Finite Group Databasebeta

← #99568

Subgroups — #99568

Order 512.

874 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 116 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#21955C211yesmaximal, characteristic, complemented
2C4×#394C211yesmaximal, characteristic
3#8310C212yesmaximal
4C4×#324C211yesmaximal, characteristic, complemented
5#7028C212yesmaximal, complemented
6C22×C4×D4C2211yescharacteristic
7C2×#432—21no—
8C2×#394C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4—21no—
11C2×C4×C22⋊C4—21no—
12#1940—24no—
13#1797—21no—
14#1797C2212yes—
15C4×C22⋊C8C2212yes—
16C2×D4⋊C8—24no—
17C2×#324—21no—
18#1442C411yescharacteristic
19#1441C411yescharacteristic
20#1440—21no—
21#1439—21no—
22#1438C411yescharacteristic
23C4×#95C2211yescharacteristic
24#1280—21no—
25C23×D4C2×C411yescharacteristic
26C24×C4—21no—
27C2×C4×D4—22no—
28C2×C4×D4—22no—
29C2×C4×D4—44no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C42—21no—
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42C2311yescharacteristic
42#432—42no—
43#432—42no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8D414yes—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×#97—22no—
50C2×C4×C8—22no—
51#394—22no—
52#394—22no—
53C4×C4⋊C4—22no—
54C4×C4⋊C4—22no—
55C4×C22⋊C4—44no—
56C4×C22⋊C4—42no—
57C4×C22⋊C4—42no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95C2×C411yescharacteristic
64C2×#95C2×C411yescharacteristic
65C2×#95C2×C411yescharacteristic
66C2×#95D411yescharacteristic
67#336—22no—
68#336—22no—
69#336—22no—
70D4⋊C8—48no—
71#324—24no—
72C25—21no—
73C22×D4—22no—
74C22×D4—21no—
75C22×D4—22no—
76C22×D4—42no—
77C22×D4—42no—
78C22×D4M1612yes—
79C23×C4—24no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—41no—
87C23×C4—41no—
88C23×C4—41no—
89C23×C4—41no—
90C23×C4—41no—
91C23×C4—41no—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C22×C8—24no—
100C22×C8—42no—
101C4×D4—28no—
102C4×D4—44no—
103C4×D4—44no—
104C4×D4—44no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—41no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—42no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42—42no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42D812yes—
142C2×C42SD1612yes—
143C2×C42C2×D412yes—
144C2×C42#4112yes—
145C4⋊C8—44no—
146C4⋊C8—44no—
147C4⋊C8—82no—
148C22⋊C8—24no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151C22⋊C8—42no—
152#97—44no—
153C4×C8—44no—
154#95—22no—
155#95—22no—
156#95—21no—
157#95—22no—
158#95—42no—
159#95—42no—
160#95—42no—
161#95—42no—
162#95—42no—
163#95M1612yes—
164C24—22no—
165C24—22no—
166C24—22no—
167C24—41no—
168C24—41no—
169C24—41no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24—41no—
174C24—41no—
175C24—41no—
176C24—41no—
177C24—41no—
178C24—41no—
179C24C2×C22⋊C411yescharacteristic
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—24no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—44no—
186C2×D4—48no—
187C2×D4—44no—
188C2×D4—42no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
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—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—44no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—82no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4#10212yes—
254C22×C4#10212yes—
255C22×C4#10412yes—
256C22×C4C4×D412yes—
257C22×C4#12011yescharacteristic
258C22×C4#12112yes—
259C22×C4#12311yescharacteristic
260C22×C4C2×M1611yescharacteristic
261C22×C4C2×M1611yescharacteristic
262C22×C4#13112yes—
263C22×C4C2×D811yescharacteristic
264C22×C4C2×SD1611yescharacteristic
265C22×C4AGL(1,ℤ/8)11yescharacteristic
266C22×C4#13711yescharacteristic
267C2×C8—44no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—44no—
271C2×C8—82no—
272C2×C8—82no—
273C4⋊C4—24no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C22⋊C4—48no—
279C22⋊C4—48no—
280C22⋊C4—48no—
281C22⋊C4—48no—
282C22⋊C4—48no—
283C42—22no—
284C42—22no—
285C42—24no—
286C42—22no—
287C42—22no—
288C42—42no—
289C42—42no—
290C42—81no—
291C42—82no—
292C42—82no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—24no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23#38611yescharacteristic
333C23#40711yescharacteristic
334C23#40811yescharacteristic
335C23C2×#10211yescharacteristic
336C23#41711yescharacteristic
337C23C2×#10411yescharacteristic
338C23#42111yescharacteristic
339D4—28no—
340D4—44no—
341D4—44no—
342D4—44no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—81no—
406C2×C4#43512yes—
407C2×C4#43612yes—
408C2×C4#44711yescharacteristic
409C2×C4#44811yescharacteristic
410C2×C4#45412yes—
411C2×C4B(4,𝔽2)12yes—
412C2×C4#48011yescharacteristic
413C2×C4#48111yescharacteristic
414C8—84no—
415C8—84no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—44no—
429C22—44no—
430C22—44no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22#148611yescharacteristic
444C22#148911yescharacteristic
445C22#149211yescharacteristic
446C22#149511yescharacteristic
447C22#187011yescharacteristic
448C22#187111yescharacteristic
449C22#187411yescharacteristic
450C22#188411yescharacteristic
451C22#188811yescharacteristic
452C22#188911yescharacteristic
453C22#189111yescharacteristic
454C4—22no—
455C4—22no—
456C4—21no—
457C4—21no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—82no—
463C4—82no—
464C4—81no—
465C4—82no—
466C4#159812yes—
467C4#159912yes—
468C2—21no—
469C2—21no—
470C2—21no—
471C2—21no—
472C2—84no—
473C2#894411yescharacteristic
474C2#896411yescharacteristic
475C2#896711yescharacteristic
476C2#900911yescharacteristic
477C2#901911yescharacteristic
478C2#903211yescharacteristic
479C2#903511yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21955C2semidirect
4. C4×#324C2semidirect
5. #7028C2semidirect

Maximal subgroups

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

subgroupindex
#219552
C4×#3942
#83102
C4×#3242
#70282