Finite Group Databasebeta

← #124256

Subgroups — #124256

Order 512.

746 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 108 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#13905C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic, complemented
3C4×#324C211yesmaximal, characteristic, complemented
4#7104C212yesmaximal, complemented
5#7103C212yesmaximal
6C22×C4×D4C411yescharacteristic
7C2×#382C411yescharacteristic
8C2×#380—21no—
9C2×#380—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1797—21no—
12C4×C22⋊C8—21no—
13C2×#324—21no—
14C2×#324—21no—
15#1445C411yescharacteristic
16#1445C2211yescharacteristic
17#1438C411yescharacteristic
18#1438C2211yescharacteristic
19#1437—21no—
20#1437—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×D4C2×C411yescharacteristic
24C24×C4—21no—
25C2×C4×D4—22no—
26C2×C4×D4—41no—
27C2×C4×D4—41no—
28C2×C4×D4—42no—
29C2×C4×D4C814yescomplemented
30C22×C4⋊C4—21no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
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—
44C2×C22⋊C8—22no—
45C2×C4×C8—41no—
46#382—42no—
47#380—44no—
48#380—42no—
49#380—42no—
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×#95C2311yescharacteristic
61#336—41no—
62#336—41no—
63#324—24no—
64#324—42no—
65C25—21no—
66C22×D4—21no—
67C22×D4—41no—
68C22×D4—41no—
69C22×D4—42no—
70C22×D4—42no—
71C22×D4C2×C812yes—
72C23×C4—21no—
73C23×C4—24no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22×C411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C22×C8—42no—
90C22×C8—42no—
91C4×D4—44no—
92C4×D4—44no—
93C4×D4—44no—
94C4×D4—44no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4C2×C812yes—
107C2×C4⋊C4D812yes—
108C2×C4⋊C4SD1612yes—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—81no—
123C2×C22⋊C4—81no—
124C2×C22⋊C4—81no—
125C2×C22⋊C4—81no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—22no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—81no—
134C2×C42—81no—
135C2×C42C2×C812yes—
136C2×C42M1612yes—
137C4⋊C8—82no—
138C22⋊C8—24no—
139C22⋊C8—42no—
140C22⋊C8—42no—
141C22⋊C8—42no—
142C4×C8—82no—
143#95—41no—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—42no—
148#95—42no—
149#95—42no—
150#95—41no—
151#95—41no—
152#95—42no—
153C24—22no—
154C24—22no—
155C24—22no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24C2×C22⋊C411yescharacteristic
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—44no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—48no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—44no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—48no—
197C22×C4—48no—
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—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—48no—
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—84no—
230C22×C4—84no—
231C22×C4—84no—
232C22×C4—84no—
233C22×C4C22⋊C812yes—
234C22×C4C22⋊C812yes—
235C22×C4#9911yescharacteristic
236C22×C4#9911yescharacteristic
237C22×C4#10012yes—
238C22×C4#10212yes—
239C22×C4#10212yes—
240C22×C4#10412yes—
241C22×C4C22×C811yescharacteristic
242C22×C4C2×M1611yescharacteristic
243C22×C4C2×D811yescharacteristic
244C22×C4C2×SD1611yescharacteristic
245C22×C4AGL(1,ℤ/8)12yes—
246C2×C8—44no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—84no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—84no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C22⋊C4—84no—
264C22⋊C4—84no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—42no—
270C42—42no—
271C42—81no—
272C42—81no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—24no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23C2×C22⋊C811yescharacteristic
313C23C2×#9911yescharacteristic
314C23C2×#10011yescharacteristic
315C23C2×#10211yescharacteristic
316C23#41811yescharacteristic
317C23C2×#10411yescharacteristic
318C23#42111yescharacteristic
319D4—44no—
320D4—44no—
321D4—44no—
322D4—44no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—816no—
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—81no—
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#32312yes—
387C2×C4#32312yes—
388C2×C4D4⋊C814yes—
389C8—84no—
390C8—162no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22C2×#32311yescharacteristic
419C22C2×D4⋊C811yescharacteristic
420C22#147811yescharacteristic
421C22#150711yescharacteristic
422C22#151911yescharacteristic
423C22#152711yescharacteristic
424C22#154511yescharacteristic
425C22#212011yescharacteristic
426C22#212011yescharacteristic
427C22#212711yescharacteristic
428C22#213311yescharacteristic
429C4—24no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—42no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—42no—
438C4—82no—
439C4—81no—
440C4—161no—
441C4—161no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2—84no—
447C2#757211yescharacteristic
448C2#1142511yescharacteristic
449C2#1143211yescharacteristic
450C2#1177411yescharacteristic
451C2#1177811yescharacteristic
452C2#1187511yescharacteristic
453C2#1187911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8558C2semidirect
3. C4×#324C2semidirect
4. #7104C2semidirect
29. C2×C4×D4C8semidirect

Maximal subgroups

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

subgroupindex
#139052
#85582
C4×#3242
#71042
#71032