Finite Group Databasebeta

← #138332

Subgroups — #138332

Order 512.

836 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C2×#1821C211yesmaximal, characteristic
2#14720C211yesmaximal, characteristic
3C2×#1354C212yesmaximal, complemented
4#8299C211yesmaximal, characteristic
5#7251C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407—21no—
9C2×#395C411yescharacteristic
10C2×#395C2211yescharacteristic
11C2×#385—21no—
12#1821—24no—
13#1779—21no—
14C2×#343—21no—
15C2×#343C2212yes—
16C2×#324—21no—
17C2×#324C2212yes—
18#1444—21no—
19#1444—21no—
20#1441C411yescharacteristic
21#1441C2211yescharacteristic
22C22⋊#95—21no—
23C22⋊#95C411yescharacteristic
24#1354—24no—
25C23×Q8C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4D411yescharacteristic
28C24×C4D411yescharacteristic
29C22×M16—22no—
30C2×C22⋊Q8—44no—
31C2×C22⋊Q8—41no—
32C2×C22⋊Q8—41no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—42no—
39C22×C4⋊C4—41no—
40C22×C4⋊C4—41no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4C2311yescharacteristic
46C22×C22⋊C4—21no—
47C22×C22⋊C4—21no—
48#407—24no—
49#407—42no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—41no—
52C2×C22⋊C8—41no—
53C2×C22⋊C8—41no—
54C2×C22⋊C8—41no—
55#395—22no—
56#395—22no—
57#395—42no—
58#385—44no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95C2×C411yescharacteristic
67C2×#95C2×C411yescharacteristic
68#343—24no—
69#343—22no—
70#343D414yes—
71#324—22no—
72#324—22no—
73#324—42no—
74C25C22⋊C411yescharacteristic
75C22×Q8—22no—
76C22×Q8—42no—
77C22×Q8—41no—
78C22×Q8—41no—
79C22×Q8—81no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4—81no—
104C23×C4—81no—
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C2×M16—24no—
112C2×M16—42no—
113C2×M16—42no—
114C2×M16—42no—
115C2×M16—42no—
116C2×M16—42no—
117C22×C8—42no—
118C22×C8—42no—
119C22⋊Q8—44no—
120C22⋊Q8—44no—
121C22⋊Q8—44no—
122C22⋊Q8—44no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—81no—
149C2×C4⋊C4—81no—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4#4112yes—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C22⋊C8—44no—
160C22⋊C8—44no—
161C22⋊C8—42no—
162C22⋊C8—42no—
163C22⋊C8—82no—
164C22⋊C8—82no—
165#95—22no—
166#95—41no—
167#95—42no—
168#95—42no—
169#95—42no—
170#95—42no—
171#95—41no—
172#95—41no—
173#95—42no—
174#95—81no—
175C24—22no—
176C24—22no—
177C24—21no—
178C24—21no—
179C24—21no—
180C24—21no—
181C24—21no—
182C24—21no—
183C24—41no—
184C24—41no—
185C24#9911yescharacteristic
186C24#9911yescharacteristic
187C24C2×C22⋊C411yescharacteristic
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—82no—
191C2×Q8—84no—
192C2×Q8—84no—
193C2×Q8—82no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
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—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
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—41no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—84no—
277C22×C4—84no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4#9912yes—
281C22×C4C4×D412yes—
282C22×C4#12011yescharacteristic
283C22×C4#12112yes—
284C22×C4#12311yescharacteristic
285C22×C4#13512yes—
286C22×C4#13712yes—
287M16—44no—
288M16—44no—
289M16—82no—
290M16—82no—
291C2×C8—44no—
292C2×C8—82no—
293C2×C8—82no—
294C2×C8—82no—
295C2×C8—82no—
296C2×C8—82no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—48no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C22⋊C4—48no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—24no—
316C23—24no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23AΣL(1,𝔽16)12yes—
344C23#35612yes—
345C23#38611yescharacteristic
346C23C2×#9911yescharacteristic
347C23C2×#9911yescharacteristic
348C23#41611yescharacteristic
349C23#41911yescharacteristic
350C23#42111yescharacteristic
351C23#42111yescharacteristic
352Q8—84no—
353Q8—84no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—84no—
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—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4#45212yes—
426C2×C4#48512yes—
427C8—84no—
428C8—162no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—44no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22#151111yescharacteristic
460C22#151111yescharacteristic
461C22#151611yescharacteristic
462C22#151611yescharacteristic
463C22#187311yescharacteristic
464C22#189711yescharacteristic
465C22#189811yescharacteristic
466C22C2×AΣL(1,𝔽16)11yescharacteristic
467C22#212111yescharacteristic
468C22C2×#35611yescharacteristic
469C22#213311yescharacteristic
470C4—22no—
471C4—22no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—82no—
477C4—81no—
478C4—81no—
479C4—82no—
480C4—82no—
481C2—21no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—42no—
486C2—42no—
487C2#915411yescharacteristic
488C2#1181511yescharacteristic
489C2#1181711yescharacteristic
490C2#1184811yescharacteristic
491C2#1185411yescharacteristic
492C2#1308211yescharacteristic
493C2#1309411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×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. C2×#1354C2semidirect
5. #7251C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18212
#147202
C2×#13542
#82992
#72512