—82no—
479C4—82no—
480C4—82no—
481C2—21no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—44no—
486C2—84no—
487C2#829811yescharacteristic
488C2#832412yes—
489C2#851712yes—
490C2#852211yescharacteristic
491C2#852411yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC23×#41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8300C2semidirect
3. #7595C2semidirect
4. #7592C2semidirect
30. C22×#41C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
#83002
#75952
#75922