Finite Group Databasebeta

← #192211

Subgroups — #192211

Order 512.

1082 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2#21948C212yesmaximal, complemented
3#21740C211yesmaximal, characteristic
4C2×#1459C211yesmaximal, characteristic
5#8860C214yesmaximal, complemented
6#8859C214yesmaximal, complemented
7C2×#1298C212yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9#2975C2212yes—
10#2974—24no—
11#2973—21no—
12#2973C2212yes—
13#2972—21no—
14D4×M16—24no—
15#2920—21no—
16C2×#407—21no—
17C2×#407C2212yes—
18C2×#407C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21#1870C2212yes—
22#1870C2212yes—
23#1869—21no—
24#1869—21no—
25#1855—24no—
26#1855C2218yescomplemented
27#1817—21no—
28#1810—24no—
29#1779—21no—
30#1459C412yes—
31#1459C412yes—
32#1459C2212yes—
33#1459C2212yes—
34C2×#324C412yes—
35C2×#323C412yes—
36#1298C2218yescomplemented
37C23×D4Q811yescharacteristic
38C24×C4D411yescharacteristic
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41C22×M16D414yes—
42C22×M16C2312yes—
43C23×C8—22no—
44C23×C8—21no—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4C2×C411yescharacteristic
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4Q811yescharacteristic
54C2×C4×D4Q811yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4Q811yescharacteristic
60C22×C42D411yescharacteristic
61#436—28no—
62#435—28no—
63#435—28no—
64C8×D4—28no—
65#432—22no—
66#432—22no—
67#431—22no—
68#431—22no—
69C2×#108—28no—
70#423—22no—
71#407—22no—
72#407—22no—
73#407—22no—
74#407—22no—
75#407—22no—
76#407—22no—
77#407—22no—
78#407D414yes—
79#407C2314yes—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8C2×C414yes—
95C2×C22⋊C8C2×C414yes—
96C2×C22⋊C8C2312yes—
97C2×C22⋊C8C2314yes—
98C4×M16—22no—
99#341—28no—
100#336—22no—
101#336—22no—
102#324C2×C414yes—
103#324C2×C414yes—
104#323C2×C414yes—
105#323C2×C414yes—
106C25C4⋊C411yescharacteristic
107C25C2×Q811yescharacteristic
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—24no—
111C22×D4—42no—
112C22×D4C4⋊C411yescharacteristic
113C22×D4C4⋊C411yescharacteristic
114C22×D4C2×Q811yescharacteristic
115C22×D4C2×Q811yescharacteristic
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—82no—
121C23×C4—82no—
122C23×C4C4⋊C411yescharacteristic
123C23×C4C4⋊C411yescharacteristic
124C23×C4C4⋊C411yescharacteristic
125C23×C4C4⋊C411yescharacteristic
126C23×C4C4⋊C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×Q811yescharacteristic
136C23×C4#4112yes—
137C23×C4#4112yes—
138C23×C4C2411yescharacteristic
139C2×M16—24no—
140C2×M16—24no—
141C2×M16—24no—
142C2×M16—82no—
143C2×M16—84no—
144C2×M16—84no—
145C2×M16C2×D418yes—
146C22×C8—24no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—24no—
150C22×C8—24no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—42no—
154C22×C8—84no—
155C22×C8—82no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C4⋊C411yescharacteristic
175C2×C4⋊C4C4⋊C411yescharacteristic
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4C4⋊C411yescharacteristic
183C2×C22⋊C4C4⋊C411yescharacteristic
184C2×C22⋊C4C4⋊C411yescharacteristic
185C2×C22⋊C4C4⋊C411yescharacteristic
186C2×C22⋊C4C22×C411yescharacteristic
187C2×C22⋊C4C22×C411yescharacteristic
188C2×C22⋊C4C2×Q811yescharacteristic
189C2×C22⋊C4C2×Q811yescharacteristic
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42C4⋊C411yescharacteristic
195C2×C42C4⋊C411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198#108—88no—
199C4⋊C8—24no—
200C4⋊C8—24no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8—24no—
204C22⋊C8—24no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208C22⋊C8—24no—
209C22⋊C8—24no—
210C22⋊C8—24no—
211C22⋊C8—41no—
212C22⋊C8—42no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216C22⋊C8—82no—
217C22⋊C8—82no—
218C22⋊C8C22×C418yes—
219#97—24no—
220C4×C8—24no—
221C24—22no—
222C24—22no—
223C24—21no—
224C24—21no—
225C24—22no—
226C24—41no—
227C24—42no—
228C24—42no—
229C24—81no—
230C24—81no—
231C24C2×C4⋊C411yescharacteristic
232C24C2×C4⋊C411yescharacteristic
233C24C2×C4⋊C411yescharacteristic
234C24C2×C4⋊C411yescharacteristic
235C24C22⋊Q812yes—
236C24C22⋊Q812yes—
237C24C22×Q811yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—48no—
250C2×D4—48no—
251C2×D4#10814yescomplemented
252C22×C4—21no—
253C22×C4—21no—
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—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—82no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4C2×C4⋊C411yescharacteristic
289C22×C4C2×C4⋊C411yescharacteristic
290C22×C4C2×C4⋊C411yescharacteristic
291C22×C4C2×C4⋊C411yescharacteristic
292C22×C4C2×C4⋊C411yescharacteristic
293C22×C4C2×C4⋊C411yescharacteristic
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C2×C4⋊C411yescharacteristic
296C22×C4C4×D412yes—
297C22×C4C4×D412yes—
298C22×C4C4×D412yes—
299C22×C4C4×D412yes—
300C22×C4#12011yescharacteristic
301C22×C4#12012yes—
302C22×C4#12011yescharacteristic
303C22×C4C22⋊Q812yes—
304C22×C4C22⋊Q812yes—
305C22×C4#12311yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4#12312yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D412yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4112yes—
312M16—48no—
313M16—48no—
314M16—48no—
315M16—84no—
316C2×C8—28no—
317C2×C8—28no—
318C2×C8—48no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—84no—
326C2×C8—84no—
327C2×C8—84no—
328C2×C8—84no—
329C2×C8—84no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C23—24no—
351C23—22no—
352C23—24no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—24no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—41no—
367C23—42no—
368C23—41no—
369C23—42no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—42no—
379C23—81no—
380C23—81no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23—84no—
385C23#38511yescharacteristic
386C23#38512yes—
387C23#38511yescharacteristic
388C23C2×#10812yes—
389C23C2×#10812yes—
390C23#43012yes—
391C23C22×C4⋊C411yescharacteristic
392C23C2×C22⋊Q812yes—
393D4—28no—
394D4—44no—
395D4—44no—
396D4—44no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4#38511yescharacteristic
443C2×C4#38511yescharacteristic
444C2×C4#38512yes—
445C2×C4C2×#10812yes—
446C2×C4C2×C4×D412yes—
447C2×C4C2×#12011yescharacteristic
448C2×C4C2×#12311yescharacteristic
449C8—48no—
450C8—48no—
451C8—82no—
452C8—82no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—21no—
457C22—21no—
458C22—22no—
459C22—21no—
460C22—22no—
461C22—22no—
462C22—21no—
463C22—22no—
464C22—21no—
465C22—22no—
466C22—42no—
467C22—42no—
468C22—48no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—84no—
482C22—84no—
483C22—81no—
484C22—84no—
485C22#185514yes—
486C22#185812yes—
487C22C2×#38511yescharacteristic
488C22C22×#10811yescharacteristic
489C22C2×#43011yescharacteristic
490C4—22no—
491C4—22no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—22no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4#302612yes—
502C2—22no—
503C2—21no—
504C2—21no—
505C2—24no—
506C2—42no—
507C2—41no—
508C2—42no—
509C2—81no—
510C2C2×#185511yescharacteristic
511C2C2×#185811yescharacteristic
512C2#2088911yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
5. #8860C2semidirect
6. #8859C2semidirect
7. C2×#1298C2semidirect
26. #1855C22semidirect
36. #1298C22semidirect
251. C2×D4#108semidirect

Maximal subgroups

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

subgroupindex
#602302
#219482
#217402
C2×#14592
#88602
#88592
C2×#12982