Finite Group Databasebeta

← #101192

Subgroups — #101192

Order 512.

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

normal subgroups only

This group has 676 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#21951C211yesmaximal, characteristic, complemented
2#15163C211yesmaximal, characteristic
3#8558C211yesmaximal, characteristic, complemented
4#7640C212yesmaximal
5#7039C212yesmaximal, complemented
6C2×#431—21no—
7C2×#398C411yescharacteristic
8C2×#396—21no—
9C2×#394C411yescharacteristic
10C2×#390C2211yescharacteristic
11C2×#388—21no—
12C2×C4×C4⋊C4C2211yescharacteristic
13#1939—24no—
14#1797—21no—
15#1797C2212yes—
16#1797C2212yes—
17C2×#324—21no—
18#1445C411yescharacteristic
19#1439—21no—
20#1438C411yescharacteristic
21#1437—21no—
22C4⋊#95C2211yescharacteristic
23#1280—21no—
24#1270—24no—
25C23×D4C2×C411yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42C2311yescharacteristic
40#431—42no—
41#431—42no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C4×C8—22no—
49#398—42no—
50#396—42no—
51#396—42no—
52#394—42no—
53#390—22no—
54#390—22no—
55#388—42no—
56#388—42no—
57C4×C4⋊C4—22no—
58C4×C4⋊C4—22no—
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#336D414yes—
70#324—42no—
71C25—21no—
72C22×D4—22no—
73C22×D4—41no—
74C22×D4—41no—
75C22×D4—42no—
76C22×D4—81no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—24no—
88C22×C8—42no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—81no—
104C2×C4⋊C4D812yes—
105C2×C4⋊C4SD1612yes—
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—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—81no—
127C2×C22⋊C4—81no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—81no—
135C2×C42—81no—
136C2×C42D812yes—
137C2×C42SD1612yes—
138C2×C42C2×D412yes—
139C2×C42#4112yes—
140C4⋊C8—44no—
141C4⋊C8—82no—
142C4⋊C8—82no—
143C4⋊C8—82no—
144C22⋊C8—42no—
145C22⋊C8—42no—
146C22⋊C8—42no—
147C22⋊C8—42no—
148C4×C8—44no—
149#95—22no—
150#95—22no—
151#95—41no—
152#95—41no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95—41no—
157#95—41no—
158C24—22no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24—41no—
169C24—41no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24C2×C22⋊C411yescharacteristic
174C2×D4—24no—
175C2×D4—42no—
176C2×D4—82no—
177C2×D4—82no—
178C2×D4—84no—
179C2×D4—84no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
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—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#10012yes—
227C22×C4#10212yes—
228C22×C4#10212yes—
229C22×C4#10212yes—
230C22×C4#10212yes—
231C22×C4C4×D412yes—
232C22×C4#12011yescharacteristic
233C22×C4#12112yes—
234C22×C4#12311yescharacteristic
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×SD1611yescharacteristic
238C22×C4C2×SD1611yescharacteristic
239C22×C4#13512yes—
240C22×C4AGL(1,ℤ/8)11yescharacteristic
241C22×C4#13711yescharacteristic
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—44no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—84no—
255C22⋊C4—48no—
256C22⋊C4—84no—
257C22⋊C4—84no—
258C22⋊C4—84no—
259C22⋊C4—81no—
260C22⋊C4—81no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C22⋊C4—81no—
264C22⋊C4—81no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42—81no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23#38611yescharacteristic
313C23C2×#10011yescharacteristic
314C23#41311yescharacteristic
315C23C2×#10211yescharacteristic
316C23C2×#10211yescharacteristic
317C23#41611yescharacteristic
318C23#41711yescharacteristic
319D4—84no—
320D4—84no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
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—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4#33114yes—
382C2×C4#44711yescharacteristic
383C2×C4#44811yescharacteristic
384C2×C4#46512yes—
385C2×C4#46612yes—
386C2×C4#48011yescharacteristic
387C2×C4#48111yescharacteristic
388C8—84no—
389C8—162no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22C2×#33111yescharacteristic
418C22#154011yescharacteristic
419C22#154711yescharacteristic
420C22#155111yescharacteristic
421C22#187411yescharacteristic
422C22#188411yescharacteristic
423C22#189111yescharacteristic
424C22#189311yescharacteristic
425C22#190011yescharacteristic
426C22#191211yescharacteristic
427C22#191311yescharacteristic
428C4—22no—
429C4—22no—
430C4—24no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—82no—
439C4—81no—
440C4—161no—
441C4—161no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2—162no—
447C2#903511yescharacteristic
448C2#913311yescharacteristic
449C2#920411yescharacteristic
450C2#938411yescharacteristic
451C2#944811yescharacteristic
452C2#948611yescharacteristic
453C2#951611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21951C2semidirect
3. #8558C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219512
#151632
#85582
#76402
#70392