Finite Group Databasebeta

← #101208

Subgroups — #101208

Order 512.

660 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 104 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
2#15163C211yesmaximal, characteristic
3#8558C211yesmaximal, characteristic, complemented
4#7640C212yesmaximal
5#7039C212yesmaximal, complemented
6C2×#432—21no—
7C2×#398C411yescharacteristic
8C2×#396—21no—
9C2×#394C411yescharacteristic
10C2×#390C2211yescharacteristic
11C2×#388—21no—
12C2×C4×C4⋊C4C2211yescharacteristic
13#1940—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#432—42no—
41#432—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×#97—22no—
49#398—42no—
50#396—44no—
51#394—42no—
52#390—22no—
53#390—22no—
54#388—42no—
55#388—42no—
56C4×C4⋊C4—22no—
57C4×C4⋊C4—22no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95C2×C411yescharacteristic
63C2×#95C2×C411yescharacteristic
64C2×#95C2×C411yescharacteristic
65C2×#95D411yescharacteristic
66#336—22no—
67#336—22no—
68#336D414yes—
69#324—42no—
70C25—21no—
71C22×D4—22no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4—42no—
75C22×D4—81no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C22×C8—24no—
87C22×C8—42no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—81no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
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—41no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—81no—
125C2×C22⋊C4—81no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—81no—
133C2×C42—81no—
134C2×C42D812yes—
135C2×C42SD1612yes—
136C2×C42C2×D412yes—
137C2×C42#4112yes—
138C4⋊C8—44no—
139C4⋊C8—82no—
140C4⋊C8—82no—
141C4⋊C8—82no—
142C22⋊C8—42no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145C22⋊C8—42no—
146#97—44no—
147#95—22no—
148#95—22no—
149#95—41no—
150#95—41no—
151#95—42no—
152#95—42no—
153#95—42no—
154#95—41no—
155#95—41no—
156C24—22no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24—41no—
169C24—41no—
170C24—41no—
171C24C2×C22⋊C411yescharacteristic
172C2×D4—24no—
173C2×D4—42no—
174C2×D4—82no—
175C2×D4—82no—
176C2×D4—84no—
177C2×D4—84no—
178C22×C4—21no—
179C22×C4—21no—
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×C4C4×D412yes—
230C22×C4#12011yescharacteristic
231C22×C4#12112yes—
232C22×C4#12311yescharacteristic
233C22×C4C2×D811yescharacteristic
234C22×C4C2×SD1611yescharacteristic
235C22×C4AGL(1,ℤ/8)11yescharacteristic
236C22×C4AGL(1,ℤ/8)12yes—
237C22×C4AGL(1,ℤ/8)11yescharacteristic
238C22×C4#13711yescharacteristic
239C22×C4#13711yescharacteristic
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—82no—
244C2×C8—82no—
245C2×C8—82no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—82no—
250C4⋊C4—84no—
251C4⋊C4—84no—
252C22⋊C4—48no—
253C22⋊C4—84no—
254C22⋊C4—84no—
255C22⋊C4—84no—
256C22⋊C4—81no—
257C22⋊C4—81no—
258C22⋊C4—82no—
259C22⋊C4—82no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—42no—
267C42—42no—
268C42—42no—
269C42—81no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—81no—
282C23—81no—
283C23—81no—
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#38611yescharacteristic
310C23C2×#10011yescharacteristic
311C23#41311yescharacteristic
312C23C2×#10211yescharacteristic
313C23#41711yescharacteristic
314C23#41711yescharacteristic
315C23#41811yescharacteristic
316D4—84no—
317D4—84no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—81no—
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#44711yescharacteristic
380C2×C4#44811yescharacteristic
381C2×C4#46812yes—
382C2×C4#46912yes—
383C2×C4#48011yescharacteristic
384C2×C4#48111yescharacteristic
385C8—84no—
386C8—162no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22#154511yescharacteristic
415C22#154711yescharacteristic
416C22#155711yescharacteristic
417C22#156011yescharacteristic
418C22#187411yescharacteristic
419C22#188411yescharacteristic
420C22#189111yescharacteristic
421C22#189511yescharacteristic
422C22#190011yescharacteristic
423C22#191211yescharacteristic
424C22#191311yescharacteristic
425C4—22no—
426C4—22no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—82no—
436C4—81no—
437C4—161no—
438C4—161no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—162no—
444C2#903511yescharacteristic
445C2#914111yescharacteristic
446C2#920411yescharacteristic
447C2#938611yescharacteristic
448C2#944811yescharacteristic
449C2#949011yescharacteristic
450C2#951511yescharacteristic
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. #21955C2semidirect
3. #8558C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219552
#151632
#85582
#76402
#70392