Finite Group Databasebeta

← #134449

Subgroups — #134449

Order 512.

736 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 114 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×#399C211yesmaximal, characteristic
2C2×#1352C212yesmaximal
3#10846C211yesmaximal, characteristic
4#8557C211yesmaximal, characteristic
5#7250C212yesmaximal
6C2×#399C2211yescharacteristic
7C2×#380—21no—
8C2×C4×C4⋊C4—21no—
9C2×C4×C4⋊C4C411yescharacteristic, complemented
10C2×C4×C22⋊C4—21no—
11C2×C4×C22⋊C4C2211yescharacteristic
12#1797—21no—
13C2×#342—21no—
14C2×#342C2212yes—
15C2×#323C2212yes—
16#1445—21no—
17#1445C411yescharacteristic, complemented
18#1444C411yescharacteristic, complemented
19#1439—21no—
20#1439C2211yescharacteristic
21C4×#95C411yescharacteristic, complemented
22#1352—24no—
23#1280—21no—
24C24×C4D411yescharacteristic
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C22⋊C4—21no—
31C22×C22⋊C4—22no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4D411yescharacteristic
34C22×C22⋊C4C2311yescharacteristic
35C22×C42—21no—
36C22×C42—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42D411yescharacteristic
39C2×C4⋊C8—41no—
40C2×C22⋊C8—22no—
41#399—21no—
42#399—21no—
43#399D412yes—
44#399Q812yes—
45#380—24no—
46#380—42no—
47C4×C4⋊C4—44no—
48C4×C4⋊C4—41no—
49C4×C4⋊C4—41no—
50C4×C22⋊C4—41no—
51C4×C22⋊C4—41no—
52C4×C22⋊C4—44no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—41no—
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95C2×C411yescharacteristic
63C2×#95D411yescharacteristic
64#342—22no—
65#342—22no—
66#342—42no—
67#336—41no—
68#323—22no—
69#323—22no—
70C25C2×D411yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—22no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4SD1612yes—
86C23×C4Q1612yes—
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C22×C8—42no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4—82no—
104C2×C4⋊C4—82no—
105C2×C4⋊C4C4⋊C412yescomplemented
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4C2×D411yescharacteristic
127C2×C22⋊C4C2×Q811yescharacteristic
128C2×C22⋊C4#4111yescharacteristic
129C2×C22⋊C4#4111yescharacteristic
130C2×C42—24no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—42no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42—42no—
139C2×C42—42no—
140C2×C42—42no—
141C2×C42—41no—
142C2×C42—41no—
143C4⋊C8—82no—
144C22⋊C8—42no—
145C22⋊C8—42no—
146#95—21no—
147#95—41no—
148#95—41no—
149#95—41no—
150#95—42no—
151#95—42no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95—44no—
157#95C4⋊C412yescomplemented
158C24—21no—
159C24—21no—
160C24—21no—
161C24—21no—
162C24—41no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24C2×C22⋊C411yescharacteristic
167C24#12111yescharacteristic
168C24C22⋊Q811yescharacteristic
169C24C2×SD1611yescharacteristic
170C24C2×Q1611yescharacteristic
171C24AGL(1,ℤ/8)11yescharacteristic
172C24#13711yescharacteristic
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—44no—
234C22×C4—84no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4#9911yescharacteristic
244C22×C4#9911yescharacteristic
245C22×C4#10312yes—
246C22×C4#10312yes—
247C22×C4#10412yes—
248C22×C4C2×C4⋊C411yescharacteristic
249C22×C4#11711yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—48no—
261C4⋊C4—42no—
262C4⋊C4—88no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—82no—
276C22⋊C4—82no—
277C42—42no—
278C42—48no—
279C42—42no—
280C42—42no—
281C42—81no—
282C42—82no—
283C42—82no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23#38011yescharacteristic
323C23C2×#9911yescharacteristic
324C23#41011yescharacteristic
325C23C2×#10311yescharacteristic
326C23#41711yescharacteristic
327C23C2×#10411yescharacteristic
328C23#42111yescharacteristic
329C23#46011yescharacteristic
330C23#46211yescharacteristic
331C23#47511yescharacteristic
332C23#47711yescharacteristic
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—162no—
402C2×C4—162no—
403C2×C4—162no—
404C2×C4—162no—
405C2×C4#32812yes—
406C2×C4#32812yes—
407C2×C4#35212yes—
408C2×C4#35312yes—
409C8—162no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22C2×#32811yescharacteristic
449C22#150911yescharacteristic
450C22#151311yescharacteristic
451C22#151811yescharacteristic
452C22#179911yescharacteristic
453C22#180311yescharacteristic
454C22#180611yescharacteristic
455C22C2×#35211yescharacteristic
456C22#212011yescharacteristic
457C22C2×#35311yescharacteristic
458C22#212611yescharacteristic
459C4—22no—
460C4—44no—
461C4—42no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—82no—
467C4—81no—
468C4—82no—
469C4—81no—
470C4—82no—
471C4—164no—
472C4#135214yescomplemented
473C2—21no—
474C2—21no—
475C2—21no—
476C2—21no—
477C2—41no—
478C2—41no—
479C2—41no—
480C2—41no—
481C2#857211yescharacteristic
482C2C2×#135211yescharacteristic
483C2#1171511yescharacteristic
484C2#1182111yescharacteristic
485C2#1182211yescharacteristic
486C2#1279411yescharacteristic
487C2#1279711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C2×C4×C4⋊C4C4semidirect
17. #1445C4semidirect
18. #1444C4semidirect
21. C4×#95C4semidirect
105. C2×C4⋊C4C4⋊C4semidirect
157. #95C4⋊C4semidirect
472. C4#1352semidirect

Maximal subgroups

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

subgroupindex
C4×#3992
C2×#13522
#108462
#85572
#72502