Finite Group Databasebeta

← #101126

Subgroups — #101126

Order 512.

650 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 106 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#21818C211yesmaximal, characteristic
2#14884C211yesmaximal, characteristic
3#10763C211yesmaximal, characteristic
4#7607C212yesmaximal
5#7037C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7C2×#431—21no—
8C22×#101—21no—
9C2×#400C411yescharacteristic
10C2×#400C411yescharacteristic
11C2×#389C2211yescharacteristic
12#1984—24no—
13#1817—21no—
14#1817C2212yes—
15C2×#343—21no—
16#1728C2212yes—
17#1446—21no—
18#1442—21no—
19#1441C411yescharacteristic
20#1441C411yescharacteristic
21#1436—21no—
22#1436—21no—
23#1434C2211yescharacteristic
24#1271—24no—
25C23×Q8D411yescharacteristic
26C22×M16—22no—
27C2×C4×Q8—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—42no—
30C2×C4×Q8—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4D411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42C2311yescharacteristic
40#431—42no—
41#431—42no—
42C2×C4⋊C8—22no—
43C2×#101—24no—
44C2×#101—42no—
45C2×#101—42no—
46C2×C22⋊C8—22no—
47C2×C4×C8—22no—
48#400—42no—
49#400—42no—
50#389—22no—
51#389—22no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95C2×C411yescharacteristic
64C2×#95C2×C411yescharacteristic
65#343—42no—
66#336—24no—
67#336—22no—
68#336—22no—
69#336—22no—
70#336D414yes—
71C22×Q8—22no—
72C22×Q8—22no—
73C22×Q8—22no—
74C22×Q8—22no—
75C22×Q8—41no—
76C22×Q8—41no—
77C22×Q8SD1612yes—
78C22×Q8Q1612yes—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C2×M16—24no—
91C2×M16—42no—
92C2×M16—42no—
93C2×M16—42no—
94C2×M16—42no—
95C2×M16—42no—
96C22×C8—24no—
97C22×C8—42no—
98C4×Q8—44no—
99C4×Q8—44no—
100C4×Q8—48no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42C2×D412yes—
130C2×C42#4112yes—
131C4⋊C8—44no—
132#101—48no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135C4×C8—44no—
136#95—41no—
137#95—41no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—42no—
149C24C2×C22⋊C411yescharacteristic
150C2×Q8—24no—
151C2×Q8—24no—
152C2×Q8—24no—
153C2×Q8—24no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—44no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
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—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#9912yes—
220C22×C4#9912yes—
221C22×C4#10312yes—
222C22×C4#10312yes—
223C22×C4#10412yes—
224C22×C4C4×D412yes—
225C22×C4#12011yescharacteristic
226C22×C4#12112yes—
227C22×C4#12311yescharacteristic
228C22×C4C2×SD1611yescharacteristic
229C22×C4C2×Q1611yescharacteristic
230C22×C4#13512yes—
231M16—44no—
232M16—44no—
233M16—82no—
234M16—82no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—82no—
240C2×C8—82no—
241C4⋊C4—24no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—42no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C42—22no—
250C42—22no—
251C42—24no—
252C42—42no—
253C42—42no—
254C42—42no—
255C42—42no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23#38611yescharacteristic
261C23C2×#9911yescharacteristic
262C23C2×#9911yescharacteristic
263C23C2×#10311yescharacteristic
264C23#41611yescharacteristic
265C23C2×#10411yescharacteristic
266C23#42111yescharacteristic
267Q8—48no—
268Q8—48no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
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#32814yes—
341C2×C4#45412yes—
342C2×C4#46512yes—
343C2×C4#46712yes—
344C8—84no—
345C8—84no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22C2×#32811yescharacteristic
359C22#150111yescharacteristic
360C22#151611yescharacteristic
361C22#152011yescharacteristic
362C22#187111yescharacteristic
363C22#188811yescharacteristic
364C22#189111yescharacteristic
365C22#189211yescharacteristic
366C22#189811yescharacteristic
367C22#191211yescharacteristic
368C22#191511yescharacteristic
369C4—22no—
370C4—24no—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—41no—
377C4—81no—
378C4—82no—
379C4—161no—
380C4—161no—
381C4—161no—
382C2—21no—
383C2—21no—
384C2—21no—
385C2—21no—
386C2#903211yescharacteristic
387C2#911111yescharacteristic
388C2#919011yescharacteristic
389C2#939611yescharacteristic
390C2#946611yescharacteristic
391C2#948411yescharacteristic
392C2#951311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#218182
#148842
#107632
#76072
#70372