Finite Group Databasebeta

← #124167

Subgroups — #124167

Order 512.

1072 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C4×#392C211yesmaximal, characteristic
2C4×#323C211yesmaximal, characteristic, complemented
3#7907C211yesmaximal, characteristic, complemented
4#7123C212yesmaximal, complemented
5#7103C212yesmaximal, complemented
6C22×C4×D4—21no—
7C22×C4×D4C411yescharacteristic
8C22×C22⋊C8—21no—
9C2×#392C411yescharacteristic
10C2×C4×C22⋊C4—21no—
11C2×C4×C22⋊C4C2211yescharacteristic
12C4×C22⋊C8—21no—
13C2×#323—21no—
14#1444C2211yescharacteristic
15#1439—21no—
16#1439C411yescharacteristic
17#1438—21no—
18#1438C2211yescharacteristic
19C4×#95C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×D4—21no—
24C23×D4C2×C411yescharacteristic
25C24×C4—21no—
26C24×C4D411yescharacteristic
27C23×C8—41no—
28C2×C4×D4—44no—
29C2×C4×D4—41no—
30C2×C4×D4—41no—
31C2×C4×D4—44no—
32C2×C4×D4—42no—
33C2×C4×D4—42no—
34C2×C4×D4—44no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C22⋊C4—21no—
40C22×C22⋊C4—21no—
41C22×C22⋊C4—21no—
42C22×C22⋊C4C2×C411yescharacteristic
43C22×C22⋊C4D411yescharacteristic
44C22×C42—21no—
45C22×C42C2×C411yescharacteristic
46C22×C42D411yescharacteristic
47C2×C22⋊C8—24no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—41no—
51C2×C22⋊C8—42no—
52C2×C4×C8—41no—
53#392—22no—
54#392C814yescomplemented
55C4×C22⋊C4—41no—
56C4×C22⋊C4—41no—
57C4×C22⋊C4—44no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95C2×C411yescharacteristic
62C2×#95C2×C411yescharacteristic
63C2×#95D411yescharacteristic
64C2×#95C2311yescharacteristic
65#336—41no—
66#323—24no—
67C25—21no—
68C25C22⋊C411yescharacteristic
69C22×D4—24no—
70C22×D4—21no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4—42no—
74C22×D4—42no—
75C22×D4—42no—
76C22×D4—42no—
77C22×D4—42no—
78C22×D4—42no—
79C22×D4—42no—
80C22×D4—42no—
81C22×D4C2×C812yescomplemented
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—24no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4D812yes—
102C23×C4SD1612yes—
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C22×C8—42no—
107C22×C8—44no—
108C22×C8—42no—
109C22×C8—82no—
110C22×C8—82no—
111C4×D4—44no—
112C4×D4—416no—
113C4×D4—416no—
114C4×D4—44no—
115C4×D4—44no—
116C4×D4—44no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4C2×C812yes—
145C2×C22⋊C4M1612yes—
146C2×C42—24no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—42no—
152C2×C42—42no—
153C2×C42—42no—
154C2×C42—42no—
155C22⋊C8—24no—
156C22⋊C8—24no—
157C22⋊C8—44no—
158C22⋊C8—42no—
159C22⋊C8—42no—
160C4×C8—82no—
161#95—21no—
162#95—41no—
163#95—41no—
164#95—41no—
165#95—42no—
166#95—42no—
167#95—42no—
168#95C2×C812yescomplemented
169C24—21no—
170C24—21no—
171C24—21no—
172C24—21no—
173C24—22no—
174C24—22no—
175C24—22no—
176C24—41no—
177C24—41no—
178C24—41no—
179C24—41no—
180C24—41no—
181C24—41no—
182C24—41no—
183C24—41no—
184C24—41no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24—41no—
189C24—41no—
190C24—41no—
191C24—41no—
192C24C22⋊C812yes—
193C24#10212yes—
194C24#10412yes—
195C24C2×C22⋊C411yescharacteristic
196C2×D4—42no—
197C2×D4—48no—
198C2×D4—42no—
199C2×D4—48no—
200C2×D4—44no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—48no—
204C2×D4—42no—
205C2×D4—48no—
206C2×D4—48no—
207C2×D4—48no—
208C2×D4—48no—
209C2×D4—42no—
210C2×D4—48no—
211C2×D4—44no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—48no—
238C22×C4—48no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—44no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—84no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—82no—
272C22×C4—84no—
273C22×C4—84no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4C22⋊C812yes—
278C22×C4#9911yescharacteristic
279C22×C4#9911yescharacteristic
280C22×C4#10212yes—
281C22×C4#10412yes—
282C22×C4C22×C811yescharacteristic
283C22×C4C2×M1611yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×SD1611yescharacteristic
286C22×C4#13512yes—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—82no—
290C2×C8—82no—
291C2×C8—82no—
292C2×C8—82no—
293C2×C8—88no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—44no—
301C22⋊C4—48no—
302C22⋊C4—48no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—48no—
306C22⋊C4—48no—
307C42—42no—
308C42—48no—
309C42—42no—
310C42—42no—
311C42—81no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—24no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23D4⋊C814yes—
374C23C2×C22⋊C811yescharacteristic
375C23C2×#9911yescharacteristic
376C23#41311yescharacteristic
377C23C2×#10211yescharacteristic
378C23#41611yescharacteristic
379C23C2×#10411yescharacteristic
380C23C2×#10411yescharacteristic
381D4—44no—
382D4—432no—
383D4—44no—
384D4—44no—
385D4—44no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—88no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—88no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4#32712yes—
447C2×C4#32712yes—
448C2×C4AΣL(1,𝔽16)12yes—
449C2×C4#35312yes—
450C8—84no—
451C8—84no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—24no—
465C22—44no—
466C22—44no—
467C22—44no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—44no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22#145911yescharacteristic
491C22C2×D4⋊C811yescharacteristic
492C22#147611yescharacteristic
493C22C2×#32711yescharacteristic
494C22#150111yescharacteristic
495C22#152411yescharacteristic
496C22#154011yescharacteristic
497C22C2×AΣL(1,𝔽16)11yescharacteristic
498C22#212011yescharacteristic
499C22C2×#35311yescharacteristic
500C22#213211yescharacteristic
501C4—22no—
502C4—44no—
503C4—41no—
504C4—41no—
505C4—42no—
506C4—41no—
507C4—41no—
508C4—42no—
509C4—82no—
510C4—82no—
511C4—81no—
512C4#134314yes—
513C2—21no—
514C2—21no—
515C2—21no—
516C2—21no—
517C2—44no—
518C2—84no—
519C2#756511yescharacteristic
520C2#1147111yescharacteristic
521C2#1147511yescharacteristic
522C2C2×#134311yescharacteristic
523C2#1169711yescharacteristic
524C2#1172611yescharacteristic
525C2#1173311yescharacteristic
subgroupG/H
centerC22×C4#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
socleC23C2×#99
prime radicalC2×#392C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#323C2semidirect
3. #7907C2semidirect
4. #7123C2semidirect
5. #7103C2semidirect
54. #392C8semidirect
81. C22×D4C2×C8semidirect
168. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3922
C4×#3232
#79072
#71232
#71032