Finite Group Databasebeta

← #120039

Subgroups — #120039

Order 512.

938 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 150 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#14339C211yesmaximal, characteristic
2#13881C211yesmaximal, characteristic
3#10727C211yesmaximal, characteristic
4#8015C212yesmaximal
5C2×#1284C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2212yes—
8C2×#382C412yescomplemented
9C2×#381C2212yes—
10C2×#380—21no—
11C2×#380C2211yescharacteristic
12C2×C4×C22⋊C4C2211yescharacteristic
13C2×#343—22no—
14#1442—22no—
15#1441—24no—
16C4⋊#95C412yescomplemented
17C22⋊#95—21no—
18C22⋊#95—21no—
19C22⋊#95—21no—
20#1294—24no—
21#1284—24no—
22C24×C4D411yescharacteristic
23C24×C4D411yescharacteristic
24C24×C4C2311yescharacteristic
25C22×M16—22no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—44no—
30C22×C4⋊C4C2×C412yes—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4D412yes—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—22no—
37C22×C42C2×C412yes—
38#407—22no—
39#407—22no—
40#407—22no—
41#407Q814yes—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44#382—44no—
45#381—22no—
46#381—22no—
47#381—22no—
48#381—22no—
49#380—22no—
50#380—22no—
51#380—44no—
52C4×C22⋊C4—21no—
53C4×C22⋊C4—21no—
54C4×C22⋊C4D412yes—
55C4×C22⋊C4Q812yes—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—21no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95D414yes—
67#343—44no—
68C25C2×D411yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—24no—
72C23×C4—22no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—24no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—81no—
84C23×C4—81no—
85C23×C4C22⋊C412yes—
86C23×C4C22⋊C412yes—
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4C2×Q811yescharacteristic
92C23×C4C2×Q812yes—
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4112yes—
96C23×C4#4112yes—
97C2×M16—24no—
98C2×M16—42no—
99C2×M16—44no—
100C2×M16—44no—
101C22×C8—42no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—48no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—82no—
117C2×C4⋊C4—82no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—44no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—42no—
129C2×C42—22no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42C4⋊C414yescomplemented
133C22⋊C8—24no—
134C22⋊C8—24no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137#95—22no—
138#95—22no—
139#95—24no—
140#95—24no—
141#95—44no—
142#95—44no—
143#95—44no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—82no—
148#95—82no—
149#95—82no—
150#95—82no—
151C24—22no—
152C24—22no—
153C24—22no—
154C24—22no—
155C24—41no—
156C24—41no—
157C24C2×C22⋊C411yescharacteristic
158C24#12011yescharacteristic
159C24#12111yescharacteristic
160C24C22⋊Q811yescharacteristic
161C24#12311yescharacteristic
162C24#12411yescharacteristic
163C24C4⋊Q811yescharacteristic
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—88no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—88no—
224C22×C4#9914yes—
225C22×C4C2×C4⋊C411yescharacteristic
226C22×C4#11711yescharacteristic
227C22×C4C4×D412yes—
228C22×C4C4×Q812yes—
229C22×C4#12111yescharacteristic
230C22×C4#12112yes—
231C22×C4#12112yes—
232C22×C4C22⋊Q812yes—
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q812yes—
235C22×C4C22⋊Q812yes—
236C22×C4#12312yes—
237C22×C4#12512yes—
238C22×C4#12512yes—
239C22×C4#12612yes—
240C22×C4C4⋊Q812yes—
241M16—48no—
242M16—84no—
243C2×C8—44no—
244C2×C8—82no—
245C2×C8—82no—
246C4⋊C4—48no—
247C4⋊C4—48no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—88no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—84no—
254C22⋊C4—84no—
255C22⋊C4—82no—
256C22⋊C4—82no—
257C42—44no—
258C42—44no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—44no—
286C23—44no—
287C23—44no—
288C23—44no—
289C23#38011yescharacteristic
290C23#38611yescharacteristic
291C23#39111yescharacteristic
292C23#39411yescharacteristic
293C23#39511yescharacteristic
294C23#39811yescharacteristic
295C23#40011yescharacteristic
296C23C2×#9912yes—
297C23#41012yes—
298C23#45412yes—
299C23#45512yes—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—82no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—84no—
355C2×C4#38712yes—
356C2×C4#38912yes—
357C2×C4#39612yes—
358C2×C4#39812yes—
359C8—84no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—44no—
383C22—44no—
384C22—44no—
385C22—44no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22#144111yescharacteristic
391C22#179912yes—
392C22#188511yescharacteristic
393C22#188811yescharacteristic
394C22#189811yescharacteristic
395C22#189911yescharacteristic
396C22#191111yescharacteristic
397C22#191211yescharacteristic
398C22#199612yes—
399C22#204012yes—
400C22#206112yes—
401C4—22no—
402C4—22no—
403C4—22no—
404C4—22no—
405C4—44no—
406C4—84no—
407C4—82no—
408C4—164no—
409C4—161no—
410C4—161no—
411C2—22no—
412C2—22no—
413C2—22no—
414C2—22no—
415C2—41no—
416C2—41no—
417C2#945911yescharacteristic
418C2#946011yescharacteristic
419C2#946511yescharacteristic
420C2#946611yescharacteristic
421C2#1120311yescharacteristic
422C2#1121111yescharacteristic
423C2#1122411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#382C4semidirect
16. C4⋊#95C4semidirect
132. C2×C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#143392
#138812
#107272
#80152
C2×#12842