Finite Group Databasebeta

← #148821

Subgroups — #148821

Order 512.

732 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 138 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×#1857C211yesmaximal, characteristic
2#14501C211yesmaximal, characteristic
3C2×#1394C211yesmaximal, characteristic
4C2×#1394C212yesmaximal
5#10750C212yesmaximal
6C2×#519C2211yescharacteristic
7C2×#428—21no—
8C22×#101C2212yes—
9C22×#101C2211yescharacteristic
10C2×#391C2212yes—
11C2×#391C2211yescharacteristic
12#1857—22no—
13#1857C414yescomplemented
14#1736—22no—
15C2×#339—21no—
16C2×#339—22no—
17#1442—22no—
18#1441—21no—
19#1436—22no—
20#1434—21no—
21#1394—24no—
22#1394—22no—
23C23×Q8C2311yescharacteristic
24C22×M16—21no—
25C22×M16—22no—
26#519—21no—
27#519—22no—
28#519—42no—
29#519C2×C412yes—
30C2×C4×Q8—41no—
31C2×C4×Q8—41no—
32C2×#117—21no—
33C2×#117—22no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4D412yes—
36C22×C4⋊C4D411yescharacteristic
37C22×C42D412yes—
38C22×C42D411yescharacteristic
39#428—24no—
40#428—42no—
41C2×#107—41no—
42C2×#106—41no—
43C2×#101—21no—
44C2×#101—21no—
45C2×#101—21no—
46C2×#101—22no—
47C2×#101—22no—
48C2×#101—22no—
49C2×#101—22no—
50C2×#101—22no—
51C2×#101C2×C412yes—
52C2×#101D412yes—
53C2×#101Q814yes—
54C2×#101Q812yes—
55#391—22no—
56#391—22no—
57#391—22no—
58#391—22no—
59#391—22no—
60#391C2×C412yes—
61#391D412yes—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70#340—42no—
71#340—42no—
72#339—24no—
73#339—44no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8—41no—
77C22×Q8—41no—
78C22×Q8C22×C411yescharacteristic
79C22×Q8C2×D411yescharacteristic
80C22×Q8C2×D411yescharacteristic
81C22×Q8C2×Q811yescharacteristic
82C22×Q8C2×Q812yes—
83C22×Q8#4112yes—
84C22×Q8#4111yescharacteristic
85C22×Q8#4112yes—
86C22×Q8#4111yescharacteristic
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—24no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—42no—
101C2×M16—42no—
102C2×M16—42no—
103C2×M16—42no—
104C2×M16—42no—
105C22×C8—41no—
106C22×C8—42no—
107C4×Q8—42no—
108C4×Q8—42no—
109C4×Q8—42no—
110C4×Q8—42no—
111C4×Q8—44no—
112C4×Q8—44no—
113#117—24no—
114#117—22no—
115#117—22no—
116#117—24no—
117#117—42no—
118#117—41no—
119#117—42no—
120#117—41no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—81no—
134C2×C4⋊C4C22⋊C412yes—
135C2×C22⋊C4—41no—
136C2×C42—22no—
137C2×C42—21no—
138C2×C42—22no—
139C2×C42—41no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42—41no—
143C2×C42—41no—
144C2×C42—42no—
145C2×C42C22⋊C412yes—
146#107—44no—
147#106—44no—
148#101—22no—
149#101—22no—
150#101—22no—
151#101—24no—
152#101—24no—
153#101—42no—
154#101—42no—
155#101C4⋊C414yescomplemented
156#95—24no—
157#95—24no—
158#95—44no—
159#95—44no—
160#95—42no—
161#95—42no—
162#95—42no—
163#95—42no—
164#95—81no—
165#95—81no—
166#95—81no—
167C24#12011yescharacteristic
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—42no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—42no—
177C2×Q8—41no—
178C2×Q8—42no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—42no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—41no—
185C2×Q8—42no—
186C2×Q8C2×C4⋊C411yescharacteristic
187C2×Q8#11711yescharacteristic
188C2×Q8C4×Q812yes—
189C2×Q8#12111yescharacteristic
190C2×Q8C22⋊Q811yescharacteristic
191C2×Q8C22⋊Q812yes—
192C2×Q8C22⋊Q812yes—
193C2×Q8#12512yes—
194C2×Q8#12512yes—
195C2×Q8#12612yes—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C4×D412yes—
232C22×C4#12111yescharacteristic
233C22×C4#12112yes—
234C22×C4#12112yes—
235C22×C4C22⋊Q811yescharacteristic
236C22×C4C22⋊Q812yes—
237C22×C4#12312yes—
238C22×C4#12311yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4C4⋊Q812yes—
241C22×C4C4⋊Q811yescharacteristic
242M16—24no—
243M16—42no—
244M16—42no—
245M16—42no—
246M16—44no—
247M16—44no—
248M16—82no—
249M16—82no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—44no—
254C2×C8—82no—
255C2×C8—82no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—81no—
267C4⋊C4—82no—
268C4⋊C4—84no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—42no—
272C42—42no—
273C42—44no—
274C42—42no—
275C42—44no—
276C42—44no—
277C42—81no—
278C42—82no—
279C23—22no—
280C23—22no—
281C23#38611yescharacteristic
282C23#39411yescharacteristic
283C23#39511yescharacteristic
284C23B(4,𝔽2)11yescharacteristic
285C23B(4,𝔽2)11yescharacteristic
286C23#45811yescharacteristic
287C23#45811yescharacteristic
288Q8—41no—
289Q8—41no—
290Q8—42no—
291Q8—41no—
292Q8—41no—
293Q8—42no—
294Q8—42no—
295Q8—42no—
296Q8—42no—
297Q8—42no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4#38011yescharacteristic
351C2×C4#38712yes—
352C2×C4#38912yes—
353C2×C4#39111yescharacteristic
354C2×C4#39612yes—
355C2×C4#39811yescharacteristic
356C2×C4#39812yes—
357C2×C4#40011yescharacteristic
358C8—44no—
359C8—84no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22#144111yescharacteristic
369C22#188911yescharacteristic
370C22#189011yescharacteristic
371C22#204411yescharacteristic
372C22#204411yescharacteristic
373C22#206211yescharacteristic
374C22#206211yescharacteristic
375C22#220312yes—
376C22#220412yes—
377C4—21no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—21no—
382C4—21no—
383C4—22no—
384C4—21no—
385C4—84no—
386C4—82no—
387C4—82no—
388C4—82no—
389C4—82no—
390C4—162no—
391C2—21no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2#1122211yescharacteristic
396C2#1305611yescharacteristic
397C2#1305711yescharacteristic
398C2#1342712yes—
399C2#1346512yes—
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
13. #1857C4semidirect
155. #101C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18572
#145012
C2×#13942
#107502