Finite Group Databasebeta

← #99832

Subgroups — #99832

Order 512.

670 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 124 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#14312C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic, complemented
4#7572C212yesmaximal
5#7024C212yesmaximal, complemented
6C2×#431—21no—
7C2×#394C411yescharacteristic
8C2×#390C2211yescharacteristic
9C2×#388—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1939—24no—
12#1797—21no—
13#1797C2212yes—
14C4×C22⋊C8C2212yes—
15C2×#324—21no—
16#1444C411yescharacteristic
17#1441C411yescharacteristic
18#1439—21no—
19#1439—21no—
20#1438C411yescharacteristic
21#1437—21no—
22#1434C2211yescharacteristic
23#1280—21no—
24#1270—24no—
25C23×D4C2×C411yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42C2311yescharacteristic
39#431—42no—
40#431—42no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C4×C8—22no—
47C2×C4×C8—22no—
48#394—22no—
49#394C814yescomplemented
50#390—22no—
51#390—22no—
52#388—42no—
53#388—42no—
54C4×C4⋊C4—22no—
55C4×C4⋊C4—22no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95C2×C411yescharacteristic
62C2×#95C2×C411yescharacteristic
63C2×#95C2×C411yescharacteristic
64C2×#95D411yescharacteristic
65#336—22no—
66#336—22no—
67#336D414yes—
68#324—24no—
69C25—21no—
70C22×D4—22no—
71C22×D4—21no—
72C22×D4—22no—
73C22×D4—81no—
74C22×D4—81no—
75C22×D4C2×C812yes—
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—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—81no—
96C2×C4⋊C4—81no—
97C2×C4⋊C4—81no—
98C2×C4⋊C4C2×C812yes—
99C2×C4⋊C4M1612yes—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—81no—
114C2×C22⋊C4—81no—
115C2×C22⋊C4—81no—
116C2×C22⋊C4—81no—
117C2×C22⋊C4—81no—
118C2×C22⋊C4—81no—
119C2×C22⋊C4—81no—
120C2×C22⋊C4—81no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—81no—
128C2×C42—81no—
129C2×C42D812yes—
130C2×C42SD1612yes—
131C2×C42C2×D412yes—
132C2×C42#4112yes—
133C4⋊C8—44no—
134C4⋊C8—82no—
135C4⋊C8—82no—
136C22⋊C8—24no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140C4×C8—44no—
141C4×C8—82no—
142#95—22no—
143#95—22no—
144#95—21no—
145#95—22no—
146#95—42no—
147#95—41no—
148#95—41no—
149#95—81no—
150#95—81no—
151#95—81no—
152#95C2×C812yes—
153C24—22no—
154C24—41no—
155C24—41no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24C2×C22⋊C411yescharacteristic
169C2×D4—24no—
170C2×D4—42no—
171C2×D4—82no—
172C2×D4—82no—
173C2×D4—82no—
174C2×D4—84no—
175C2×D4—82no—
176C22×C4—21no—
177C22×C4—21no—
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—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—41no—
191C22×C4—41no—
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—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4C22⋊C812yes—
223C22×C4C22⋊C812yes—
224C22×C4#10212yes—
225C22×C4#10212yes—
226C22×C4#10412yes—
227C22×C4C4×D412yes—
228C22×C4#12011yescharacteristic
229C22×C4#12112yes—
230C22×C4#12311yescharacteristic
231C22×C4C22×C811yescharacteristic
232C22×C4C2×M1611yescharacteristic
233C22×C4#13112yes—
234C22×C4C2×D811yescharacteristic
235C22×C4C2×SD1611yescharacteristic
236C22×C4AGL(1,ℤ/8)11yescharacteristic
237C22×C4#13711yescharacteristic
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—82no—
243C2×C8—82no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—44no—
247C4⋊C4—82no—
248C4⋊C4—84no—
249C22⋊C4—48no—
250C22⋊C4—84no—
251C22⋊C4—84no—
252C22⋊C4—84no—
253C22⋊C4—84no—
254C22⋊C4—81no—
255C22⋊C4—81no—
256C22⋊C4—81no—
257C22⋊C4—81no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—42no—
263C42—42no—
264C42—42no—
265C42—81no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
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#38611yescharacteristic
306C23C2×C22⋊C811yescharacteristic
307C23#40811yescharacteristic
308C23C2×#10211yescharacteristic
309C23#41711yescharacteristic
310C23C2×#10411yescharacteristic
311C23#42111yescharacteristic
312D4—84no—
313D4—84no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
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—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
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×C4D4⋊C814yes—
375C2×C4C8×D412yes—
376C2×C4#43512yes—
377C2×C4#44711yescharacteristic
378C2×C4#44811yescharacteristic
379C2×C4#48011yescharacteristic
380C2×C4#48111yescharacteristic
381C8—84no—
382C8—84no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22C2×D4⋊C811yescharacteristic
411C22#147711yescharacteristic
412C22#148011yescharacteristic
413C22#148311yescharacteristic
414C22#186911yescharacteristic
415C22#187111yescharacteristic
416C22#187411yescharacteristic
417C22#191011yescharacteristic
418C22#191211yescharacteristic
419C22#191311yescharacteristic
420C22#191411yescharacteristic
421C4—22no—
422C4—22no—
423C4—24no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—82no—
432C4—81no—
433C4—161no—
434C4—161no—
435C2—21no—
436C2—21no—
437C2—21no—
438C2—21no—
439C2—162no—
440C2#893911yescharacteristic
441C2#910211yescharacteristic
442C2#910711yescharacteristic
443C2#918611yescharacteristic
444C2#918811yescharacteristic
445C2#919711yescharacteristic
446C2#920411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. C4×#324C2semidirect
5. #7024C2semidirect
49. #394C8semidirect

Maximal subgroups

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

subgroupindex
#219512
#143122
C4×#3242
#75722
#70242