Finite Group Databasebeta

← #146864

Subgroups — #146864

Order 512.

1078 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C4×#390C211yesmaximal, characteristic
2C2×#1393C212yesmaximal
3C2×#1393C212yesmaximal
4#10761C212yesmaximal
5C22×C4×D4C2211yescharacteristic
6C22×#99C2214yes—
7C2×#390C2211yescharacteristic
8C2×#390C2211yescharacteristic
9C2×C4×C4⋊C4C411yescharacteristic, complemented
10C2×C4×C22⋊C4—22no—
11C2×C43C411yescharacteristic, complemented
12C2×#342—22no—
13#1437—22no—
14C4⋊#95C411yescharacteristic, complemented
15C4⋊#95C411yescharacteristic, complemented
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—41no—
25C2×C4×D4—41no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—22no—
30C22×C22⋊C4—22no—
31C22×C22⋊C4—24no—
32C22×C22⋊C4D412yes—
33C22×C42—21no—
34C22×C42—21no—
35C22×C42—21no—
36C22×C42C2×C411yescharacteristic
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C411yescharacteristic
39C2×#99—24no—
40C2×#99—24no—
41C2×#99—24no—
42C2×#99—24no—
43C2×#99—24no—
44C2×#99C2×C418yes—
45#390—22no—
46#390—22no—
47#390C2×C414yes—
48#390D412yes—
49#390Q812yes—
50C4×C4⋊C4—41no—
51C4×C4⋊C4—41no—
52C4×C22⋊C4—48no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—21no—
56C2×#95—42no—
57C2×#95—42no—
58C43—42no—
59#342—44no—
60C25C2×D412yes—
61C22×D4—41no—
62C22×D4—41no—
63C22×D4—41no—
64C22×D4—41no—
65C22×D4C22×C412yes—
66C22×D4C2×D411yescharacteristic
67C22×D4C2×Q811yescharacteristic
68C22×D4#4112yes—
69C22×D4#4112yes—
70C22×D4#4112yes—
71C22×D4#4112yes—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—24no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—42no—
80C23×C4—44no—
81C23×C4—44no—
82C23×C4—44no—
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22×C411yescharacteristic
87C4×D4—44no—
88C4×D4—44no—
89C4×D4—44no—
90C4×D4—44no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4C4212yes—
102C2×C4⋊C4C4⋊C412yes—
103C2×C22⋊C4—28no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—28no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—44no—
111C2×C22⋊C4—44no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4—44no—
114C2×C22⋊C4—44no—
115C2×C22⋊C4—44no—
116C2×C22⋊C4—44no—
117C2×C22⋊C4—44no—
118C2×C22⋊C4—44no—
119C2×C42—22no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—24no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—42no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42—44no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42C4212yes—
140C2×C42C4⋊C412yes—
141#99—28no—
142#99—28no—
143#99—44no—
144#99—44no—
145#95—44no—
146#95—44no—
147#95—42no—
148#95—84no—
149#95—84no—
150C24—22no—
151C24—22no—
152C24—22no—
153C24—22no—
154C24—42no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24C2×C22⋊C411yescharacteristic
159C24C4×D414yes—
160C24#12112yes—
161C24C22⋊Q812yes—
162C24#12314yes—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4#11714yes—
178C2×D4C4×D412yes—
179C2×D4C4×Q812yes—
180C2×D4#12412yes—
181C2×D4#12512yes—
182C2×D4#12614yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—416no—
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—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—48no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—88no—
233C22×C4—84no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4—84no—
237C22×C4—84no—
238C22×C4—84no—
239C22×C4#9911yescharacteristic
240C22×C4#9911yescharacteristic
241C22×C4#9912yes—
242C22×C4C2×C4211yescharacteristic
243C22×C4C2×C4⋊C411yescharacteristic
244C22×C4#11711yescharacteristic
245C22×C4#11711yescharacteristic
246C22×C4#11711yescharacteristic
247C22×C4#11711yescharacteristic
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C22⋊C4—48no—
255C22⋊C4—416no—
256C22⋊C4—48no—
257C22⋊C4—48no—
258C22⋊C4—48no—
259C22⋊C4—48no—
260C22⋊C4—84no—
261C22⋊C4—84no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—42no—
267C42—42no—
268C42—48no—
269C42—42no—
270C42—42no—
271C42—48no—
272C42—82no—
273C42—82no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—22no—
279C23—22no—
280C23—24no—
281C23—24no—
282C23—22no—
283C23—22no—
284C23—24no—
285C23—24no—
286C23—44no—
287C23—42no—
288C23—42no—
289C23—44no—
290C23—44no—
291C23—42no—
292C23—42no—
293C23—44no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—82no—
299C23—82no—
300C23—82no—
301C23—82no—
302C23—82no—
303C23—82no—
304C23C4×C22⋊C411yescharacteristic
305C23#38011yescharacteristic
306C23#38111yescharacteristic
307C23#38714yes—
308C23#38814yes—
309C23C2×#9911yescharacteristic
310C23C2×#9911yescharacteristic
311C23#41011yescharacteristic
312C23#41011yescharacteristic
313D4—44no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—42no—
318D4—44no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—88no—
365C2×C4—88no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—88no—
374C2×C4—88no—
375C2×C4—88no—
376C2×C4—164no—
377C2×C4—164no—
378C2×C4—164no—
379C2×C4—164no—
380C2×C4#35312yes—
381C2×C4#35412yes—
382C2×C4#37611yescharacteristic
383C2×C4C4×C4⋊C411yescharacteristic
384C2×C4#38211yescharacteristic
385C2×C4#38311yescharacteristic
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—24no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—82no—
411C22—82no—
412C22—84no—
413C22—84no—
414C22#143411yescharacteristic
415C22C4×#9911yescharacteristic
416C22C4×#9911yescharacteristic
417C22#179911yescharacteristic
418C22#179911yescharacteristic
419C22#181911yescharacteristic
420C22#182011yescharacteristic
421C22C2×#35311yescharacteristic
422C22C2×#35411yescharacteristic
423C22#212611yescharacteristic
424C22#212611yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—22no—
430C4—22no—
431C4—44no—
432C4—44no—
433C4—42no—
434C4—82no—
435C4—82no—
436C4—82no—
437C4—168no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—44no—
443C2—44no—
444C2#1080111yescharacteristic
445C2C4×#35311yescharacteristic
446C2C4×#35411yescharacteristic
447C2#1279711yescharacteristic
448C2#1279811yescharacteristic
449C2#1285611yescharacteristic
450C2#1285711yescharacteristic
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
11. C2×C43C4semidirect
14. C4⋊#95C4semidirect
15. C4⋊#95C4semidirect
18. #1393C4semidirect

Maximal subgroups

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

subgroupindex
C4×#3902
C2×#13932
#107612