Finite Group Databasebeta

← #100405

Subgroups — #100405

Order 512.

567 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 567 conjugacy classes of subgroups — too many to draw. Only its 104 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#33107C211yesmaximal, characteristic
2#13936C211yesmaximal, characteristic, complemented
3#9712C211yesmaximal, characteristic
4#9680C211yesmaximal, characteristic
5#8707C211yesmaximal, characteristic, complemented
6#7039C211yesmaximal, characteristic, complemented
7#7028C211yesmaximal, characteristic, complemented
8C2×#481—21no—
9#2856C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1982C412yescomplemented
12#1948C412yescomplemented
13#1926—22no—
14#1874—21no—
15#1797C2211yescharacteristic
16#1797C2211yescharacteristic
17C4×C22⋊C8C2211yescharacteristic
18C2×#327—22no—
19C2×#327—21no—
20C2×#327—21no—
21#1443—22no—
22C4⋊#95—21no—
23C4⋊#95—21no—
24C4⋊#95C2211yescharacteristic
25#1434—21no—
26#1434—21no—
27C4×#95C2211yescharacteristic
28#1279—22no—
29C2×#127C2×C411yescharacteristic
30C2×#121—21no—
31C2×#121—21no—
32C2×#121—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4D411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C42—21no—
37C22×C42—21no—
38C22×C42C2311yescharacteristic
39#481—42no—
40#481—42no—
41C2×#106—22no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8C2×C412yes—
44C2×#102—22no—
45C2×#102—22no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C4×C8—21no—
50#390—21no—
51#390—21no—
52#390—21no—
53#384C2×C411yescharacteristic
54C4×C4⋊C4—21no—
55C4×C4⋊C4—21no—
56C4×C4⋊C4C2×C412yes—
57C4×C4⋊C4D412yes—
58C2×#95—21no—
59C2×#95—22no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95D411yescharacteristic
66C2×#95D411yescharacteristic
67C2×#95D412yes—
68#340—22no—
69#336—21no—
70#336—21no—
71#336—21no—
72#327—42no—
73#327—44no—
74#327—42no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—41no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—22no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—22no—
88C22×C8—41no—
89C22×C8—41no—
90#127—42no—
91#121—42no—
92#121—42no—
93#121—41no—
94#121—41no—
95#121—41no—
96#121—41no—
97#121—41no—
98#121—41no—
99#121—41no—
100#121—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4C22⋊C412yes—
116C2×C4⋊C4C22⋊C412yes—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42—42no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42—82no—
128C2×C42C22×C411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42#4111yescharacteristic
131C2×C42#4111yescharacteristic
132#106—44no—
133C4⋊C8—42no—
134C4⋊C8—42no—
135#102—44no—
136#102—44no—
137C22⋊C8—22no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C4×C8—42no—
144#95—22no—
145#95—22no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—42no—
150#95—42no—
151#95—44no—
152#95—41no—
153#95—41no—
154#95—82no—
155#95—82no—
156#95—82no—
157#95SD1612yes—
158C24—41no—
159C24#12011yescharacteristic
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—42no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4—81no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
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—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4C2×SD1611yescharacteristic
224C22×C4#13511yescharacteristic
225C22×C4#13511yescharacteristic
226C22×C4#13511yescharacteristic
227C22×C4AGL(1,ℤ/8)11yescharacteristic
228C22×C4AGL(1,ℤ/8)11yescharacteristic
229C22×C4AGL(1,ℤ/8)11yescharacteristic
230C22×C4#13711yescharacteristic
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—81no—
236C2×C8—81no—
237C2×C8—81no—
238C2×C8—81no—
239C2×C8—81no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C22⋊C4—82no—
255C22⋊C4—81no—
256C22⋊C4—81no—
257C22⋊C4—81no—
258C22⋊C4—81no—
259C42—22no—
260C42—22no—
261C42—41no—
262C42—41no—
263C42—82no—
264C42—82no—
265C42—82no—
266C42—82no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23#38611yescharacteristic
279C23#44811yescharacteristic
280C23#44911yescharacteristic
281C23#44911yescharacteristic
282C23#44911yescharacteristic
283C23#45311yescharacteristic
284C23#45411yescharacteristic
285D4—82no—
286D4—82no—
287D4—161no—
288D4—161no—
289D4—162no—
290D4—161no—
291D4—161no—
292D4—161no—
293D4—161no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
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—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4#41012yes—
351C2×C4#41611yescharacteristic
352C2×C4#41811yescharacteristic
353C2×C4#42112yes—
354C2×C4C4×SD1611yescharacteristic
355C2×C4#44011yescharacteristic
356C2×C4#46011yescharacteristic
357C2×C4#46411yescharacteristic
358C2×C4#48211yescharacteristic
359C2×C4#48211yescharacteristic
360C8—82no—
361C8—82no—
362C8—161no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—161no—
376C22—161no—
377C22—161no—
378C22—161no—
379C22—161no—
380C22—161no—
381C22—161no—
382C22#162011yescharacteristic
383C22#162211yescharacteristic
384C22#162611yescharacteristic
385C22#162811yescharacteristic
386C22#187511yescharacteristic
387C22#188011yescharacteristic
388C22#188311yescharacteristic
389C22#189711yescharacteristic
390C22#189911yescharacteristic
391C22#190411yescharacteristic
392C22#190611yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—41no—
401C4—82no—
402C4—82no—
403C4—82no—
404C4—82no—
405C4—162no—
406C4—161no—
407C4#151412yes—
408C4#152112yes—
409C2—21no—
410C2—21no—
411C2—21no—
412C2—21no—
413C2—321no—
414C2#897811yescharacteristic
415C2#915711yescharacteristic
416C2#918111yescharacteristic
417C2#923811yescharacteristic
418C2#926111yescharacteristic
419C2#932911yescharacteristic
420C2#936711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13936C2semidirect
5. #8707C2semidirect
6. #7039C2semidirect
7. #7028C2semidirect
11. #1982C4semidirect
12. #1948C4semidirect

Maximal subgroups

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

subgroupindex
#331072
#139362
#97122
#96802
#87072
#70392
#70282