Finite Group Databasebeta

← #124191

Subgroups — #124191

Order 512.

736 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 116 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×#393C211yesmaximal, characteristic
2C4×#323C211yesmaximal, characteristic
3#7907C211yesmaximal, characteristic
4#7131C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C22×C4×Q8C411yescharacteristic
7C22×C22⋊C8—21no—
8C2×#393C411yescharacteristic
9C2×C4×C22⋊C4—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11C4×C22⋊C8—21no—
12C2×#323—21no—
13#1446C2211yescharacteristic
14#1442—21no—
15#1442C2211yescharacteristic
16#1439—21no—
17#1439C411yescharacteristic
18C4×#95—21no—
19C4×#95C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C23×C8—41no—
26C2×C4×Q8—44no—
27C2×C4×Q8—41no—
28C2×C4×Q8—41no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4C2×C411yescharacteristic
34C22×C22⋊C4D411yescharacteristic
35C22×C42—21no—
36C22×C42—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42D411yescharacteristic
39C2×C22⋊C8—24no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—41no—
43C2×C22⋊C8—42no—
44C2×C4×C8—41no—
45#393—22no—
46#393C814yescomplemented
47C4×C22⋊C4—41no—
48C4×C22⋊C4—41no—
49C4×C22⋊C4—44no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60C2×#95C2311yescharacteristic
61#336—41no—
62#323—24no—
63C25C22⋊C411yescharacteristic
64C22×Q8—21no—
65C22×Q8—42no—
66C22×Q8—42no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8C2×C812yescomplemented
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4SD1612yes—
83C23×C4Q1612yes—
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—42no—
88C22×C8—44no—
89C22×C8—42no—
90C22×C8—82no—
91C22×C8—82no—
92C4×Q8—44no—
93C4×Q8—44no—
94C4×Q8—44no—
95C4×Q8—44no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4C2×C812yes—
113C2×C22⋊C4M1612yes—
114C2×C42—24no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—42no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—42no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42—42no—
127C22⋊C8—24no—
128C22⋊C8—24no—
129C22⋊C8—44no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132C4×C8—82no—
133#95—24no—
134#95—21no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95C2×C812yescomplemented
145C24—21no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24C22⋊C812yes—
154C24#10312yes—
155C24#10412yes—
156C24C2×C22⋊C411yescharacteristic
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—48no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
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—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
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—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—84no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4C22⋊C812yes—
229C22×C4#9911yescharacteristic
230C22×C4#9911yescharacteristic
231C22×C4#10312yes—
232C22×C4#10412yes—
233C22×C4C22×C811yescharacteristic
234C22×C4C2×M1611yescharacteristic
235C22×C4C2×SD1611yescharacteristic
236C22×C4C2×Q1611yescharacteristic
237C22×C4#13512yes—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—88no—
245C4⋊C4—42no—
246C4⋊C4—48no—
247C4⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—44no—
251C22⋊C4—48no—
252C22⋊C4—48no—
253C42—44no—
254C42—42no—
255C42—44no—
256C42—42no—
257C42—42no—
258C42—81no—
259C42—84no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23—81no—
286C23Q8⋊C814yes—
287C23C2×C22⋊C811yescharacteristic
288C23C2×#9911yescharacteristic
289C23#41311yescharacteristic
290C23C2×#10311yescharacteristic
291C23#41611yescharacteristic
292C23C2×#10411yescharacteristic
293C23C2×#10411yescharacteristic
294Q8—44no—
295Q8—44no—
296Q8—44no—
297Q8—44no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4#32812yes—
364C2×C4#32812yes—
365C2×C4AΣL(1,𝔽16)12yes—
366C2×C4#35412yes—
367C8—84no—
368C8—84no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—24no—
382C22—44no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22#145911yescharacteristic
393C22C2×Q8⋊C811yescharacteristic
394C22#147611yescharacteristic
395C22C2×#32811yescharacteristic
396C22#150111yescharacteristic
397C22#152411yescharacteristic
398C22#154211yescharacteristic
399C22C2×AΣL(1,𝔽16)11yescharacteristic
400C22#212011yescharacteristic
401C22C2×#35411yescharacteristic
402C22#213211yescharacteristic
403C4—22no—
404C4—41no—
405C4—41no—
406C4—44no—
407C4—41no—
408C4—41no—
409C4—42no—
410C4—42no—
411C4—82no—
412C4—82no—
413C4—81no—
414C4—82no—
415C4#135114yes—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—44no—
421C2#756611yescharacteristic
422C2#1147111yescharacteristic
423C2#1147711yescharacteristic
424C2C2×#135111yescharacteristic
425C2#1172111yescharacteristic
426C2#1172811yescharacteristic
427C2#1173311yescharacteristic
subgroupG/H
centerC22×C4#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
socleC23C2×#99
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7131C2semidirect
5. #7122C2semidirect
46. #393C8semidirect
69. C22×Q8C2×C8semidirect
144. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3932
C4×#3232
#79072
#71312
#71222