Finite Group Databasebeta

← #146865

Subgroups — #146865

Order 512.

972 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 158 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#13785C211yesmaximal, characteristic
2C2×#1393C212yesmaximal
3C2×#1393C212yesmaximal
4#10761C212yesmaximal
5C22×C4×D4C2211yescharacteristic
6C22×#99C2214yes—
7C2×#390C2211yescharacteristic
8C2×#390C2211yescharacteristic
9C2×C4×C4⋊C4C411yescharacteristic, complemented
10C2×#376C411yescharacteristic, complemented
11C2×#342—22no—
12#1437—22no—
13C4⋊#95C411yescharacteristic, complemented
14C4⋊#95C411yescharacteristic, complemented
15C22⋊#95—22no—
16C22⋊#95—22no—
17#1393—24no—
18#1393C418yescomplemented
19C23×D4C2311yescharacteristic
20C24×C4D412yes—
21C2×C4×D4—22no—
22C2×C4×D4—22no—
23C2×C4×D4—42no—
24C2×C4×D4—42no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C22⋊C4—22no—
29C22×C22⋊C4—22no—
30C22×C22⋊C4—24no—
31C22×C22⋊C4D412yes—
32C22×C42—21no—
33C22×C42C2×C411yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42C2×C411yescharacteristic
36C2×#99—24no—
37C2×#99—24no—
38C2×#99—24no—
39C2×#99—24no—
40C2×#99—24no—
41C2×#99C2×C418yes—
42#390—22no—
43#390—22no—
44#390C2×C414yes—
45#390D412yes—
46#390Q812yes—
47C4×C4⋊C4—42no—
48#376—42no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—42no—
55C2×#95—42no—
56#342—44no—
57C25C2×D412yes—
58C22×D4—41no—
59C22×D4—41no—
60C22×D4—41no—
61C22×D4—41no—
62C22×D4C22×C412yes—
63C22×D4C2×D411yescharacteristic
64C22×D4C2×Q811yescharacteristic
65C22×D4#4112yes—
66C22×D4#4112yes—
67C22×D4#4112yes—
68C22×D4#4112yes—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—24no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—42no—
77C23×C4—44no—
78C23×C4—82no—
79C23×C4—82no—
80C23×C4C22⋊C412yes—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C4×D4—48no—
85C4×D4—48no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4C4212yes—
97C2×C4⋊C4C4⋊C412yes—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—28no—
101C2×C22⋊C4—28no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—44no—
106C2×C22⋊C4—44no—
107C2×C22⋊C4—44no—
108C2×C22⋊C4—44no—
109C2×C22⋊C4—44no—
110C2×C22⋊C4—82no—
111C2×C22⋊C4—82no—
112C2×C22⋊C4—82no—
113C2×C22⋊C4—82no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—42no—
125C2×C42—82no—
126C2×C42C4212yes—
127C2×C42C4⋊C412yes—
128#99—28no—
129#99—28no—
130#99—44no—
131#99—44no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—82no—
136#95—82no—
137#95—84no—
138#95—84no—
139C24—22no—
140C24—22no—
141C24—22no—
142C24—22no—
143C24—42no—
144C24—42no—
145C24—42no—
146C24—42no—
147C24C2×C22⋊C411yescharacteristic
148C24C4×D414yes—
149C24#12112yes—
150C24C22⋊Q812yes—
151C24#12314yes—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4#11714yes—
167C2×D4C4×D412yes—
168C2×D4C4×Q812yes—
169C2×D4#12412yes—
170C2×D4#12512yes—
171C2×D4#12614yes—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
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—48no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—88no—
221C22×C4—88no—
222C22×C4—84no—
223C22×C4—84no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4—84no—
228C22×C4#9912yes—
229C22×C4#9912yes—
230C22×C4C2×C4211yescharacteristic
231C22×C4C2×C4⋊C411yescharacteristic
232C22×C4#11711yescharacteristic
233C22×C4#11711yescharacteristic
234C22×C4#11711yescharacteristic
235C22×C4#11711yescharacteristic
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C22⋊C4—48no—
245C22⋊C4—88no—
246C22⋊C4—84no—
247C22⋊C4—84no—
248C22⋊C4—84no—
249C22⋊C4—84no—
250C42—24no—
251C42—24no—
252C42—42no—
253C42—42no—
254C42—42no—
255C42—42no—
256C42—84no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—24no—
264C23—24no—
265C23—22no—
266C23—22no—
267C23—24no—
268C23—24no—
269C23—44no—
270C23—42no—
271C23—42no—
272C23—44no—
273C23—44no—
274C23—42no—
275C23—42no—
276C23—44no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—82no—
282C23—82no—
283C23—82no—
284C23—82no—
285C23—82no—
286C23—82no—
287C23C4×C22⋊C411yescharacteristic
288C23#38011yescharacteristic
289C23#38111yescharacteristic
290C23#38714yes—
291C23#38814yes—
292C23C2×#9911yescharacteristic
293C23C2×#9911yescharacteristic
294C23#41011yescharacteristic
295C23#41011yescharacteristic
296D4—44no—
297D4—42no—
298D4—42no—
299D4—42no—
300D4—42no—
301D4—44no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
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—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—88no—
339C2×C4—88no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—88no—
356C2×C4—88no—
357C2×C4—88no—
358C2×C4—164no—
359C2×C4—164no—
360C2×C4—164no—
361C2×C4—164no—
362C2×C4#37611yescharacteristic
363C2×C4C4×C4⋊C411yescharacteristic
364C2×C4#38211yescharacteristic
365C2×C4#38311yescharacteristic
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—24no—
380C22—44no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22—44no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—42no—
389C22—42no—
390C22—82no—
391C22—82no—
392C22—84no—
393C22—84no—
394C22#143411yescharacteristic
395C22C4×#9911yescharacteristic
396C22C4×#9911yescharacteristic
397C22#179911yescharacteristic
398C22#179911yescharacteristic
399C22#181911yescharacteristic
400C22#182011yescharacteristic
401C22#212711yescharacteristic
402C22#212711yescharacteristic
403C22#212811yescharacteristic
404C22#212911yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—42no—
410C4—42no—
411C4—82no—
412C4—82no—
413C4—82no—
414C4—82no—
415C4—82no—
416C4—168no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—44no—
422C2—44no—
423C2#1080111yescharacteristic
424C2#1262711yescharacteristic
425C2#1262811yescharacteristic
426C2#1279911yescharacteristic
427C2#1280011yescharacteristic
428C2#1285811yescharacteristic
429C2#1285911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, prime radical, cosocleC23×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C2×C4×C4⋊C4C4semidirect
10. C2×#376C4semidirect
13. C4⋊#95C4semidirect
14. C4⋊#95C4semidirect
18. #1393C4semidirect

Maximal subgroups

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

subgroupindex
#137852
C2×#13932
#107612