Finite Group Databasebeta

← #100077

Subgroups — #100077

Order 512.

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

normal subgroups only

This group has 616 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#21902C211yesmaximal, characteristic
2#14229C211yesmaximal, characteristic
3#8559C211yesmaximal, characteristic
4#7573C212yesmaximal, complemented
5#7028C212yesmaximal, complemented
6C2×#397C411yescharacteristic
7C2×#389C2211yescharacteristic
8C2×#388—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1923—24no—
11#1797—21no—
12#1797C2212yes—
13#1779—21no—
14C4×C22⋊C8C2212yes—
15C2×#323—21no—
16#1445C411yescharacteristic
17#1442C411yescharacteristic
18#1439C411yescharacteristic
19#1436—21no—
20#1434—21no—
21C22⋊#95—21no—
22C4×#95C2211yescharacteristic
23#1280—21no—
24#1271—24no—
25C24×C4D411yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4D411yescharacteristic
33C22×C42—21no—
34C22×C42D411yescharacteristic
35C22×C42C2311yescharacteristic
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C4×C8—22no—
43#397—22no—
44#397—22no—
45#389—22no—
46#389—22no—
47#388—42no—
48#388—42no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95D411yescharacteristic
63#336—22no—
64#336—22no—
65#336D414yes—
66#323—42no—
67C25C22⋊C411yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4—42no—
77C23×C4—81no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—24no—
85C22×C8—42no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—81no—
95C2×C4⋊C4M1612yes—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—81no—
108C2×C22⋊C4D812yes—
109C2×C22⋊C4SD1612yes—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42—41no—
115C2×C42—41no—
116C2×C42—81no—
117C2×C42—81no—
118C2×C42C2×D412yes—
119C2×C42#4112yes—
120C4⋊C8—82no—
121C4⋊C8—82no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—82no—
127C22⋊C8—82no—
128C4×C8—82no—
129#95—24no—
130#95—22no—
131#95—22no—
132#95—21no—
133#95—22no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—41no—
140#95—41no—
141#95—81no—
142#95M1612yes—
143C24—21no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24#10212yes—
154C24C2×C22⋊C411yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
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—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—42no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—84no—
212C22×C4—84no—
213C22×C4#10212yes—
214C22×C4C4×D412yes—
215C22×C4#12011yescharacteristic
216C22×C4#12112yes—
217C22×C4#12311yescharacteristic
218C22×C4C2×M1611yescharacteristic
219C22×C4C2×M1611yescharacteristic
220C22×C4#13112yes—
221C22×C4C2×D811yescharacteristic
222C22×C4C2×SD1611yescharacteristic
223C22×C4#13512yes—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—44no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C4⋊C4—24no—
231C4⋊C4—42no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—48no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—84no—
241C42—42no—
242C42—42no—
243C42—42no—
244C42—81no—
245C42—82no—
246C42—82no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23#38611yescharacteristic
275C23#40711yescharacteristic
276C23#40811yescharacteristic
277C23C2×#10211yescharacteristic
278C23#41611yescharacteristic
279C23#42111yescharacteristic
280C23#42111yescharacteristic
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—84no—
320C2×C4—81no—
321C2×C4—84no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
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#43512yes—
344C2×C4#43612yes—
345C2×C4#45512yes—
346C2×C4#46512yes—
347C2×C4#46612yes—
348C8—84no—
349C8—162no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—44no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22#148011yescharacteristic
374C22#148211yescharacteristic
375C22#148611yescharacteristic
376C22#148811yescharacteristic
377C22#187011yescharacteristic
378C22#188411yescharacteristic
379C22#188511yescharacteristic
380C22#189311yescharacteristic
381C22#189711yescharacteristic
382C22#191311yescharacteristic
383C22#191511yescharacteristic
384C4—22no—
385C4—22no—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—42no—
390C4—41no—
391C4—82no—
392C4—82no—
393C4—81no—
394C4—82no—
395C4—81no—
396C4—161no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—82no—
402C2#896111yescharacteristic
403C2#907611yescharacteristic
404C2#911011yescharacteristic
405C2#938411yescharacteristic
406C2#939011yescharacteristic
407C2#939311yescharacteristic
408C2#941111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×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. #7573C2semidirect
5. #7028C2semidirect

Maximal subgroups

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

subgroupindex
#219022
#142292
#85592
#75732
#70282