Finite Group Databasebeta

← #124258

Subgroups — #124258

Order 512.

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

normal subgroups only

This group has 904 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#14154C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic, complemented
3C4×#324C211yesmaximal, characteristic, complemented
4#7103C214yesmaximal, complemented
5C22×#127C411yescharacteristic
6C2×#390—21no—
7C2×#390—21no—
8C2×#390—21no—
9C2×#390—21no—
10C2×C4×C4⋊C4C411yescharacteristic
11C2×C4×C4⋊C4C2211yescharacteristic
12#1797—21no—
13C4×C22⋊C8—21no—
14C2×#324—21no—
15C2×#324—21no—
16#1438C412yes—
17#1438C2212yes—
18#1280C2212yes—
19#1280C2212yes—
20C23×D4—21no—
21C23×D4—21no—
22C23×D4C2×C412yes—
23C2×#127—22no—
24C2×#127—41no—
25C2×#127—41no—
26C2×#127—42no—
27C2×#127C814yescomplemented
28C22×C4⋊C4—21no—
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C22⋊C4—22no—
32C22×C22⋊C4—22no—
33C22×C22⋊C4—22no—
34C22×C22⋊C4—22no—
35C22×C42—21no—
36C22×C42C2×C411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C2×C4⋊C8—41no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C4×C8—41no—
43#390—44no—
44#390—44no—
45#390—44no—
46#390—42no—
47#390—42no—
48C4×C4⋊C4—41no—
49C4×C4⋊C4—41no—
50C4×C4⋊C4—42no—
51C2×#95C2×C412yes—
52C2×#95C2×C411yescharacteristic
53C2×#95C2311yescharacteristic
54#336—41no—
55#336—41no—
56#324—24no—
57#324—42no—
58C25—22no—
59C22×D4—24no—
60C22×D4—24no—
61C22×D4—22no—
62C22×D4—42no—
63C22×D4—42no—
64C22×D4—42no—
65C22×D4—42no—
66C22×D4—42no—
67C22×D4—42no—
68C22×D4—42no—
69C22×D4—42no—
70C22×D4—42no—
71C22×D4—42no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4—42no—
75C22×D4—42no—
76C22×D4—42no—
77C22×D4—82no—
78C22×D4—82no—
79C22×D4C2×C814yes—
80C23×C4—22no—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C22×C8—42no—
89C22×C8—42no—
90#127—44no—
91#127—44no—
92#127—44no—
93#127—44no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4D812yes—
103C2×C4⋊C4SD1612yes—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—44no—
106C2×C22⋊C4—44no—
107C2×C22⋊C4—44no—
108C2×C22⋊C4—44no—
109C2×C22⋊C4—44no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4—82no—
114C2×C22⋊C4—82no—
115C2×C22⋊C4—82no—
116C2×C22⋊C4—82no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—22no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42—42no—
127C2×C42—42no—
128C2×C42—42no—
129C2×C42C2×C812yes—
130C2×C42M1612yes—
131C4⋊C8—82no—
132C22⋊C8—24no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136C4×C8—82no—
137#95—41no—
138#95—41no—
139#95—42no—
140C24—24no—
141C24—42no—
142C24—42no—
143C24—42no—
144C24—42no—
145C24—42no—
146C24—42no—
147C24—42no—
148C24—42no—
149C24—42no—
150C24—42no—
151C24—42no—
152C24—42no—
153C24—42no—
154C24—42no—
155C24C2×C22⋊C411yescharacteristic
156C2×D4—44no—
157C2×D4—48no—
158C2×D4—44no—
159C2×D4—48no—
160C2×D4—48no—
161C2×D4—48no—
162C2×D4—44no—
163C2×D4—44no—
164C2×D4—48no—
165C2×D4—84no—
166C2×D4—82no—
167C2×D4—82no—
168C2×D4—84no—
169C2×D4—84no—
170C2×D4—84no—
171C2×D4—84no—
172C2×D4—84no—
173C2×D4—84no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4—88no—
177C2×D4—84no—
178C2×D4—82no—
179C2×D4—82no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
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—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4C22⋊C812yes—
215C22×C4C22⋊C812yes—
216C22×C4#9911yescharacteristic
217C22×C4#9911yescharacteristic
218C22×C4#10012yes—
219C22×C4#10212yes—
220C22×C4#10212yes—
221C22×C4#10412yes—
222C22×C4C22×C811yescharacteristic
223C22×C4C2×M1611yescharacteristic
224C22×C4C2×D811yescharacteristic
225C22×C4C2×SD1611yescharacteristic
226C22×C4AGL(1,ℤ/8)12yes—
227C2×C8—44no—
228C2×C8—82no—
229C2×C8—82no—
230C2×C8—82no—
231C2×C8—82no—
232C2×C8—82no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—48no—
237C22⋊C4—88no—
238C22⋊C4—84no—
239C22⋊C4—84no—
240C22⋊C4—88no—
241C22⋊C4—88no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—42no—
247C42—42no—
248C42—81no—
249C42—81no—
250C42—84no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—44no—
256C23—44no—
257C23—44no—
258C23—44no—
259C23—44no—
260C23—44no—
261C23—44no—
262C23—82no—
263C23—82no—
264C23—82no—
265C23—82no—
266C23—82no—
267C23—82no—
268C23—82no—
269C23—82no—
270C23—82no—
271C23—82no—
272C23—82no—
273C23—82no—
274C23—82no—
275C23—82no—
276C23—82no—
277C23—82no—
278C23—82no—
279C23—82no—
280C23—82no—
281C23—82no—
282C23—82no—
283C23—82no—
284C23—82no—
285C23—82no—
286C23—82no—
287C23—82no—
288C23—82no—
289C23—82no—
290C23C2×C22⋊C811yescharacteristic
291C23C2×#9911yescharacteristic
292C23C2×#10011yescharacteristic
293C23C2×#10211yescharacteristic
294C23#41811yescharacteristic
295C23C2×#10411yescharacteristic
296C23#42111yescharacteristic
297D4—816no—
298D4—88no—
299D4—88no—
300D4—88no—
301D4—88no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4#32312yes—
349C2×C4#32312yes—
350C2×C4D4⋊C814yes—
351C2×C4#35312yes—
352C2×C4#35512yes—
353C8—84no—
354C8—162no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—84no—
368C22—84no—
369C22—84no—
370C22—84no—
371C22—84no—
372C22—84no—
373C22—84no—
374C22—162no—
375C22—162no—
376C22—162no—
377C22—162no—
378C22—162no—
379C22—162no—
380C22—162no—
381C22—162no—
382C22C2×#32311yescharacteristic
383C22C2×D4⋊C811yescharacteristic
384C22#147811yescharacteristic
385C22#150711yescharacteristic
386C22#151911yescharacteristic
387C22#152711yescharacteristic
388C22#154511yescharacteristic
389C22#212211yescharacteristic
390C22#212211yescharacteristic
391C22C2×#35311yescharacteristic
392C22C2×#35511yescharacteristic
393C4—24no—
394C4—22no—
395C4—42no—
396C4—41no—
397C4—42no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—84no—
402C4—81no—
403C4#131814yes—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2—164no—
409C2#757211yescharacteristic
410C2#1142711yescharacteristic
411C2C2×#131811yescharacteristic
412C2#1177611yescharacteristic
413C2#1178111yescharacteristic
414C2#1187711yescharacteristic
415C2#1188211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC22×#127C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8558C2semidirect
3. C4×#324C2semidirect
4. #7103C2semidirect
27. C2×#127C8semidirect

Maximal subgroups

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

subgroupindex
#141542
#85582
C4×#3242
#71032