Finite Group Databasebeta

← #124278

Subgroups — #124278

Order 512.

640 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 108 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#13908C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7123C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C2×#380—21no—
7C2×#380C411yescharacteristic
8C2×C4×C22⋊C4C2211yescharacteristic
9C22×#95C411yescharacteristic
10#1797—21no—
11C4×C22⋊C8—21no—
12C2×#323—21no—
13C2×#323—21no—
14#1444C411yescharacteristic
15#1444C2211yescharacteristic
16#1442C411yescharacteristic
17#1442C2211yescharacteristic
18#1436—21no—
19#1436—21no—
20C22⋊#95—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C4×C8—41no—
38#380—42no—
39#380—42no—
40#380—41no—
41#380—41no—
42C4×C22⋊C4—41no—
43C4×C22⋊C4—41no—
44C2×#95—21no—
45C2×#95—22no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95C814yescomplemented
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2311yescharacteristic
58#336—41no—
59#336—41no—
60#323—24no—
61#323—42no—
62C25C22⋊C411yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4—21no—
68C23×C4—24no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—42no—
72C23×C4—42no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—81no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4—81no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C2×C812yes—
83C23×C4M1612yes—
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—42no—
88C22×C8—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—81no—
97C2×C4⋊C4—81no—
98C2×C4⋊C4—81no—
99C2×C4⋊C4—81no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4D812yes—
109C2×C22⋊C4SD1612yes—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—81no—
115C2×C42—81no—
116C4⋊C8—82no—
117C22⋊C8—24no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C4×C8—82no—
122#95—24no—
123#95—21no—
124#95—21no—
125#95—41no—
126#95—41no—
127#95—42no—
128#95—42no—
129#95—41no—
130#95—41no—
131#95—42no—
132#95—41no—
133#95—41no—
134#95—81no—
135#95—81no—
136#95C2×C812yes—
137#95C2×C812yes—
138C24—21no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—41no—
143C24—41no—
144C24—41no—
145C24—41no—
146C24C22⋊C812yes—
147C24#10212yes—
148C24#10412yes—
149C24C2×C22⋊C411yescharacteristic
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—48no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—48no—
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—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—84no—
212C22×C4—84no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—84no—
216C22×C4—84no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4C22⋊C812yes—
220C22×C4#9911yescharacteristic
221C22×C4#9911yescharacteristic
222C22×C4#10112yes—
223C22×C4#10212yes—
224C22×C4C22×C811yescharacteristic
225C22×C4C2×M1611yescharacteristic
226C22×C4C2×D811yescharacteristic
227C22×C4C2×SD1611yescharacteristic
228C22×C4#13712yes—
229C2×C8—44no—
230C2×C8—82no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—82no—
235C4⋊C4—84no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—84no—
244C42—42no—
245C42—42no—
246C42—81no—
247C42—81no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23D4⋊C814yes—
275C23C2×C22⋊C811yescharacteristic
276C23C2×#9911yescharacteristic
277C23C2×#10111yescharacteristic
278C23C2×#10211yescharacteristic
279C23#41911yescharacteristic
280C23C2×#10411yescharacteristic
281C23#42111yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—24no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—816no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4#32412yes—
351C2×C4#32412yes—
352C8—84no—
353C8—162no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—24no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22C2×#32411yescharacteristic
378C22C2×D4⋊C811yescharacteristic
379C22#147911yescharacteristic
380C22#150811yescharacteristic
381C22#151911yescharacteristic
382C22#152811yescharacteristic
383C22#154611yescharacteristic
384C22#212111yescharacteristic
385C22#212111yescharacteristic
386C22#212611yescharacteristic
387C22#213211yescharacteristic
388C4—22no—
389C4—22no—
390C4—44no—
391C4—42no—
392C4—41no—
393C4—41no—
394C4—42no—
395C4—41no—
396C4—41no—
397C4—82no—
398C4—81no—
399C4—161no—
400C4—161no—
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—44no—
406C2#757311yescharacteristic
407C2#1144711yescharacteristic
408C2#1145311yescharacteristic
409C2#1178911yescharacteristic
410C2#1179311yescharacteristic
411C2#1187611yescharacteristic
412C2#1188411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7123C2semidirect
5. #7122C2semidirect
53. C2×#95C8semidirect

Maximal subgroups

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

subgroupindex
#139082
#85572
C4×#3232
#71232
#71222