Finite Group Databasebeta

← #100404

Subgroups — #100404

Order 512.

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

normal subgroups only

This group has 617 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#33106C211yesmaximal, characteristic
2#13918C211yesmaximal, characteristic, complemented
3#9712C211yesmaximal, characteristic
4#9680C211yesmaximal, characteristic
5#8706C211yesmaximal, characteristic, complemented
6#7039C211yesmaximal, characteristic, complemented
7#7028C211yesmaximal, characteristic, complemented
8C2×#480—21no—
9#2856C2211yescharacteristic
10C2×#389—21no—
11C2×#382—21no—
12C2×C4×C4⋊C4C2211yescharacteristic
13#1982C412yescomplemented
14#1947C412yescomplemented
15#1925—22no—
16#1874—21no—
17#1797C2211yescharacteristic
18#1797C2211yescharacteristic
19C4×C22⋊C8C2211yescharacteristic
20C2×#327—22no—
21C2×#327—21no—
22C2×#327—21no—
23#1442—22no—
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#480—42no—
41#480—42no—
42C2×#107—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
57#382—42no—
58#382—42no—
59C4×C4⋊C4—21no—
60C4×C4⋊C4—21no—
61C4×C4⋊C4C2×C412yes—
62C4×C4⋊C4D412yes—
63C2×#95—21no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95D411yescharacteristic
70C2×#95D411yescharacteristic
71C2×#95D412yes—
72#340—22no—
73#336—21no—
74#336—21no—
75#336—21no—
76#327—42no—
77#327—44no—
78#327—42no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—41no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C22×C8—22no—
92C22×C8—41no—
93C22×C8—41no—
94#127—42no—
95#121—42no—
96#121—42no—
97#121—41no—
98#121—41no—
99#121—41no—
100#121—41no—
101#121—41no—
102#121—41no—
103#121—41no—
104#121—41no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—82no—
120C2×C4⋊C4C22⋊C412yes—
121C2×C4⋊C4C22⋊C412yes—
122C2×C4⋊C4D812yes—
123C2×C4⋊C4SD1612yes—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—42no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42C22×C411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4111yescharacteristic
141C2×C42#4111yescharacteristic
142#107—44no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145#102—44no—
146#102—44no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C4×C8—42no—
154#95—22no—
155#95—22no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—42no—
160#95—42no—
161#95—44no—
162#95—44no—
163#95—44no—
164#95—41no—
165#95—41no—
166#95D812yes—
167C24—41no—
168C24#12011yescharacteristic
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4—42no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4—81no—
179C2×D4—81no—
180C2×D4—81no—
181C2×D4—81no—
182C2×D4—81no—
183C2×D4—81no—
184C2×D4—81no—
185C2×D4—81no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
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—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4C2×D811yescharacteristic
235C22×C4C2×D811yescharacteristic
236C22×C4C2×SD1611yescharacteristic
237C22×C4#13511yescharacteristic
238C22×C4AGL(1,ℤ/8)11yescharacteristic
239C22×C4AGL(1,ℤ/8)11yescharacteristic
240C22×C4AGL(1,ℤ/8)11yescharacteristic
241C22×C4#13711yescharacteristic
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—81no—
247C2×C8—81no—
248C2×C8—81no—
249C2×C8—81no—
250C2×C8—81no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—88no—
265C4⋊C4#10214yes—
266C22⋊C4—82no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C42—22no—
272C42—22no—
273C42—41no—
274C42—41no—
275C42—42no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—82no—
280C42—82no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23#38611yescharacteristic
293C23#44711yescharacteristic
294C23#44711yescharacteristic
295C23#44811yescharacteristic
296C23#44911yescharacteristic
297C23#45311yescharacteristic
298C23#45411yescharacteristic
299D4—82no—
300D4—82no—
301D4—161no—
302D4—161no—
303D4—162no—
304D4—161no—
305D4—161no—
306D4—161no—
307D4—161no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4#41012yes—
367C2×C4C2×#10211yescharacteristic
368C2×C4#41711yescharacteristic
369C2×C4#42112yes—
370C2×C4C4×D811yescharacteristic
371C2×C4#44211yescharacteristic
372C2×C4#45911yescharacteristic
373C2×C4#46311yescharacteristic
374C2×C4#48011yescharacteristic
375C2×C4#48111yescharacteristic
376C8—82no—
377C8—82no—
378C8—161no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—161no—
392C22—161no—
393C22—161no—
394C22—161no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22#161911yescharacteristic
399C22#162211yescharacteristic
400C22#162511yescharacteristic
401C22#162711yescharacteristic
402C22#187411yescharacteristic
403C22#187911yescharacteristic
404C22#188311yescharacteristic
405C22#189711yescharacteristic
406C22#189911yescharacteristic
407C22#190511yescharacteristic
408C22#190611yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—44no—
416C4—41no—
417C4—41no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—162no—
422C4—161no—
423C4#151212yes—
424C4#151712yes—
425C2—21no—
426C2—21no—
427C2—21no—
428C2—21no—
429C2—321no—
430C2#897711yescharacteristic
431C2#915611yescharacteristic
432C2#918011yescharacteristic
433C2#923711yescharacteristic
434C2#926011yescharacteristic
435C2#933011yescharacteristic
436C2#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. #13918C2semidirect
5. #8706C2semidirect
6. #7039C2semidirect
7. #7028C2semidirect
13. #1982C4semidirect
14. #1947C4semidirect

Maximal subgroups

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

subgroupindex
#331062
#139182
#97122
#96802
#87062
#70392
#70282