Finite Group Databasebeta

← #101118

Subgroups — #101118

Order 512.

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

normal subgroups only

This group has 688 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#14881C211yesmaximal, characteristic
3#10763C211yesmaximal, characteristic
4#7609C212yesmaximal
5#7037C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7C2×#431—21no—
8C22×#101—21no—
9C2×#395C411yescharacteristic
10C2×#395C411yescharacteristic
11C2×#391C2211yescharacteristic
12#1984—24no—
13#1817—21no—
14#1817C2212yes—
15C2×#343—21no—
16#1728C2212yes—
17#1446—21no—
18#1444—21no—
19#1441C411yescharacteristic
20#1441C411yescharacteristic
21#1436—21no—
22#1436—21no—
23#1434C2211yescharacteristic
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C23×Q8D411yescharacteristic
27C22×M16—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—22no—
30C2×C4×Q8—42no—
31C2×C4×Q8—42no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4D411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42C2311yescharacteristic
41#431—42no—
42#431—42no—
43C2×C4⋊C8—22no—
44C2×#101—24no—
45C2×#101—42no—
46C2×#101—42no—
47C2×C22⋊C8—22no—
48C2×C4×C8—22no—
49#395—42no—
50#395—42no—
51#391—22no—
52#391—22no—
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—22no—
76C22×Q8—41no—
77C22×Q8—41no—
78C22×Q8—42no—
79C22×Q8—41no—
80C22×Q8—41no—
81C22×Q8—81no—
82C22×Q8SD1612yes—
83C22×Q8Q1612yes—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C2×M16—24no—
96C2×M16—42no—
97C2×M16—42no—
98C2×M16—42no—
99C2×M16—42no—
100C2×M16—42no—
101C22×C8—24no—
102C22×C8—42no—
103C4×Q8—44no—
104C4×Q8—44no—
105C4×Q8—48no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—81no—
121C2×C4⋊C4—81no—
122C2×C4⋊C4—81no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42C2×D412yes—
134C2×C42#4112yes—
135C4⋊C8—44no—
136#101—48no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139C4×C8—44no—
140#95—24no—
141#95—41no—
142#95—41no—
143#95—42no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—41no—
148#95—42no—
149#95—41no—
150#95—42no—
151#95—81no—
152#95—81no—
153C24C2×C22⋊C411yescharacteristic
154C2×Q8—24no—
155C2×Q8—24no—
156C2×Q8—24no—
157C2×Q8—24no—
158C2×Q8—24no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—44no—
162C2×Q8—44no—
163C2×Q8—44no—
164C2×Q8—44no—
165C2×Q8—42no—
166C2×Q8—82no—
167C2×Q8—82no—
168C2×Q8—84no—
169C2×Q8—84no—
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—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
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—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4#9912yes—
230C22×C4#9912yes—
231C22×C4#10312yes—
232C22×C4#10312yes—
233C22×C4#10412yes—
234C22×C4C4×D412yes—
235C22×C4#12011yescharacteristic
236C22×C4#12112yes—
237C22×C4#12311yescharacteristic
238C22×C4C2×SD1611yescharacteristic
239C22×C4C2×Q1611yescharacteristic
240C22×C4#13512yes—
241M16—44no—
242M16—44no—
243M16—82no—
244M16—82no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—82no—
250C2×C8—82no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—82no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C42—22no—
258C42—22no—
259C42—24no—
260C42—42no—
261C42—42no—
262C42—42no—
263C42—42no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#38611yescharacteristic
269C23C2×#9911yescharacteristic
270C23C2×#9911yescharacteristic
271C23C2×#10311yescharacteristic
272C23#41611yescharacteristic
273C23C2×#10411yescharacteristic
274C23#42111yescharacteristic
275Q8—48no—
276Q8—48no—
277Q8—84no—
278Q8—84no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
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—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4#32814yes—
351C2×C4#45612yes—
352C2×C4#46512yes—
353C2×C4#46712yes—
354C8—84no—
355C8—84no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22C2×#32811yescharacteristic
369C22#150111yescharacteristic
370C22#151611yescharacteristic
371C22#152011yescharacteristic
372C22#187111yescharacteristic
373C22#188611yescharacteristic
374C22#189111yescharacteristic
375C22#189211yescharacteristic
376C22#189811yescharacteristic
377C22#191211yescharacteristic
378C22#191411yescharacteristic
379C4—22no—
380C4—24no—
381C4—22no—
382C4—22no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—41no—
387C4—82no—
388C4—82no—
389C4—81no—
390C4—82no—
391C4—161no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2—21no—
396C2#903011yescharacteristic
397C2#911111yescharacteristic
398C2#918811yescharacteristic
399C2#939211yescharacteristic
400C2#946211yescharacteristic
401C2#948411yescharacteristic
402C2#951111yescharacteristic
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
#148812
#107632
#76092
#70372