Finite Group Databasebeta

← #136155

Subgroups — #136155

Order 512.

469 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 88 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#14603C211yesmaximal, characteristic
2C2×#1353C211yesmaximal, characteristic
3#11296C211yesmaximal, characteristic
4#9023C211yesmaximal, characteristic
5#8856C211yesmaximal, characteristic
6#7251C211yesmaximal, characteristic
7#7189C211yesmaximal, characteristic
8C2×#400C2211yescharacteristic
9#2316C2211yescharacteristic
10#1852—21no—
11#1824—21no—
12C2×#343C2211yescharacteristic
13#1736—21no—
14C2×#339—21no—
15C2×#324C2211yescharacteristic
16C2×#324C2211yescharacteristic
17#1442—21no—
18#1441—21no—
19#1441C2211yescharacteristic
20#1436—21no—
21#1436—21no—
22#1436—21no—
23C4⋊#95C2211yescharacteristic
24#1434—21no—
25#1353C414yescomplemented
26#1294—21no—
27#1294—21no—
28C22×M16—21no—
29C2×#117—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4D411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
34C22×C4⋊C4C2311yescharacteristic
35C22×C42—21no—
36C22×C42D411yescharacteristic
37C2×#107—41no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40#400—21no—
41#400—21no—
42#400C2×C412yes—
43#400Q812yes—
44#389—41no—
45#389—41no—
46#380—21no—
47#380—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95D411yescharacteristic
61C2×#95D411yescharacteristic
62#343—21no—
63#343C2×C412yes—
64#340—41no—
65#340—41no—
66#340—41no—
67#340—41no—
68#340—41no—
69#339—42no—
70#324—21no—
71#324—21no—
72#324—21no—
73#324C2×C412yes—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×M16—22no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16—41no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—41no—
92#117—41no—
93#117—41no—
94#117—41no—
95#117—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—81no—
114C2×C4⋊C4—81no—
115C2×C4⋊C4C4⋊C412yes—
116C2×C4⋊C4Q1612yes—
117C2×C4⋊C4C22×C411yescharacteristic
118C2×C4⋊C4C2×Q811yescharacteristic
119C2×C4⋊C4#4111yescharacteristic
120C2×C4⋊C4#4111yescharacteristic
121C2×C22⋊C4—41no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42—42no—
128C2×C42—81no—
129C2×C42—81no—
130#107—82no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135#95—22no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—21no—
140#95—41no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—81no—
149#95—81no—
150#95—81no—
151#95—81no—
152#95—81no—
153#95—81no—
154#95C4⋊C412yes—
155C24#12011yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—41no—
170C22×C4—41no—
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—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
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×C4C2×C4⋊C411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4C4×D411yescharacteristic
215C22×C4C22⋊Q811yescharacteristic
216C22×C4C22⋊Q811yescharacteristic
217C22×C4#12311yescharacteristic
218C22×C4C2×Q1611yescharacteristic
219C22×C4#13511yescharacteristic
220C22×C4AGL(1,ℤ/8)11yescharacteristic
221C22×C4#13711yescharacteristic
222M16—42no—
223M16—42no—
224M16—81no—
225M16—81no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—81no—
229C2×C8—81no—
230C2×C8—81no—
231C2×C8—81no—
232C2×C8—81no—
233C2×C8—81no—
234C2×C8—81no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C4⋊C4—84no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C22⋊C4—81no—
255C22⋊C4—81no—
256C42—42no—
257C42—81no—
258C42—81no—
259C42—81no—
260C42—84no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23#38511yescharacteristic
266C23#44911yescharacteristic
267C23#45111yescharacteristic
268C23#45411yescharacteristic
269C23#45511yescharacteristic
270C23B(4,𝔽2)11yescharacteristic
271C23#45811yescharacteristic
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4C4×Q1611yescharacteristic
347C2×C4#44011yescharacteristic
348C2×C4#47711yescharacteristic
349C2×C4#47911yescharacteristic
350C8—82no—
351C8—82no—
352C8—161no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22#159711yescharacteristic
366C22#160211yescharacteristic
367C22#161211yescharacteristic
368C22#161711yescharacteristic
369C22#186011yescharacteristic
370C22#186311yescharacteristic
371C22#186711yescharacteristic
372C22#220011yescharacteristic
373C22#220111yescharacteristic
374C22#220411yescharacteristic
375C22#220411yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—82no—
385C4—82no—
386C4—82no—
387C4—82no—
388C4—82no—
389C4—161no—
390C4—161no—
391C4—161no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2—21no—
396C2#886711yescharacteristic
397C2#1214411yescharacteristic
398C2#1214711yescharacteristic
399C2#1222311yescharacteristic
400C2#1222511yescharacteristic
401C2#1296111yescharacteristic
402C2#1297211yescharacteristic
subgroupG/H
center, socleC23B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1353C4semidirect

Maximal subgroups

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

subgroupindex
#146032
C2×#13532
#112962
#90232
#88562
#72512
#71892