Finite Group Databasebeta

← #142174

Subgroups — #142174

Order 512.

718 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 94 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#14916C211yesmaximal, characteristic
2C4×#342C211yesmaximal, characteristic
3#8559C211yesmaximal, characteristic
4#7250C212yesmaximal
5#7250C212yesmaximal
6C22×#125C411yescharacteristic, complemented
7C2×#385—21no—
8C2×C4×C22⋊C4—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1444C411yescharacteristic, complemented
15#1442C411yescharacteristic, complemented
16#1441—21no—
17#1439—21no—
18#1439C2211yescharacteristic
19#1439C2211yescharacteristic
20#1436—21no—
21#1434C411yescharacteristic, complemented
22#1280—21no—
23C24×C4D411yescharacteristic
24C2×#125—41no—
25C2×#125—41no—
26C2×#125—44no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C22⋊C4—22no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4D411yescharacteristic
36C22×C22⋊C4C2311yescharacteristic
37C22×C42—41no—
38C22×C42C2×C411yescharacteristic
39C22×C42D411yescharacteristic
40C2×C4⋊C8—41no—
41C2×C22⋊C8—22no—
42#385—44no—
43C4×C22⋊C4—24no—
44C4×C22⋊C4—41no—
45C4×C22⋊C4—41no—
46C4×C22⋊C4—42no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—41no—
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95D411yescharacteristic
59#342—24no—
60#342—22no—
61#342Q814yes—
62#336—41no—
63#323—24no—
64C25C2×D411yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4—81no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C22×C8—42no—
84#125—44no—
85#125—44no—
86#125—44no—
87#125—44no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—81no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—81no—
120C2×C22⋊C4C2×Q812yes—
121C2×C22⋊C4#4112yes—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—42no—
127C2×C42—44no—
128C2×C42—81no—
129C2×C42—82no—
130C2×C42—82no—
131C4⋊C8—82no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—44no—
143#95—42no—
144#95—81no—
145#95—81no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24C2×C22⋊C411yescharacteristic
155C24C22⋊Q812yes—
156C24#13512yes—
157C24AGL(1,ℤ/8)11yescharacteristic
158C24#13711yescharacteristic
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—42no—
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—44no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—44no—
198C22×C4—44no—
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×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—41no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—84no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4#9911yescharacteristic
231C22×C4#9911yescharacteristic
232C22×C4#10412yes—
233C22×C4#10412yes—
234C22×C4C4×Q812yes—
235C22×C4#12411yescharacteristic
236C22×C4C4⋊Q811yescharacteristic
237C2×C8—82no—
238C2×C8—82no—
239C2×C8—82no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—48no—
246C4⋊C4—44no—
247C4⋊C4—84no—
248C22⋊C4—28no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—48no—
255C22⋊C4—44no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C42—42no—
259C42—42no—
260C42—42no—
261C42—81no—
262C42—88no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—22no—
268C23—22no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—22no—
273C23—21no—
274C23—21no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23#39111yescharacteristic
294C23C2×#9911yescharacteristic
295C23#41011yescharacteristic
296C23#41611yescharacteristic
297C23#41711yescharacteristic
298C23C2×#10411yescharacteristic
299C23C2×#10411yescharacteristic
300C23#47812yes—
301C23#47912yes—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—48no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4AΣL(1,𝔽16)12yes—
376C8—162no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—22no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—42no—
400C22—42no—
401C22—82no—
402C22—81no—
403C22—81no—
404C22—82no—
405C22#150011yescharacteristic
406C22#150111yescharacteristic
407C22#150411yescharacteristic
408C22#150911yescharacteristic
409C22#198911yescharacteristic
410C22#199411yescharacteristic
411C22#199611yescharacteristic
412C22C2×AΣL(1,𝔽16)11yescharacteristic
413C22#212311yescharacteristic
414C22#212611yescharacteristic
415C22#212711yescharacteristic
416C4—22no—
417C4—42no—
418C4—41no—
419C4—42no—
420C4—42no—
421C4—41no—
422C4—41no—
423C4—82no—
424C4—88no—
425C4—81no—
426C4—82no—
427C4—82no—
428C4—161no—
429C2—21no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—42no—
434C2—42no—
435C2#991011yescharacteristic
436C2#1172511yescharacteristic
437C2#1173211yescharacteristic
438C2#1175311yescharacteristic
439C2#1176011yescharacteristic
440C2#1329611yescharacteristic
441C2#1330011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C22×#125C4semidirect
14. #1444C4semidirect
15. #1442C4semidirect
21. #1434C4semidirect

Maximal subgroups

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

subgroupindex
#149162
C4×#3422
#85592
#72502