Finite Group Databasebeta

← #96911

Subgroups — #96911

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2#10712C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1271C214yesmaximal, complemented
5C23×C4⋊C4—21no—
6C23×C4⋊C4C411yescharacteristic
7C23×C4⋊C4C411yescharacteristic
8C23×C42C2211yescharacteristic
9C22×C22⋊C8—21no—
10C22×C22⋊C8—21no—
11C2×#389—21no—
12C2×#389—21no—
13C2×#389—28no—
14C2×#389C412yes—
15C2×#389C2212yes—
16C22×#95—21no—
17C2×#336—21no—
18C2×#336—21no—
19C2×#336C2212yes—
20C2×#336C2212yes—
21#1271—28no—
22C24×C4—21no—
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C23×C8—22no—
28C23×C8—22no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—24no—
32C22×C4⋊C4—24no—
33C22×C4⋊C4—28no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—24no—
38C22×C4⋊C4—42no—
39C22×C4⋊C4—44no—
40C22×C4⋊C4—42no—
41C22×C4⋊C4—42no—
42C22×C4⋊C4—42no—
43C22×C4⋊C4C814yescomplemented
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C412yes—
46C22×C42—24no—
47C22×C42—42no—
48C22×C42—42no—
49C22×C42C2×C411yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42C2311yescharacteristic
53C2×C22⋊C8—24no—
54C2×C22⋊C8—24no—
55C2×C22⋊C8—42no—
56C2×C22⋊C8—42no—
57C2×C22⋊C8—42no—
58C2×C22⋊C8—42no—
59#389—28no—
60#389—28no—
61#389—24no—
62#389—24no—
63#389—24no—
64#389D418yes—
65C2×#95—28no—
66C2×#95—22no—
67C2×#95—22no—
68#336—24no—
69#336—24no—
70#336—42no—
71#336—42no—
72C25C22⋊C411yescharacteristic
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—24no—
80C23×C4—24no—
81C23×C4—24no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—22no—
93C23×C4—41no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4—41no—
97C23×C4—42no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—42no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C2×C812yes—
111C23×C4M1612yes—
112C23×C4D812yes—
113C23×C4SD1612yes—
114C23×C4SD1612yes—
115C23×C4Q1612yes—
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C22×C8—24no—
120C22×C8—24no—
121C22×C8—42no—
122C22×C8—42no—
123C22×C8—42no—
124C22×C8—42no—
125C22×C8—42no—
126C22×C8—42no—
127C22×C8—42no—
128C22×C8—42no—
129C22×C8—42no—
130C22×C8—42no—
131C22×C8—42no—
132C22×C8—42no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—216no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—216no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—416no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—416no—
151C2×C4⋊C4—416no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C22⋊C414yes—
159C2×C4⋊C4C2×C814yes—
160C2×C4⋊C4M1614yes—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42—416no—
173C2×C42C2×D414yes—
174C22⋊C8—48no—
175C22⋊C8—48no—
176#95—28no—
177#95—28no—
178C24—21no—
179C24—21no—
180C24—21no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24C22⋊C812yes—
187C24#9912yes—
188C24#10112yes—
189C24#10212yes—
190C24#10312yes—
191C24#10412yes—
192C24#10412yes—
193C24C2×C22⋊C411yescharacteristic
194C22×C4—28no—
195C22×C4—28no—
196C22×C4—28no—
197C22×C4—28no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—216no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—48no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—48no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—84no—
277C22×C4—84no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4C22⋊C812yes—
281C22×C4#9912yes—
282C22×C4#10112yes—
283C22×C4#10212yes—
284C22×C4#10312yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4#12012yes—
287C22×C4#12012yes—
288C22×C4C22×C811yescharacteristic
289C22×C4C2×M1612yes—
290C22×C4C2×M1611yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×SD1611yescharacteristic
293C22×C4C2×SD1611yescharacteristic
294C22×C4C2×Q1611yescharacteristic
295C22×C4AGL(1,ℤ/8)12yes—
296C22×C4#13712yes—
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—44no—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—44no—
303C2×C8—82no—
304C2×C8—82no—
305C2×C8—82no—
306C2×C8—82no—
307C2×C8—82no—
308C2×C8—82no—
309C2×C8—82no—
310C2×C8—82no—
311C4⋊C4—216no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—44no—
316C4⋊C4—48no—
317C4⋊C4—44no—
318C4⋊C4—416no—
319C4⋊C4—48no—
320C4⋊C4—48no—
321C4⋊C4C22⋊C818yescomplemented
322C42—44no—
323C42—44no—
324C42—44no—
325C42—44no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23#32414yes—
355C23D4⋊C814yes—
356C23Q8⋊C814yes—
357C23#32714yes—
358C23#32814yes—
359C23#33014yes—
360C23#33214yes—
361C23#37911yescharacteristic
362C23C2×C22⋊C811yescharacteristic
363C23#40711yescharacteristic
364C23C2×#9911yescharacteristic
365C23C2×#9911yescharacteristic
366C23C2×#10111yescharacteristic
367C23C2×#10111yescharacteristic
368C23C2×#10211yescharacteristic
369C23C2×#10311yescharacteristic
370C23#41811yescharacteristic
371C23#41911yescharacteristic
372C23C2×#10411yescharacteristic
373C23C2×#10411yescharacteristic
374C23#42111yescharacteristic
375C23#42111yescharacteristic
376C2×C4—216no—
377C2×C4—216no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
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—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—28no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—432no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—816no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4#32414yes—
443C2×C4C2×C22⋊C812yes—
444C2×C4#40712yes—
445C2×C4#44712yes—
446C2×C4#44812yes—
447C2×C4#45012yes—
448C2×C4#45112yes—
449C8—84no—
450C8—84no—
451C22—24no—
452C22—24no—
453C22—24no—
454C22—24no—
455C22—24no—
456C22—24no—
457C22—24no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22#127118yescomplemented
466C22C2×#32411yescharacteristic
467C22C2×#32411yescharacteristic
468C22#146311yescharacteristic
469C22#146311yescharacteristic
470C22C2×D4⋊C811yescharacteristic
471C22C2×Q8⋊C811yescharacteristic
472C22#147811yescharacteristic
473C22#147911yescharacteristic
474C22#148611yescharacteristic
475C22#148711yescharacteristic
476C22#149011yescharacteristic
477C22#149111yescharacteristic
478C22C2×#32711yescharacteristic
479C22C2×#32811yescharacteristic
480C22#150711yescharacteristic
481C22#150811yescharacteristic
482C22#151011yescharacteristic
483C22#151111yescharacteristic
484C22#151911yescharacteristic
485C22#152011yescharacteristic
486C22C2×#33011yescharacteristic
487C22#152811yescharacteristic
488C22#152811yescharacteristic
489C22#153411yescharacteristic
490C22C2×#33211yescharacteristic
491C22#154611yescharacteristic
492C22#154911yescharacteristic
493C22#155611yescharacteristic
494C22#177911yescharacteristic
495C22#178111yescharacteristic
496C22#178411yescharacteristic
497C22#178611yescharacteristic
498C22#178711yescharacteristic
499C22#178911yescharacteristic
500C22#178911yescharacteristic
501C4—24no—
502C4—22no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—48no—
509C4—42no—
510C4—42no—
511C4—84no—
512C2—28no—
513C2C2×#127111yescharacteristic
514C2#757311yescharacteristic
515C2#757611yescharacteristic
516C2#759311yescharacteristic
517C2#760711yescharacteristic
518C2#761011yescharacteristic
519C2#762711yescharacteristic
520C2#764111yescharacteristic
521C2#829911yescharacteristic
522C2#831011yescharacteristic
523C2#831111yescharacteristic
524C2#832811yescharacteristic
525C2#832911yescharacteristic
526C2#833511yescharacteristic
527C2#834011yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1271C2semidirect
43. C22×C4⋊C4C8semidirect
321. C4⋊C4C22⋊C8semidirect
465. C22#1271semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
#107122
C2×#12712