Finite Group Databasebeta

← #134451

Subgroups — #134451

Order 512.

634 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 106 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
1#14317C211yesmaximal, characteristic
2C2×#1352C212yesmaximal
3#10846C211yesmaximal, characteristic
4#8557C211yesmaximal, characteristic
5#7250C212yesmaximal
6C2×#399C2211yescharacteristic
7C2×#389C411yescharacteristic, complemented
8C2×#380—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1445C411yescharacteristic, complemented
15#1442C411yescharacteristic, complemented
16#1441—21no—
17#1439—21no—
18#1439C2211yescharacteristic
19#1436—21no—
20#1434C411yescharacteristic, complemented
21C22⋊#95—21no—
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×C42C2×C411yescharacteristic
36C22×C42D411yescharacteristic
37C2×C4⋊C8—41no—
38C2×C22⋊C8—22no—
39#399—21no—
40#399—21no—
41#399D412yes—
42#399Q812yes—
43#389—41no—
44#389—41no—
45#380—24no—
46#380—42no—
47C4×C22⋊C4—41no—
48C4×C22⋊C4—41no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—41no—
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95D411yescharacteristic
62#342—22no—
63#342—22no—
64#342—42no—
65#336—41no—
66#323—22no—
67#323—22no—
68C25C2×D411yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—42no—
77C23×C4—81no—
78C23×C4—81no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4SD1612yes—
84C23×C4Q1612yes—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—42no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—81no—
97C2×C4⋊C4—81no—
98C2×C4⋊C4—81no—
99C2×C4⋊C4—81no—
100C2×C4⋊C4—82no—
101C2×C4⋊C4—82no—
102C2×C4⋊C4—81no—
103C2×C4⋊C4C4⋊C412yescomplemented
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—81no—
121C2×C22⋊C4—81no—
122C2×C22⋊C4—81no—
123C2×C22⋊C4—81no—
124C2×C22⋊C4C2×D411yescharacteristic
125C2×C22⋊C4C2×Q811yescharacteristic
126C2×C22⋊C4#4111yescharacteristic
127C2×C22⋊C4#4111yescharacteristic
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—81no—
133C2×C42—81no—
134C4⋊C8—82no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137#95—21no—
138#95—41no—
139#95—41no—
140#95—41no—
141#95—42no—
142#95—44no—
143#95—41no—
144#95—41no—
145#95—42no—
146#95—81no—
147#95—81no—
148#95—81no—
149#95—81no—
150#95—81no—
151#95C4⋊C412yescomplemented
152C24—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24C2×C22⋊C411yescharacteristic
161C24#12111yescharacteristic
162C24C22⋊Q811yescharacteristic
163C24C2×SD1611yescharacteristic
164C24C2×Q1611yescharacteristic
165C24AGL(1,ℤ/8)11yescharacteristic
166C24#13711yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—44no—
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—81no—
227C22×C4—81no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—84no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4#9911yescharacteristic
238C22×C4#9911yescharacteristic
239C22×C4#10312yes—
240C22×C4#10312yes—
241C22×C4#10412yes—
242C22×C4C2×C4⋊C411yescharacteristic
243C22×C4#11711yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4C22⋊Q811yescharacteristic
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—88no—
255C4⋊C4—84no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—82no—
265C22⋊C4—82no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C22⋊C4—82no—
269C22⋊C4—82no—
270C42—42no—
271C42—81no—
272C42—81no—
273C42—81no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23#38011yescharacteristic
313C23C2×#9911yescharacteristic
314C23#41011yescharacteristic
315C23C2×#10311yescharacteristic
316C23#41711yescharacteristic
317C23C2×#10411yescharacteristic
318C23#42111yescharacteristic
319C23#46011yescharacteristic
320C23#46211yescharacteristic
321C23#47511yescharacteristic
322C23#47711yescharacteristic
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—84no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—162no—
394C2×C4—162no—
395C2×C4—162no—
396C2×C4—162no—
397C2×C4#32812yes—
398C2×C4#32812yes—
399C8—162no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22C2×#32811yescharacteristic
439C22#150911yescharacteristic
440C22#151311yescharacteristic
441C22#151811yescharacteristic
442C22#179911yescharacteristic
443C22#180311yescharacteristic
444C22#180611yescharacteristic
445C22#212111yescharacteristic
446C22#212311yescharacteristic
447C22#212711yescharacteristic
448C22#212811yescharacteristic
449C4—22no—
450C4—22no—
451C4—42no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—82no—
457C4—81no—
458C4—81no—
459C4—82no—
460C4—161no—
461C4—164no—
462C4—161no—
463C2—21no—
464C2—21no—
465C2—21no—
466C2—21no—
467C2—41no—
468C2—41no—
469C2—41no—
470C2—41no—
471C2#857211yescharacteristic
472C2#1171611yescharacteristic
473C2#1171811yescharacteristic
474C2#1182411yescharacteristic
475C2#1182711yescharacteristic
476C2#1279611yescharacteristic
477C2#1279911yescharacteristic
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
7. C2×#389C4semidirect
14. #1445C4semidirect
15. #1442C4semidirect
20. #1434C4semidirect
103. C2×C4⋊C4C4⋊C4semidirect
151. #95C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#143172
C2×#13522
#108462
#85572
#72502