Finite Group Databasebeta

← #134497

Subgroups — #134497

Order 512.

802 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 98 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×#394C211yesmaximal, characteristic
2C2×#1345C212yesmaximal
3#10802C211yesmaximal, characteristic, complemented
4#8560C211yesmaximal, characteristic, complemented
5#7251C212yesmaximal, complemented
6C22×C4×D4C411yescharacteristic
7C2×#394C2211yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×C4×C22⋊C4—21no—
10C2×C4×C22⋊C4—21no—
11#1797—21no—
12C2×#343—21no—
13C2×#343C2212yes—
14#1728—21no—
15C2×#324C2212yes—
16#1442C411yescharacteristic
17#1441C2211yescharacteristic
18#1440—21no—
19#1439—21no—
20#1438C411yescharacteristic
21C4×#95C411yescharacteristic
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⋊C4C2×C411yescharacteristic
31C22×C4⋊C4D411yescharacteristic
32C22×C4⋊C4C2311yescharacteristic
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C42—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—
48C4×C4⋊C4—41no—
49C4×C4⋊C4—41no—
50C4×C22⋊C4—44no—
51C4×C22⋊C4—44no—
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—24no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—41no—
86C23×C4—41no—
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C2×M16—24no—
95C2×M16—42no—
96C2×M16—42no—
97C2×M16—42no—
98C2×M16—42no—
99C2×M16—42no—
100C22×C8—42no—
101C22×C8—42no—
102C4×D4—44no—
103C4×D4—44no—
104C4×D4—44no—
105C4×D4—44no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4C2×D411yescharacteristic
114C2×C4⋊C4C2×Q811yescharacteristic
115C2×C4⋊C4#4111yescharacteristic
116C2×C4⋊C4#4111yescharacteristic
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—21no—
141C2×C42—41no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42—42no—
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—42no—
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—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—44no—
209C22×C4—48no—
210C22×C4—48no—
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—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—44no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—84no—
241C22×C4—84no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4#9911yescharacteristic
247C22×C4#9911yescharacteristic
248C22×C4#10412yes—
249C22×C4C2×C4⋊C411yescharacteristic
250C22×C4#11711yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4#13512yes—
256C22×C4AGL(1,ℤ/8)12yes—
257M16—44no—
258M16—44no—
259M16—82no—
260M16—82no—
261C2×C8—44no—
262C2×C8—82no—
263C2×C8—82no—
264C2×C8—82no—
265C2×C8—82no—
266C2×C8—82no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—48no—
277C22⋊C4—48no—
278C22⋊C4—48no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—42no—
285C42—42no—
286C42—81no—
287C42—84no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—24no—
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—42no—
310C23—42no—
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—81no—
326C23—81no—
327C23#38011yescharacteristic
328C23C2×#9911yescharacteristic
329C23#41011yescharacteristic
330C23#41611yescharacteristic
331C23#41811yescharacteristic
332C23C2×#10411yescharacteristic
333C23#42111yescharacteristic
334D4—44no—
335D4—44no—
336D4—44no—
337D4—44no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
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—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—88no—
386C2×C4—88no—
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×C4—81no—
403C2×C4—81no—
404C2×C4AΣL(1,𝔽16)12yes—
405C2×C4#35512yes—
406C2×C4#46312yes—
407C2×C4#47812yes—
408C8—84no—
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—44no—
423C22—44no—
424C22—44no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22—82no—
437C22#150111yescharacteristic
438C22#150511yescharacteristic
439C22#151011yescharacteristic
440C22#152111yescharacteristic
441C22#179911yescharacteristic
442C22#180411yescharacteristic
443C22#180611yescharacteristic
444C22C2×AΣL(1,𝔽16)11yescharacteristic
445C22#212011yescharacteristic
446C22C2×#35511yescharacteristic
447C22#213211yescharacteristic
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—42no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4—81no—
461C4—81no—
462C2—21no—
463C2—21no—
464C2—21no—
465C2—21no—
466C2—84no—
467C2#857511yescharacteristic
468C2#1173311yescharacteristic
469C2#1173411yescharacteristic
470C2#1180211yescharacteristic
471C2#1180411yescharacteristic
472C2#1279311yescharacteristic
473C2#1280111yescharacteristic
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. #8560C2semidirect
5. #7251C2semidirect

Maximal subgroups

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

subgroupindex
C4×#3942
C2×#13452
#108022
#85602
#72512