Finite Group Databasebeta

← #100063

Subgroups — #100063

Order 512.

736 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 128 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#20523C211yesmaximal, characteristic
2#14230C211yesmaximal, characteristic
3#8559C211yesmaximal, characteristic
4#7576C212yesmaximal, complemented
5#7028C212yesmaximal, complemented
6C22×C22⋊C8—21no—
7C2×#399C411yescharacteristic
8C2×#389C2211yescharacteristic
9C2×#387—21no—
10C2×#385—21no—
11C2×C4×C22⋊C4C2211yescharacteristic
12#1797C2212yes—
13C8×C22⋊C4—24no—
14C4×C22⋊C8—21no—
15C4×C22⋊C8C2212yes—
16C2×#323—21no—
17#1445C411yescharacteristic
18#1444C411yescharacteristic
19#1439C411yescharacteristic
20#1437—21no—
21C4⋊#95—21no—
22C4×#95C2211yescharacteristic
23#1280—21no—
24#1271—24no—
25C24×C4D411yescharacteristic
26C23×C8—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4D411yescharacteristic
36C22×C42—21no—
37C22×C42D411yescharacteristic
38C22×C42C2311yescharacteristic
39C2×C4⋊C8—22no—
40C2×C22⋊C8—24no—
41C2×C22⋊C8—24no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C4×C8—22no—
46C2×C4×C8—22no—
47#399—22no—
48#399C814yescomplemented
49#389—22no—
50#389—22no—
51#387—44no—
52#385—44no—
53C4×C22⋊C4—22no—
54C4×C22⋊C4—22no—
55C2×#95—21no—
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#323—42no—
69C25C22⋊C411yescharacteristic
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—42no—
79C23×C4—81no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C22×C8—24no—
87C22×C8—28no—
88C22×C8—42no—
89C22×C8—82no—
90C22×C8—82no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—81no—
105C2×C4⋊C4—81no—
106C2×C4⋊C4C2×C812yescomplemented
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—81no—
117C2×C22⋊C4C2×C812yes—
118C2×C22⋊C4M1612yes—
119C2×C22⋊C4SD1612yes—
120C2×C22⋊C4Q1612yes—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42—81no—
127C2×C42—81no—
128C2×C42C2×D412yes—
129C2×C42#4112yes—
130C4⋊C8—82no—
131C22⋊C8—28no—
132C22⋊C8—24no—
133C22⋊C8—24no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136C4×C8—82no—
137C4×C8—82no—
138#95—24no—
139#95—22no—
140#95—22no—
141#95—21no—
142#95—22no—
143#95—42no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95C2×C812yescomplemented
149C24—21no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—41no—
154C24—41no—
155C24—41no—
156C24—41no—
157C24C22⋊C812yes—
158C24#10312yes—
159C24#10412yes—
160C24C2×C22⋊C411yescharacteristic
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—24no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—41no—
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—81no—
215C22×C4—81no—
216C22×C4—84no—
217C22×C4—84no—
218C22×C4C22⋊C812yes—
219C22×C4#10312yes—
220C22×C4#10412yes—
221C22×C4C4×D412yes—
222C22×C4#12011yescharacteristic
223C22×C4#12112yes—
224C22×C4#12311yescharacteristic
225C22×C4C22×C811yescharacteristic
226C22×C4C2×M1611yescharacteristic
227C22×C4#13112yes—
228C22×C4C2×SD1611yescharacteristic
229C22×C4C2×Q1611yescharacteristic
230C22×C4#13512yes—
231C2×C8—28no—
232C2×C8—44no—
233C2×C8—44no—
234C2×C8—88no—
235C2×C8—82no—
236C2×C8—82no—
237C2×C8—82no—
238C4⋊C4—24no—
239C4⋊C4—48no—
240C4⋊C4—42no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—84no—
244C4⋊C4—82no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—48no—
250C22⋊C4—84no—
251C42—42no—
252C42—42no—
253C42—42no—
254C42—81no—
255C42—84no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23Q8⋊C814yes—
283C23#38611yescharacteristic
284C23C2×C22⋊C811yescharacteristic
285C23#40811yescharacteristic
286C23C2×#10311yescharacteristic
287C23#41611yescharacteristic
288C23C2×#10411yescharacteristic
289C23C2×#10411yescharacteristic
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—84no—
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—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4C8×D412yes—
354C2×C4#43512yes—
355C2×C4#45312yes—
356C2×C4#46512yes—
357C2×C4#46712yes—
358C8—48no—
359C8—162no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—24no—
373C22—44no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22C2×Q8⋊C811yescharacteristic
384C22#147611yescharacteristic
385C22#149311yescharacteristic
386C22#149411yescharacteristic
387C22#186911yescharacteristic
388C22#188411yescharacteristic
389C22#188711yescharacteristic
390C22#189211yescharacteristic
391C22#189611yescharacteristic
392C22#191311yescharacteristic
393C22#191511yescharacteristic
394C4—22no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—44no—
400C4—42no—
401C4—41no—
402C4—42no—
403C4—42no—
404C4—82no—
405C4—82no—
406C4—161no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—44no—
412C2#895911yescharacteristic
413C2#907111yescharacteristic
414C2#910811yescharacteristic
415C2#938311yescharacteristic
416C2#938811yescharacteristic
417C2#939511yescharacteristic
418C2#940911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊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
4. #7576C2semidirect
5. #7028C2semidirect
48. #399C8semidirect
106. C2×C4⋊C4C2×C8semidirect
148. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
#205232
#142302
#85592
#75762
#70282