Finite Group Databasebeta

← #124340

Subgroups — #124340

Order 512.

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

normal subgroups only

This group has 704 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#13959C211yesmaximal, characteristic
2#8699C211yesmaximal, characteristic, complemented
3#7906C211yesmaximal, characteristic, complemented
4#7132C212yesmaximal, complemented
5#7103C212yesmaximal
6C22×C4×D4C411yescharacteristic
7C2×#431—21no—
8C2×#383C411yescharacteristic
9C2×#376C2211yescharacteristic
10#1817—21no—
11#1445C411yescharacteristic
12#1445C2211yescharacteristic
13#1438C411yescharacteristic
14#1438C2211yescharacteristic
15#1437—21no—
16#1437—21no—
17C22⋊#95—21no—
18C22⋊#95—21no—
19#1280—21no—
20#1280—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⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C42C2×C411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39#431—24no—
40#431—42no—
41C2×C4⋊C8—41no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C4×C8—41no—
45#383—42no—
46#376—41no—
47#376—41no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95D411yescharacteristic
58C2×#95D411yescharacteristic
59C2×#95C2311yescharacteristic
60#336—41no—
61#336—41no—
62C25—21no—
63C22×D4—21no—
64C22×D4—41no—
65C22×D4—41no—
66C22×D4—42no—
67C22×D4—42no—
68C22×D4C2×C812yes—
69C23×C4—21no—
70C23×C4—24no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—81no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4—81no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C22×C8—42no—
87C22×C8—42no—
88C4×D4—44no—
89C4×D4—44no—
90C4×D4—44no—
91C4×D4—44no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4C2×C812yes—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—81no—
110C2×C22⋊C4—81no—
111C2×C22⋊C4—81no—
112C2×C22⋊C4—81no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—22no—
124C2×C42—41no—
125C2×C42C2×C812yes—
126C2×C42M1612yes—
127C4⋊C8—44no—
128C22⋊C8—24no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132C4×C8—44no—
133#95—41no—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95—41no—
139#95—42no—
140#95—42no—
141#95—81no—
142#95—81no—
143#95—81no—
144#95—81no—
145C24—22no—
146C24—22no—
147C24—22no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24—41no—
155C24—41no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24C2×C22⋊C411yescharacteristic
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—44no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—48no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—44no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—48no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
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—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4—84no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4C22⋊C812yes—
228C22×C4C22⋊C812yes—
229C22×C4#9911yescharacteristic
230C22×C4#9911yescharacteristic
231C22×C4#10012yes—
232C22×C4#10412yes—
233C22×C4C22×C811yescharacteristic
234C22×C4C2×M1611yescharacteristic
235C22×C4#13512yes—
236C22×C4AGL(1,ℤ/8)11yescharacteristic
237C22×C4#13711yescharacteristic
238C2×C8—44no—
239C2×C8—82no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—84no—
249C22⋊C4—84no—
250C22⋊C4—82no—
251C22⋊C4—82no—
252C22⋊C4—84no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—42no—
262C42—42no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—24no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23C2×C22⋊C811yescharacteristic
303C23C2×#9911yescharacteristic
304C23C2×#10011yescharacteristic
305C23#41611yescharacteristic
306C23#41711yescharacteristic
307C23C2×#10411yescharacteristic
308C23#42111yescharacteristic
309D4—44no—
310D4—44no—
311D4—44no—
312D4—44no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—816no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
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#32312yes—
378C2×C4#32312yes—
379C8—84no—
380C8—162no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—44no—
394C22—44no—
395C22—44no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22C2×#32311yescharacteristic
409C22#147611yescharacteristic
410C22#147711yescharacteristic
411C22#150411yescharacteristic
412C22#151611yescharacteristic
413C22#153011yescharacteristic
414C22#155111yescharacteristic
415C22#212011yescharacteristic
416C22#212011yescharacteristic
417C22#212811yescharacteristic
418C22#213411yescharacteristic
419C4—22no—
420C4—22no—
421C4—42no—
422C4—41no—
423C4—41no—
424C4—42no—
425C4—41no—
426C4—41no—
427C4—42no—
428C4—81no—
429C4—82no—
430C4—161no—
431C4—161no—
432C2—21no—
433C2—21no—
434C2—21no—
435C2—21no—
436C2—84no—
437C2#757811yescharacteristic
438C2#1142511yescharacteristic
439C2#1143311yescharacteristic
440C2#1175411yescharacteristic
441C2#1176111yescharacteristic
442C2#1184011yescharacteristic
443C2#1184711yescharacteristic
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×#376 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8699C2semidirect
3. #7906C2semidirect
4. #7132C2semidirect
29. C2×C4×D4C8semidirect

Maximal subgroups

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

subgroupindex
#139592
#86992
#79062
#71322
#71032