Finite Group Databasebeta

← #142009

Subgroups — #142009

Order 512.

758 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 118 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#15125C211yesmaximal, characteristic
2#10846C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7250C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic, complemented
6C2×#393—21no—
7C2×#380—21no—
8C2×#380—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10C4×C22⋊C8—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1446—21no—
15#1442C412yescomplemented
16#1439C2212yes—
17C4⋊#95—21no—
18C4⋊#95C411yescharacteristic, complemented
19#1280—21no—
20C23×Q8—21no—
21C24×C4D411yescharacteristic
22C2×C4⋊Q8—41no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—44no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4C2×C412yes—
29C22×C22⋊C4—22no—
30C22×C22⋊C4—22no—
31C22×C22⋊C4D411yescharacteristic
32C22×C22⋊C4C2311yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C22⋊C8—22no—
36C2×C4×C8—41no—
37#393—42no—
38#393—42no—
39#380—24no—
40#380—44no—
41#380—42no—
42C4×C22⋊C4—41no—
43C4×C22⋊C4—41no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—41no—
49C2×#95C2×C412yes—
50C2×#95C2×C411yescharacteristic
51C2×#95D411yescharacteristic
52#342—24no—
53#342—42no—
54#336—41no—
55#323—22no—
56#323Q814yescomplemented
57C25C2×D411yescharacteristic
58C22×Q8—42no—
59C22×Q8—44no—
60C22×Q8—42no—
61C22×Q8—42no—
62C22×Q8—82no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4SD1612yes—
76C23×C4Q1612yes—
77C23×C4C22×C411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C22×C8—42no—
80C4⋊Q8—48no—
81C4⋊Q8—44no—
82C4⋊Q8—44no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—82no—
94C2×C4⋊C4—82no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—82no—
109C2×C22⋊C4D812yes—
110C2×C22⋊C4SD1612yes—
111C2×C22⋊C4C2×Q812yes—
112C2×C22⋊C4#4112yes—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—42no—
118C2×C42—42no—
119C22⋊C8—24no—
120C22⋊C8—42no—
121C4×C8—82no—
122#95—41no—
123#95—41no—
124#95—42no—
125#95—44no—
126#95—44no—
127#95—44no—
128#95—42no—
129#95—82no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—41no—
135C24—41no—
136C24—41no—
137C24—41no—
138C24C2×C22⋊C411yescharacteristic
139C24#12411yescharacteristic
140C24C4⋊Q811yescharacteristic
141C24C2×D811yescharacteristic
142C24C2×SD1611yescharacteristic
143C24C2×SD1611yescharacteristic
144C24C2×Q1611yescharacteristic
145C2×Q8—48no—
146C2×Q8—84no—
147C2×Q8—88no—
148C2×Q8—88no—
149C2×Q8—84no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—84no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4#9911yescharacteristic
205C22×C4#9911yescharacteristic
206C22×C4#10212yes—
207C22×C4#10212yes—
208C22×C4#10312yes—
209C22×C4#10312yes—
210C22×C4#10412yes—
211C22×C4C4×Q812yes—
212C22×C4C22⋊Q812yes—
213C2×C8—44no—
214C2×C8—82no—
215C2×C8—82no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—84no—
221C4⋊C4—88no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C22⋊C4—84no—
229C22⋊C4—84no—
230C22⋊C4—82no—
231C22⋊C4—82no—
232C42—42no—
233C42—42no—
234C42—42no—
235C42—81no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23#39111yescharacteristic
275C23C2×#9911yescharacteristic
276C23#41011yescharacteristic
277C23C2×#10211yescharacteristic
278C23C2×#10311yescharacteristic
279C23C2×#10411yescharacteristic
280C23#42111yescharacteristic
281C23#48611yescharacteristic
282C23#48711yescharacteristic
283C23#49811yescharacteristic
284C23#50011yescharacteristic
285Q8—816no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—82no—
336C2×C4—162no—
337C2×C4—162no—
338C2×C4—162no—
339C2×C4—162no—
340C2×C4#32812yes—
341C2×C4#32812yes—
342C2×C4AΣL(1,𝔽16)12yes—
343C2×C4#35412yes—
344C8—84no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22C2×#32811yescharacteristic
384C22#150211yescharacteristic
385C22#151811yescharacteristic
386C22#151911yescharacteristic
387C22#198611yescharacteristic
388C22#199011yescharacteristic
389C22#199611yescharacteristic
390C22C2×AΣL(1,𝔽16)11yescharacteristic
391C22#212311yescharacteristic
392C22C2×#35411yescharacteristic
393C22#212811yescharacteristic
394C4—22no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—84no—
401C4—162no—
402C4—164no—
403C4#135114yes—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2—41no—
409C2—41no—
410C2—41no—
411C2—41no—
412C2#988411yescharacteristic
413C2C2×#135111yescharacteristic
414C2#1171811yescharacteristic
415C2#1187311yescharacteristic
416C2#1187811yescharacteristic
417C2#1329311yescharacteristic
418C2#1329911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C22×C4⋊Q8C4semidirect
15. #1442C4semidirect
18. C4⋊#95C4semidirect
56. #323Q8semidirect

Maximal subgroups

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

subgroupindex
#151252
#108462
C4×#3232
#72502