Finite Group Databasebeta

← #136273

Subgroups — #136273

Order 512.

451 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 84 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#22204C211yesmaximal, characteristic
2#13001C211yesmaximal, characteristic
3#12929C211yesmaximal, characteristic, complemented
4#11704C211yesmaximal, characteristic
5#11327C211yesmaximal, characteristic, complemented
6#7264C211yesmaximal, characteristic, complemented
7#7184C211yesmaximal, characteristic, complemented
8#3050C2211yescharacteristic
9#2929—21no—
10C22×#106C2211yescharacteristic
11C2×#389—21no—
12C2×#365—21no—
13C2×#360—21no—
14#2112C2211yescharacteristic
15#2112C2211yescharacteristic
16#1929—21no—
17#1922—21no—
18#1896—21no—
19#1876—21no—
20#1866—21no—
21#1810C2211yescharacteristic
22C2×#340—21no—
23C2×#340C2211yescharacteristic
24C2×#336C2211yescharacteristic
25#1386—21no—
26#1383—21no—
27#1348C412yescomplemented
28#1347C412yescomplemented
29C2×#135D411yescharacteristic
30C2×#131—21no—
31C23×C8D411yescharacteristic
32C23×C8C2311yescharacteristic
33#525—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—41no—
37C22×C42D411yescharacteristic
38C2×M32—21no—
39C2×M32—21no—
40#467C2×C411yescharacteristic
41#467Q811yescharacteristic
42#466C2×C411yescharacteristic
43#466Q811yescharacteristic
44#465—21no—
45#465—21no—
46#435—41no—
47#435—41no—
48C8×D4—41no—
49C8×D4—41no—
50C2×#108D411yescharacteristic
51C2×#108D411yescharacteristic
52#427D411yescharacteristic
53C2×#106—21no—
54C2×#106—21no—
55C2×#106—42no—
56#424—21no—
57C2×#104—21no—
58C2×#104—21no—
59#416—21no—
60#408—21no—
61#407—21no—
62#405—21no—
63#389—42no—
64#389—42no—
65C2×#95—41no—
66#365—42no—
67#360—42no—
68C22⋊C16—21no—
69C22⋊C16—21no—
70C22⋊C16—21no—
71C22⋊C16C2×C412yes—
72#341—21no—
73#340—21no—
74#340—42no—
75#340C2×C412yes—
76#339—21no—
77#339—21no—
78#336—21no—
79#336—21no—
80C2×#41—21no—
81C23×C4—21no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4C2×D411yescharacteristic
85#135—41no—
86#135—41no—
87C2×Q16C4⋊C411yescharacteristic, complemented
88C2×SD16—21no—
89C2×D8C4⋊C411yescharacteristic, complemented
90#131—81no—
91#131—81no—
92C2×M16—21no—
93C2×M16—21no—
94C2×M16—41no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8SD1611yescharacteristic
103C22×C8SD1611yescharacteristic
104C22×C8C22×C411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×Q811yescharacteristic
108C22×C8#4111yescharacteristic
109C22×C8#4111yescharacteristic
110#123—41no—
111C22⋊Q8—21no—
112C22⋊Q8—21no—
113#121—21no—
114#121—21no—
115#120—41no—
116C4×D4—41no—
117C4×D4—41no—
118#117—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—82no—
133C2×C4⋊C4—82no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—41no—
137C2×C42—41no—
138M32—41no—
139M32—41no—
140M32—41no—
141M32—41no—
142C2×C16—22no—
143C2×C16—41no—
144#108—41no—
145#108—41no—
146#107C4⋊C411yescharacteristic, complemented
147#107C4⋊C411yescharacteristic, complemented
148#106—21no—
149#106—22no—
150#106—22no—
151#106—41no—
152#106—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155#104—81no—
156#104—81no—
157#103—21no—
158#103—21no—
159#102—21no—
160#102—21no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C22⋊C8—41no—
165#97—41no—
166C4×C8—41no—
167#95—44no—
168C24AGL(1,ℤ/8)11yescharacteristic
169#41—41no—
170#41—41no—
171#41—41no—
172#41—41no—
173C2×Q8—21no—
174C2×D4—21no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4C4×D411yescharacteristic
205C22×C4#12011yescharacteristic
206C22×C4C22⋊Q811yescharacteristic
207C22×C4C2×SD1611yescharacteristic
208C22×C4#13511yescharacteristic
209C22×C4AGL(1,ℤ/8)11yescharacteristic
210Q16—41no—
211SD16—81no—
212D8—41no—
213M16—41no—
214M16—41no—
215M16—41no—
216M16—41no—
217M16—82no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—81no—
235C2×C8—82no—
236C2×C8C2×C4⋊C411yescharacteristic
237C2×C8C4×D411yescharacteristic
238C2×C8C22⋊Q811yescharacteristic
239C2×C8#12311yescharacteristic
240C4⋊C4—24no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—81no—
251C4⋊C4—88no—
252C4⋊C4—81no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42—41no—
259C42—41no—
260C42—41no—
261C42—81no—
262C16—42no—
263C16—81no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23#44211yescharacteristic
272C23#44911yescharacteristic
273C23#47611yescharacteristic
274Q8—41no—
275Q8—41no—
276D4—41no—
277D4—41no—
278D4—81no—
279D4—81no—
280D4—81no—
281D4—81no—
282D4—81no—
283D4—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—161no—
317C2×C4—161no—
318C2×C4—161no—
319C2×C4—161no—
320C2×C4#38511yescharacteristic
321C2×C4C4×SD1611yescharacteristic
322C2×C4#45011yescharacteristic
323C2×C4#45311yescharacteristic
324C2×C4#45611yescharacteristic
325C2×C4B(4,𝔽2)11yescharacteristic
326C2×C4#45811yescharacteristic
327C2×C4#47811yescharacteristic
328C8—21no—
329C8—21no—
330C8—21no—
331C8—21no—
332C8—41no—
333C8—41no—
334C8—81no—
335C8—81no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22#160011yescharacteristic
354C22#160911yescharacteristic
355C22#186411yescharacteristic
356C22#218011yescharacteristic
357C22AGL(1,ℤ/16)11yescharacteristic
358C22#223011yescharacteristic
359C22#223111yescharacteristic
360C4—21no—
361C4—21no—
362C4—82no—
363C4—84no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—162no—
368C4#160411yescharacteristic
369C4#161311yescharacteristic
370C4#185911yescharacteristic
371C4#186711yescharacteristic
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—161no—
379C2#887311yescharacteristic
380C2#1295011yescharacteristic
381C2#1295511yescharacteristic
subgroupG/H
center, socleC22#1864
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radical#3050C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12929C2semidirect
5. #11327C2semidirect
6. #7264C2semidirect
7. #7184C2semidirect
27. #1348C4semidirect
28. #1347C4semidirect
87. C2×Q16C4⋊C4semidirect
89. C2×D8C4⋊C4semidirect
146. #107C4⋊C4semidirect
147. #107C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#222042
#130012
#129292
#117042
#113272
#72642
#71842