Finite Group Databasebeta

← #124168

Subgroups — #124168

Order 512.

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

normal subgroups only

This group has 806 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
1C4×#394C211yesmaximal, characteristic
2C4×#324C211yesmaximal, characteristic, complemented
3#7908C211yesmaximal, characteristic, complemented
4#7122C212yesmaximal, complemented
5#7103C212yesmaximal
6C22×C4×D4C411yescharacteristic
7C2×#431—21no—
8C2×#394C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4—21no—
11C2×C4×C22⋊C4—21no—
12C4×C22⋊C8—21no—
13C2×#324—21no—
14#1442C2211yescharacteristic
15#1441C411yescharacteristic
16#1440—21no—
17#1439—21no—
18#1438C2211yescharacteristic
19C4×#95C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×D4C2×C411yescharacteristic
24C24×C4—21no—
25C2×C4×D4—41no—
26C2×C4×D4—41no—
27C2×C4×D4—44no—
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×C42—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40#431—24no—
41#431—42no—
42C2×C4⋊C8—41no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C4×C8—41no—
46C2×C4×C8—41no—
47#394—22no—
48#394C814yescomplemented
49C4×C4⋊C4—41no—
50C4×C4⋊C4—41no—
51C4×C22⋊C4—44no—
52C4×C22⋊C4—42no—
53C4×C22⋊C4—42no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95D411yescharacteristic
62C2×#95C2311yescharacteristic
63#336—41no—
64#324—24no—
65C25—21no—
66C22×D4—21no—
67C22×D4—41no—
68C22×D4—41no—
69C22×D4—42no—
70C22×D4—42no—
71C22×D4C2×C812yes—
72C23×C4—24no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C22×C8—42no—
91C22×C8—42no—
92C4×D4—44no—
93C4×D4—44no—
94C4×D4—44no—
95C4×D4—44no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4C2×C812yes—
104C2×C4⋊C4M1612yes—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
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—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—42no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—22no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—42no—
129C2×C42—42no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42D812yes—
134C2×C42SD1612yes—
135C4⋊C8—44no—
136C22⋊C8—24no—
137C22⋊C8—24no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140C4×C8—44no—
141C4×C8—82no—
142#95—21no—
143#95—41no—
144#95—41no—
145#95—41no—
146#95—42no—
147#95—42no—
148#95—42no—
149#95—42no—
150#95—41no—
151#95—41no—
152#95C2×C812yes—
153C24—22no—
154C24—22no—
155C24—22no—
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—42no—
170C2×D4—42no—
171C2×D4—44no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—48no—
176C2×D4—42no—
177C2×D4—44no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—48no—
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—41no—
227C22×C4—44no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—82no—
231C22×C4—84no—
232C22×C4—84no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4C22⋊C812yes—
237C22×C4C22⋊C812yes—
238C22×C4#9911yescharacteristic
239C22×C4#9911yescharacteristic
240C22×C4#10212yes—
241C22×C4#10212yes—
242C22×C4#10412yes—
243C22×C4#10412yes—
244C22×C4C22×C811yescharacteristic
245C22×C4C2×M1611yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C2×SD1611yescharacteristic
248C22×C4#13512yes—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C22⋊C4—48no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—48no—
264C22⋊C4—48no—
265C22⋊C4—48no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42—81no—
274C42—84no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—24no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
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—81no—
313C23—81no—
314C23C2×C22⋊C811yescharacteristic
315C23C2×#9911yescharacteristic
316C23#41311yescharacteristic
317C23C2×#10211yescharacteristic
318C23#41611yescharacteristic
319C23C2×#10411yescharacteristic
320C23C2×#10411yescharacteristic
321D4—44no—
322D4—44no—
323D4—44no—
324D4—44no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—88no—
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—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—88no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4D4⋊C814yes—
388C2×C4#32712yes—
389C2×C4#32712yes—
390C2×C4AΣL(1,𝔽16)12yes—
391C2×C4#35512yes—
392C8—84no—
393C8—84no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—44no—
407C22—44no—
408C22—44no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22#145911yescharacteristic
422C22C2×D4⋊C811yescharacteristic
423C22#147611yescharacteristic
424C22C2×#32711yescharacteristic
425C22#150111yescharacteristic
426C22#152411yescharacteristic
427C22#154011yescharacteristic
428C22C2×AΣL(1,𝔽16)11yescharacteristic
429C22#212011yescharacteristic
430C22#212611yescharacteristic
431C22C2×#35511yescharacteristic
432C4—24no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—42no—
437C4—41no—
438C4—41no—
439C4—42no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—81no—
444C4#134514yes—
445C2—21no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—84no—
450C2#756511yescharacteristic
451C2#1147111yescharacteristic
452C2#1147611yescharacteristic
453C2#1169111yescharacteristic
454C2C2×#134511yescharacteristic
455C2#1172511yescharacteristic
456C2#1173411yescharacteristic
subgroupG/H
centerC22×C4#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
socleC23C2×#99
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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. C4×#324C2semidirect
3. #7908C2semidirect
4. #7122C2semidirect
48. #394C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3942
C4×#3242
#79082
#71222
#71032