Finite Group Databasebeta

← #140877

Subgroups — #140877

Order 512.

684 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 110 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#15161C211yesmaximal, characteristic
2C2×#1352C212yesmaximal
3C2×#1351C212yesmaximal
4#10846C211yesmaximal, characteristic
5C4×#323C211yesmaximal, characteristic
6C2×#399C2211yescharacteristic
7C2×#393C2211yescharacteristic
8C2×#391C411yescharacteristic, complemented
9C2×#380—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11C4×C22⋊C8—21no—
12C2×#342—21no—
13C2×#342C2212yes—
14C2×#323C2212yes—
15#1446C411yescharacteristic, complemented
16#1445C411yescharacteristic, complemented
17#1441—21no—
18#1439—21no—
19#1436—21no—
20C4⋊#95C411yescharacteristic, complemented
21C22⋊#95—21no—
22#1352—24no—
23#1351—24no—
24#1280—21no—
25C23×Q8C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—41no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—22no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4D411yescharacteristic
35C22×C22⋊C4C2311yescharacteristic
36C22×C42C2×C411yescharacteristic
37C22×C42D411yescharacteristic
38C2×C22⋊C8—22no—
39C2×C4×C8—41no—
40#399—22no—
41#399—22no—
42#393—22no—
43#393—22no—
44#391—41no—
45#391—41no—
46#380—24no—
47#380—42no—
48C4×C22⋊C4—41no—
49C4×C22⋊C4—41no—
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—22no—
64#342—22no—
65#342—42no—
66#336—41no—
67#323—22no—
68#323D414yes—
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—22no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—42no—
83C23×C4—81no—
84C23×C4—81no—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4SD1612yes—
90C23×C4Q1612yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—42no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—44no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—81no—
103C2×C4⋊C4—81no—
104C2×C4⋊C4—82no—
105C2×C4⋊C4—82no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
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—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—81no—
123C2×C22⋊C4—81no—
124C2×C22⋊C4SD1612yes—
125C2×C22⋊C4Q1612yes—
126C2×C22⋊C4C2×D412yes—
127C2×C22⋊C4#4112yes—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—81no—
133C2×C42—81no—
134C22⋊C8—24no—
135C22⋊C8—42no—
136C4×C8—82no—
137#95—22no—
138#95—22no—
139#95—41no—
140#95—41no—
141#95—42no—
142#95—42no—
143#95—44no—
144#95—41no—
145#95—41no—
146#95—42no—
147#95—42no—
148#95—81no—
149#95—81no—
150#95—81no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—41no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24C2×C22⋊C411yescharacteristic
160C24#12411yescharacteristic
161C24#12711yescharacteristic
162C24C2×SD1611yescharacteristic
163C24C2×SD1611yescharacteristic
164C24C2×Q1611yescharacteristic
165C24C2×Q1611yescharacteristic
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—82no—
169C2×Q8—84no—
170C2×Q8—84no—
171C2×Q8—82no—
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—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
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—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—44no—
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—41no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—84no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4#9911yescharacteristic
241C22×C4#9911yescharacteristic
242C22×C4#10312yes—
243C22×C4#10312yes—
244C22×C4#10312yes—
245C22×C4#10312yes—
246C22×C4#10412yes—
247C22×C4C4×D412yes—
248C22×C4#12112yes—
249C2×C8—44no—
250C2×C8—82no—
251C2×C8—82no—
252C4⋊C4—82no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C4⋊C4—88no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—82no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C22⋊C4—82no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C42—42no—
269C42—81no—
270C42—81no—
271C42—81no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23#39011yescharacteristic
311C23C2×#9911yescharacteristic
312C23#41011yescharacteristic
313C23C2×#10311yescharacteristic
314C23C2×#10311yescharacteristic
315C23C2×#10411yescharacteristic
316C23#42111yescharacteristic
317C23#48711yescharacteristic
318C23#48711yescharacteristic
319C23#49211yescharacteristic
320C23#49411yescharacteristic
321Q8—84no—
322Q8—84no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
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—84no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—162no—
390C2×C4—162no—
391C2×C4—162no—
392C2×C4—162no—
393C2×C4#32812yes—
394C2×C4#32812yes—
395C8—84no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22C2×#32811yescharacteristic
435C22#150311yescharacteristic
436C22#151811yescharacteristic
437C22#152011yescharacteristic
438C22#195611yescharacteristic
439C22#196411yescharacteristic
440C22#198011yescharacteristic
441C22#212011yescharacteristic
442C22#212111yescharacteristic
443C22#212611yescharacteristic
444C22#212711yescharacteristic
445C4—22no—
446C4—22no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—82no—
453C4—82no—
454C4—161no—
455C4—164no—
456C4—161no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—41no—
462C2—41no—
463C2—41no—
464C2—41no—
465C2#967911yescharacteristic
466C2#1171511yescharacteristic
467C2#1171611yescharacteristic
468C2#1189711yescharacteristic
469C2#1190211yescharacteristic
470C2#1321911yescharacteristic
471C2#1322511yescharacteristic
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. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#151612
C2×#13522
C2×#13512
#108462
C4×#3232