Finite Group Databasebeta

← #100408

Subgroups — #100408

Order 512.

603 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 112 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#13933C211yesmaximal, 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×#389—21no—
11C2×C4×C4⋊C4C2211yescharacteristic
12#1982C412yescomplemented
13#1948C412yescomplemented
14#1926—22no—
15#1874—21no—
16#1797C2211yescharacteristic
17#1797C2211yescharacteristic
18C4×C22⋊C8C2211yescharacteristic
19C2×#327—22no—
20C2×#327—21no—
21C2×#327—21no—
22#1442—22no—
23C4⋊#95—21no—
24C4⋊#95—21no—
25C4⋊#95C2211yescharacteristic
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⋊C4—22no—
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4D411yescharacteristic
37C22×C42—21no—
38C22×C42—21no—
39C22×C42C2311yescharacteristic
40#481—42no—
41#481—42no—
42C2×#106—22no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8C2×C412yes—
45C2×#102—22no—
46C2×#102—22no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C4×C8—21no—
51#390—21no—
52#390—21no—
53#390—21no—
54#389—42no—
55#389—42no—
56#384C2×C411yescharacteristic
57C4×C4⋊C4—21no—
58C4×C4⋊C4—21no—
59C4×C4⋊C4C2×C412yes—
60C4×C4⋊C4D412yes—
61C2×#95—21no—
62C2×#95—22no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95D411yescharacteristic
68C2×#95D411yescharacteristic
69C2×#95D412yes—
70#340—22no—
71#336—21no—
72#336—21no—
73#336—21no—
74#327—42no—
75#327—44no—
76#327—42no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—41no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C22×C8—22no—
90C22×C8—41no—
91C22×C8—41no—
92#127—42no—
93#121—42no—
94#121—42no—
95#121—41no—
96#121—41no—
97#121—41no—
98#121—41no—
99#121—41no—
100#121—41no—
101#121—41no—
102#121—41no—
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—44no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—82no—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4D812yes—
121C2×C4⋊C4SD1612yes—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—42no—
128C2×C42—42no—
129C2×C42—42no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42—82no—
133C2×C42C22×C411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42#4111yescharacteristic
136C2×C42#4111yescharacteristic
137#106—44no—
138C4⋊C8—42no—
139C4⋊C8—42no—
140#102—44no—
141#102—44no—
142C22⋊C8—22no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C4×C8—42no—
149#95—22no—
150#95—22no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—42no—
155#95—42no—
156#95—44no—
157#95—41no—
158#95—41no—
159#95—82no—
160#95—82no—
161#95SD1612yes—
162C24—41no—
163C24#12011yescharacteristic
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—42no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4—81no—
179C2×D4—81no—
180C2×D4—81no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
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—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12311yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4C2×SD1611yescharacteristic
229C22×C4C2×SD1611yescharacteristic
230C22×C4#13511yescharacteristic
231C22×C4AGL(1,ℤ/8)11yescharacteristic
232C22×C4AGL(1,ℤ/8)11yescharacteristic
233C22×C4#13711yescharacteristic
234C22×C4#13711yescharacteristic
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—81no—
240C2×C8—81no—
241C2×C8—81no—
242C2×C8—81no—
243C2×C8—81no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—88no—
258C4⋊C4#10214yes—
259C22⋊C4—82no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C22⋊C4—81no—
263C22⋊C4—81no—
264C42—22no—
265C42—22no—
266C42—41no—
267C42—41no—
268C42—82no—
269C42—82no—
270C42—82no—
271C42—82no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23#38611yescharacteristic
284C23#44711yescharacteristic
285C23#44811yescharacteristic
286C23#44811yescharacteristic
287C23#44911yescharacteristic
288C23#45311yescharacteristic
289C23#45411yescharacteristic
290D4—82no—
291D4—82no—
292D4—161no—
293D4—161no—
294D4—162no—
295D4—161no—
296D4—161no—
297D4—161no—
298D4—161no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
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—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4#41012yes—
356C2×C4C2×#10211yescharacteristic
357C2×C4#41711yescharacteristic
358C2×C4#42112yes—
359C2×C4C4×SD1611yescharacteristic
360C2×C4#44011yescharacteristic
361C2×C4#46011yescharacteristic
362C2×C4#46411yescharacteristic
363C2×C4#48011yescharacteristic
364C2×C4#48111yescharacteristic
365C8—82no—
366C8—82no—
367C8—161no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—161no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22—161no—
385C22—161no—
386C22—161no—
387C22#162111yescharacteristic
388C22#162311yescharacteristic
389C22#162611yescharacteristic
390C22#162711yescharacteristic
391C22#187411yescharacteristic
392C22#188011yescharacteristic
393C22#188311yescharacteristic
394C22#189711yescharacteristic
395C22#189911yescharacteristic
396C22#190411yescharacteristic
397C22#190611yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—41no—
405C4—41no—
406C4—44no—
407C4—82no—
408C4—82no—
409C4—82no—
410C4—162no—
411C4—161no—
412C4#151212yes—
413C4#151712yes—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2—321no—
419C2#897911yescharacteristic
420C2#915611yescharacteristic
421C2#917911yescharacteristic
422C2#923811yescharacteristic
423C2#926111yescharacteristic
424C2#932911yescharacteristic
425C2#936611yescharacteristic
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. #13933C2semidirect
5. #8707C2semidirect
6. #7039C2semidirect
7. #7028C2semidirect
12. #1982C4semidirect
13. #1948C4semidirect

Maximal subgroups

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

subgroupindex
#331072
#139332
#97122
#96802
#87072
#70392
#70282