Finite Group Databasebeta

← #134428

Subgroups — #134428

Order 512.

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

normal subgroups only

This group has 826 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×#394C211yesmaximal, characteristic
2C2×#1345C212yesmaximal
3#10847C211yesmaximal, characteristic, complemented
4#8558C211yesmaximal, characteristic, complemented
5#7251C212yesmaximal, complemented
6C22×C4×D4C411yescharacteristic
7C2×#423—21no—
8C2×#394C2211yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4—21no—
11C2×C4×C22⋊C4—21no—
12#1797—21no—
13C2×#343—21no—
14C2×#343C2212yes—
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
42#423—24no—
43#423—42no—
44C2×C4⋊C8—41no—
45C2×C4⋊C8—41no—
46C2×C4⋊C8—41no—
47C2×C22⋊C8—22no—
48#394—21no—
49#394—21no—
50#394D412yes—
51#394Q812yes—
52C4×C4⋊C4—41no—
53C4×C4⋊C4—41no—
54C4×C22⋊C4—44no—
55C4×C22⋊C4—42no—
56C4×C22⋊C4—42no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95C2×C411yescharacteristic
63C2×#95C2×C411yescharacteristic
64C2×#95C2×C411yescharacteristic
65C2×#95D411yescharacteristic
66#343—22no—
67#343—22no—
68#343—42no—
69#336—41no—
70#324—22no—
71#324—22no—
72C25—21no—
73C22×D4—21no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4—42no—
77C22×D4—42no—
78C22×D4C4⋊C412yes—
79C23×C4—24no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—41no—
87C23×C4—41no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C2×M16—24no—
98C2×M16—42no—
99C2×M16—42no—
100C2×M16—42no—
101C2×M16—42no—
102C2×M16—42no—
103C22×C8—42no—
104C22×C8—42no—
105C4×D4—44no—
106C4×D4—44no—
107C4×D4—44no—
108C4×D4—44no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4C2×D411yescharacteristic
118C2×C4⋊C4C2×Q811yescharacteristic
119C2×C4⋊C4#4111yescharacteristic
120C2×C4⋊C4#4111yescharacteristic
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
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×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—42no—
148C2×C42—42no—
149C2×C42—42no—
150C2×C42—42no—
151C2×C42—42no—
152C2×C42D812yes—
153C2×C42SD1612yes—
154C4⋊C8—44no—
155C4⋊C8—44no—
156C4⋊C8—82no—
157C22⋊C8—42no—
158C22⋊C8—42no—
159#95—21no—
160#95—41no—
161#95—41no—
162#95—41no—
163#95—42no—
164#95—42no—
165#95—42no—
166#95—42no—
167#95—41no—
168#95—41no—
169#95C4⋊C412yes—
170C24—22no—
171C24—22no—
172C24—22no—
173C24—41no—
174C24—41no—
175C24—41no—
176C24—41no—
177C24—41no—
178C24—41no—
179C24—41no—
180C24—41no—
181C24—41no—
182C24—41no—
183C24—41no—
184C24—41no—
185C24C2×C22⋊C411yescharacteristic
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—44no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—48no—
193C2×D4—42no—
194C2×D4—44no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—44no—
217C22×C4—48no—
218C22×C4—48no—
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—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—44no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—82no—
249C22×C4—84no—
250C22×C4—84no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4#9911yescharacteristic
255C22×C4#9911yescharacteristic
256C22×C4#10212yes—
257C22×C4#10212yes—
258C22×C4#10412yes—
259C22×C4C2×C4⋊C411yescharacteristic
260C22×C4#11711yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×SD1611yescharacteristic
267C22×C4AGL(1,ℤ/8)11yescharacteristic
268C22×C4#13711yescharacteristic
269M16—44no—
270M16—44no—
271M16—82no—
272M16—82no—
273C2×C8—44no—
274C2×C8—82no—
275C2×C8—82no—
276C2×C8—82no—
277C2×C8—82no—
278C2×C8—82no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—48no—
290C22⋊C4—48no—
291C22⋊C4—48no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—81no—
304C42—84no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—24no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23#38011yescharacteristic
345C23C2×#9911yescharacteristic
346C23#41011yescharacteristic
347C23C2×#10211yescharacteristic
348C23#41711yescharacteristic
349C23C2×#10411yescharacteristic
350C23#42111yescharacteristic
351D4—44no—
352D4—44no—
353D4—44no—
354D4—44no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—88no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—88no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4#32712yes—
424C2×C4#32712yes—
425C2×C4AΣL(1,𝔽16)12yes—
426C2×C4#35512yes—
427C2×C4#45911yescharacteristic
428C2×C4#46111yescharacteristic
429C2×C4#47411yescharacteristic
430C2×C4#47611yescharacteristic
431C8—84no—
432C8—162no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22C2×#32711yescharacteristic
461C22#150911yescharacteristic
462C22#151311yescharacteristic
463C22#151711yescharacteristic
464C22#179911yescharacteristic
465C22#180211yescharacteristic
466C22#180611yescharacteristic
467C22C2×AΣL(1,𝔽16)11yescharacteristic
468C22#212011yescharacteristic
469C22C2×#35511yescharacteristic
470C22#213211yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—22no—
476C4—42no—
477C4—41no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—82no—
482C4—82no—
483C4—81no—
484C4—81no—
485C4—82no—
486C4#134514yescomplemented
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—84no—
492C2#857111yescharacteristic
493C2C2×#134511yescharacteristic
494C2#1169711yescharacteristic
495C2#1182811yescharacteristic
496C2#1182911yescharacteristic
497C2#1279311yescharacteristic
498C2#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. #10847C2semidirect
4. #8558C2semidirect
5. #7251C2semidirect
486. C4#1345semidirect

Maximal subgroups

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

subgroupindex
C4×#3942
C2×#13452
#108472
#85582
#72512