Finite Group Databasebeta

← #134471

Subgroups — #134471

Order 512.

770 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 96 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#14176C211yesmaximal, characteristic
2C2×#1345C212yesmaximal
3#10802C211yesmaximal, characteristic, complemented
4#8558C211yesmaximal, characteristic, complemented
5#7251C212yesmaximal, complemented
6C22×C4×D4C411yescharacteristic
7C2×#394C2211yescharacteristic
8C2×#388—21no—
9C2×#385—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1797—21no—
12C2×#343—21no—
13C2×#343C2212yes—
14#1728—21no—
15C2×#324C2212yes—
16#1444C411yescharacteristic
17#1441C2211yescharacteristic
18#1438C411yescharacteristic
19#1437—21no—
20#1434C411yescharacteristic
21C22⋊#95—21no—
22#1345—24no—
23#1280—21no—
24C23×D4C2×C411yescharacteristic
25C24×C4—21no—
26C22×M16—22no—
27C2×C4×D4—41no—
28C2×C4×D4—41no—
29C2×C4×D4—44no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4C2311yescharacteristic
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C42C2×C411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42C2×C4⋊C8—41no—
43C2×C22⋊C8—22no—
44#394—21no—
45#394—21no—
46#394D412yes—
47#394Q812yes—
48#388—44no—
49#385—44no—
50C4×C4⋊C4—41no—
51C4×C4⋊C4—41no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95D411yescharacteristic
61#343—24no—
62#343—22no—
63#343—22no—
64#336—41no—
65#336—41no—
66#336—41no—
67#324—22no—
68#324—22no—
69C25—21no—
70C22×D4—21no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4—42no—
74C22×D4—42no—
75C22×D4C4⋊C412yes—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—81no—
85C23×C4—81no—
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C2×M16—24no—
94C2×M16—42no—
95C2×M16—42no—
96C2×M16—42no—
97C2×M16—42no—
98C2×M16—42no—
99C22×C8—42no—
100C22×C8—42no—
101C4×D4—44no—
102C4×D4—44no—
103C4×D4—44no—
104C4×D4—44no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—81no—
115C2×C4⋊C4C2×D411yescharacteristic
116C2×C4⋊C4C2×Q811yescharacteristic
117C2×C4⋊C4#4111yescharacteristic
118C2×C4⋊C4#4111yescharacteristic
119C2×C22⋊C4—24no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
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—81no—
136C2×C22⋊C4—81no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—21no—
144C2×C42—41no—
145C2×C42—42no—
146C2×C42—42no—
147C4⋊C8—82no—
148C22⋊C8—42no—
149C22⋊C8—42no—
150#95—21no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—42no—
155#95—42no—
156#95—42no—
157#95—42no—
158#95—81no—
159#95C4⋊C412yes—
160C24—22no—
161C24—22no—
162C24—22no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24—41no—
169C24—41no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24—41no—
174C24—41no—
175C24C2×C22⋊C411yescharacteristic
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—44no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—48no—
183C2×D4—42no—
184C2×D4—44no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
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—48no—
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—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—84no—
238C22×C4—84no—
239C22×C4—84no—
240C22×C4—84no—
241C22×C4—84no—
242C22×C4—84no—
243C22×C4#9911yescharacteristic
244C22×C4#9911yescharacteristic
245C22×C4#10412yes—
246C22×C4C2×C4⋊C411yescharacteristic
247C22×C4#11711yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4#13512yes—
253C22×C4AGL(1,ℤ/8)12yes—
254M16—44no—
255M16—44no—
256M16—82no—
257M16—82no—
258C2×C8—44no—
259C2×C8—82no—
260C2×C8—82no—
261C2×C8—82no—
262C2×C8—82no—
263C2×C8—82no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—84no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—84no—
279C22⋊C4—84no—
280C42—42no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—42no—
285C42—81no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—24no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23#38011yescharacteristic
326C23C2×#9911yescharacteristic
327C23#41011yescharacteristic
328C23#41611yescharacteristic
329C23#41811yescharacteristic
330C23C2×#10411yescharacteristic
331C23#42111yescharacteristic
332D4—44no—
333D4—44no—
334D4—44no—
335D4—44no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—816no—
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—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4AΣL(1,𝔽16)12yes—
403C2×C4#46312yes—
404C2×C4#47812yes—
405C8—84no—
406C8—162no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—44no—
420C22—44no—
421C22—44no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22#150011yescharacteristic
435C22#150711yescharacteristic
436C22#151411yescharacteristic
437C22#151611yescharacteristic
438C22#179911yescharacteristic
439C22#180411yescharacteristic
440C22#180611yescharacteristic
441C22C2×AΣL(1,𝔽16)11yescharacteristic
442C22#212011yescharacteristic
443C22#213311yescharacteristic
444C22#213411yescharacteristic
445C4—22no—
446C4—22no—
447C4—22no—
448C4—42no—
449C4—41no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—82no—
455C4—81no—
456C4—82no—
457C4—81no—
458C4—161no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—21no—
463C2—84no—
464C2#857311yescharacteristic
465C2#1177811yescharacteristic
466C2#1177911yescharacteristic
467C2#1184711yescharacteristic
468C2#1184811yescharacteristic
469C2#1279311yescharacteristic
470C2#1280311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10802C2semidirect
4. #8558C2semidirect
5. #7251C2semidirect

Maximal subgroups

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

subgroupindex
#141762
C2×#13452
#108022
#85582
#72512