Finite Group Databasebeta

← #141152

Subgroups — #141152

Order 512.

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

normal subgroups only

This group has 686 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#15050C211yesmaximal, characteristic
2C2×#1351C212yesmaximal
3C2×#1344C212yesmaximal
4C4×#342C211yesmaximal, characteristic
5#8559C211yesmaximal, characteristic
6C2×#397C2211yescharacteristic
7C2×#393C2211yescharacteristic
8C2×#391C411yescharacteristic, complemented
9C2×C4×C22⋊C4—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1797—21no—
12C2×#342—21no—
13C2×#342C2212yes—
14C2×#323C2212yes—
15#1446C411yescharacteristic, complemented
16#1445C411yescharacteristic, complemented
17#1443—21no—
18#1440—21no—
19#1434—21no—
20#1434C411yescharacteristic, complemented
21C22⋊#95—21no—
22#1351—24no—
23#1344—24no—
24#1280—21no—
25C23×Q8C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C22⋊C4—22no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4D411yescharacteristic
32C22×C22⋊C4C2311yescharacteristic
33C22×C42—41no—
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C4⋊C8—41no—
37C2×C22⋊C8—22no—
38#397—22no—
39#397—22no—
40#393—22no—
41#393—22no—
42#391—41no—
43#391—41no—
44C4×C22⋊C4—24no—
45C4×C22⋊C4—41no—
46C4×C22⋊C4—41no—
47C4×C22⋊C4—42no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—41no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95D411yescharacteristic
63#342—24no—
64#342—22no—
65#342D414yes—
66#336—41no—
67#323—22no—
68#323—22no—
69C25C2×D411yescharacteristic
70C22×Q8—22no—
71C22×Q8—42no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—81no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—22no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—81no—
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—42no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—81no—
115C2×C22⋊C4C2×D412yes—
116C2×C22⋊C4#4112yes—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—42no—
122C2×C42—44no—
123C2×C42—82no—
124C2×C42—82no—
125C2×C42—81no—
126C4⋊C8—82no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#95—22no—
130#95—22no—
131#95—41no—
132#95—41no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—44no—
137#95—41no—
138#95—41no—
139#95—42no—
140#95—42no—
141#95—41no—
142#95—41no—
143#95—41no—
144#95—41no—
145#95—81no—
146#95—81no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—21no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24—41no—
155C24C2×C22⋊C411yescharacteristic
156C24#12112yes—
157C24#13512yes—
158C24AGL(1,ℤ/8)11yescharacteristic
159C24#13711yescharacteristic
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—82no—
163C2×Q8—84no—
164C2×Q8—84no—
165C2×Q8—82no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—44no—
199C22×C4—44no—
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—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—84no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4#9911yescharacteristic
238C22×C4#9911yescharacteristic
239C22×C4#10412yes—
240C22×C4#10412yes—
241C22×C4C4×D412yes—
242C22×C4#12411yescharacteristic
243C22×C4#12711yescharacteristic
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—82no—
247C4⋊C4—82no—
248C22⋊C4—28no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—84no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C42—42no—
259C42—81no—
260C42—81no—
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#39011yescharacteristic
294C23C2×#9911yescharacteristic
295C23#41011yescharacteristic
296C23#41611yescharacteristic
297C23#41711yescharacteristic
298C23C2×#10411yescharacteristic
299C23C2×#10411yescharacteristic
300C23#46312yes—
301C23#46412yes—
302Q8—84no—
303Q8—84no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—48no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
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×C4—81no—
376C2×C4—81no—
377C2×C4#35412yes—
378C8—162no—
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—22no—
392C22—22no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22#150011yescharacteristic
408C22#150111yescharacteristic
409C22#150411yescharacteristic
410C22#150911yescharacteristic
411C22#196211yescharacteristic
412C22#196411yescharacteristic
413C22#198311yescharacteristic
414C22#212011yescharacteristic
415C22#212111yescharacteristic
416C22C2×#35411yescharacteristic
417C22#212911yescharacteristic
418C4—22no—
419C4—42no—
420C4—41no—
421C4—41no—
422C4—42no—
423C4—41no—
424C4—42no—
425C4—88no—
426C4—82no—
427C4—81no—
428C4—161no—
429C4—161no—
430C4—161no—
431C2—21no—
432C2—21no—
433C2—21no—
434C2—21no—
435C2—42no—
436C2—42no—
437C2#971311yescharacteristic
438C2#1172811yescharacteristic
439C2#1173111yescharacteristic
440C2#1175611yescharacteristic
441C2#1175911yescharacteristic
442C2#1322211yescharacteristic
443C2#1322411yescharacteristic
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
8. C2×#391C4semidirect
15. #1446C4semidirect
16. #1445C4semidirect
20. #1434C4semidirect

Maximal subgroups

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

subgroupindex
#150502
C2×#13512
C2×#13442
C4×#3422
#85592