Finite Group Databasebeta

← #149282

Subgroups — #149282

Order 512.

790 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 146 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×#1857C212yesmaximal
2#14467C211yesmaximal, characteristic
3C2×#1394C212yesmaximal
4C2×#1394C211yesmaximal, characteristic
5#10750C211yesmaximal, characteristic
6C2×#519C2211yescharacteristic
7C2×#428—22no—
8C22×#101C2211yescharacteristic
9C22×#101C2212yes—
10C2×#391C2212yes—
11C2×#391C2211yescharacteristic
12#1857—24no—
13#1857—24no—
14#1736—21no—
15C2×#339—22no—
16C2×#339—21no—
17#1444—22no—
18#1441—21no—
19#1436—22no—
20C4⋊#95—21no—
21#1394—24no—
22#1394C414yescomplemented
23C23×Q8C2311yescharacteristic
24C22×M16—22no—
25C22×M16—21no—
26#519—22no—
27#519—21no—
28#519—42no—
29#519D412yes—
30C2×C4×Q8—41no—
31C2×C4×Q8—41no—
32C2×#117—22no—
33C2×#117—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4D412yes—
38C22×C4⋊C4D411yescharacteristic
39C22×C42D412yes—
40C22×C42D411yescharacteristic
41#428—28no—
42#428—44no—
43C2×#107—42no—
44C2×#106—42no—
45C2×#101—22no—
46C2×#101—22no—
47C2×#101—22no—
48C2×#101—22no—
49C2×#101—22no—
50C2×#101—21no—
51C2×#101C2×C414yes—
52C2×#101D414yes—
53C2×#101Q812yes—
54C2×#101Q814yes—
55#391—22no—
56#391—22no—
57#391—22no—
58#391—22no—
59#391—22no—
60#391C2×C412yes—
61#391Q812yes—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—21no—
67C2×#95—22no—
68#340—41no—
69#340—41no—
70#339—28no—
71#339—42no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8C22×C411yescharacteristic
77C22×Q8C2×D411yescharacteristic
78C22×Q8C2×D412yes—
79C22×Q8C2×Q811yescharacteristic
80C22×Q8C2×Q812yes—
81C22×Q8C2×Q811yescharacteristic
82C22×Q8#4112yes—
83C22×Q8#4111yescharacteristic
84C22×Q8#4111yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D412yes—
91C2×M16—24no—
92C2×M16—24no—
93C2×M16—24no—
94C2×M16—22no—
95C2×M16—42no—
96C2×M16—42no—
97C2×M16—42no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C22×C8—42no—
104C22×C8—41no—
105C4×Q8—42no—
106C4×Q8—42no—
107C4×Q8—44no—
108C4×Q8—44no—
109C4×Q8—42no—
110C4×Q8—42no—
111#117—22no—
112#117—24no—
113#117—24no—
114#117—22no—
115#117—41no—
116#117—42no—
117#117—42no—
118#117—41no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—81no—
133C2×C4⋊C4—81no—
134C2×C4⋊C4—81no—
135C2×C4⋊C4—81no—
136C2×C4⋊C4—82no—
137C2×C4⋊C4—81no—
138C2×C4⋊C4C4⋊C412yes—
139C2×C22⋊C4—41no—
140C2×C42—22no—
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—42no—
144C2×C42—41no—
145C2×C42—41no—
146C2×C42—41no—
147C2×C42—42no—
148C2×C42—42no—
149C2×C42C4⋊C412yes—
150#107—48no—
151#106—48no—
152#101—24no—
153#101—24no—
154#101—24no—
155#101—24no—
156#101—42no—
157#101C4⋊C418yescomplemented
158#95—24no—
159#95—24no—
160#95—44no—
161#95—42no—
162#95—42no—
163#95—42no—
164#95—42no—
165#95—82no—
166#95—81no—
167C24#12011yescharacteristic
168C2×Q8—41no—
169C2×Q8—42no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—41no—
177C2×Q8—42no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—41no—
186C2×Q8C2×C4⋊C412yes—
187C2×Q8C4×D411yescharacteristic
188C2×Q8C4×Q811yescharacteristic
189C2×Q8C22⋊Q812yes—
190C2×Q8C22⋊Q812yes—
191C2×Q8#12312yes—
192C2×Q8#12511yescharacteristic
193C2×Q8#12512yes—
194C2×Q8C4⋊Q812yes—
195C2×Q8C4⋊Q811yescharacteristic
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—41no—
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×C4C2×C4⋊C411yescharacteristic
233C22×C4C4×D412yes—
234C22×C4#12012yes—
235C22×C4#12111yescharacteristic
236C22×C4#12112yes—
237C22×C4C22⋊Q812yes—
238C22×C4C22⋊Q811yescharacteristic
239C22×C4C22⋊Q812yes—
240C22×C4#12311yescharacteristic
241C22×C4#12711yescharacteristic
242C22×C4C4⋊Q811yescharacteristic
243C22×C4C4⋊Q812yes—
244M16—28no—
245M16—44no—
246M16—44no—
247M16—44no—
248M16—42no—
249M16—42no—
250M16—81no—
251M16—81no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—81no—
257C2×C8—81no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—44no—
268C4⋊C4—81no—
269C4⋊C4—82no—
270C4⋊C4—84no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C42—42no—
276C42—44no—
277C42—42no—
278C42—44no—
279C42—42no—
280C42—44no—
281C42—81no—
282C42—82no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23#38511yescharacteristic
288C23#39211yescharacteristic
289C23#39511yescharacteristic
290C23B(4,𝔽2)11yescharacteristic
291C23B(4,𝔽2)11yescharacteristic
292C23#45811yescharacteristic
293C23#45811yescharacteristic
294Q8—42no—
295Q8—41no—
296Q8—41no—
297Q8—42no—
298Q8—42no—
299Q8—42no—
300Q8—41no—
301Q8—42no—
302Q8—42no—
303Q8—41no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—24no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—82no—
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—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4#38411yescharacteristic
361C2×C4#38512yes—
362C2×C4#38911yescharacteristic
363C2×C4#38912yes—
364C2×C4#39512yes—
365C2×C4#39812yes—
366C2×C4#39811yescharacteristic
367C2×C4#39911yescharacteristic
368C8—48no—
369C8—82no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22#144411yescharacteristic
383C22#186711yescharacteristic
384C22#186711yescharacteristic
385C22#202111yescharacteristic
386C22#202211yescharacteristic
387C22#206211yescharacteristic
388C22#206211yescharacteristic
389C22#220211yescharacteristic
390C22#220311yescharacteristic
391C22#220411yescharacteristic
392C22#220511yescharacteristic
393C4—22no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—21no—
401C4—44no—
402C4—84no—
403C4—82no—
404C4—82no—
405C4—82no—
406C4—162no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2#1134411yescharacteristic
412C2#1297011yescharacteristic
413C2#1297311yescharacteristic
414C2#1335711yescharacteristic
415C2#1335911yescharacteristic
416C2#1346411yescharacteristic
417C2#1346511yescharacteristic
subgroupG/H
center, socleC23B(4,𝔽2)
commutatorC22×Q8C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC24#120
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1394C4semidirect
157. #101C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18572
#144672
C2×#13942
#107502