Finite Group Databasebeta

← #99564

Subgroups — #99564

Order 512.

722 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 116 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#21955C211yesmaximal, characteristic
2C4×#395C211yesmaximal, characteristic
3#8311C212yesmaximal
4C4×#324C211yesmaximal, characteristic
5#7028C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7C2×#432—21no—
8C2×#395C411yescharacteristic
9C2×C4×C4⋊C4—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1940—24no—
12#1797—21no—
13#1797C2212yes—
14C4×C22⋊C8C2212yes—
15C2×Q8⋊C8—24no—
16C2×#324—21no—
17#1446C411yescharacteristic
18#1444—21no—
19#1444C411yescharacteristic
20#1441—21no—
21#1441C411yescharacteristic
22C4×#95—21no—
23C4×#95C2211yescharacteristic
24#1280—21no—
25C23×Q8C2×C411yescharacteristic
26C2×C4×Q8—22no—
27C2×C4×Q8—22no—
28C2×C4×Q8—44no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4D411yescharacteristic
35C22×C42—21no—
36C22×C42—21no—
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42C2311yescharacteristic
40#432—42no—
41#432—42no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8D414yes—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×#97—22no—
48C2×C4×C8—22no—
49#395—22no—
50#395—22no—
51C4×C4⋊C4—22no—
52C4×C4⋊C4—22no—
53C4×C4⋊C4—44no—
54C2×#95—21no—
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#336—22no—
68Q8⋊C8—48no—
69#324—24no—
70C22×Q8—21no—
71C22×Q8—22no—
72C22×Q8—22no—
73C22×Q8—42no—
74C22×Q8—42no—
75C22×Q8M1612yes—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—24no—
88C22×C8—42no—
89C4×Q8—28no—
90C4×Q8—44no—
91C4×Q8—44no—
92C4×Q8—44no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—24no—
116C2×C42—24no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—42no—
120C2×C42—42no—
121C2×C42—42no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42—42no—
127C2×C42SD1612yes—
128C2×C42Q1612yes—
129C2×C42C2×D412yes—
130C2×C42#4112yes—
131C4⋊C8—44no—
132C4⋊C8—44no—
133C4⋊C8—82no—
134C22⋊C8—24no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138#97—44no—
139C4×C8—44no—
140#95—24no—
141#95—22no—
142#95—22no—
143#95—21no—
144#95—22no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—42no—
149#95—42no—
150#95—42no—
151#95M1612yes—
152C24C2×C22⋊C411yescharacteristic
153C2×Q8—24no—
154C2×Q8—24no—
155C2×Q8—24no—
156C2×Q8—44no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—44no—
160C2×Q8—48no—
161C2×Q8—42no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
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—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
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#10312yes—
218C22×C4#10312yes—
219C22×C4#10412yes—
220C22×C4C4×D412yes—
221C22×C4#12011yescharacteristic
222C22×C4#12112yes—
223C22×C4#12311yescharacteristic
224C22×C4C2×M1611yescharacteristic
225C22×C4C2×M1611yescharacteristic
226C22×C4#13112yes—
227C22×C4C2×SD1611yescharacteristic
228C22×C4C2×Q1611yescharacteristic
229C22×C4AGL(1,ℤ/8)11yescharacteristic
230C22×C4#13711yescharacteristic
231C2×C8—44no—
232C2×C8—44no—
233C2×C8—44no—
234C2×C8—44no—
235C2×C8—82no—
236C2×C8—82no—
237C4⋊C4—24no—
238C4⋊C4—44no—
239C4⋊C4—48no—
240C4⋊C4—42no—
241C4⋊C4—48no—
242C4⋊C4—48no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C42—22no—
246C42—22no—
247C42—24no—
248C42—22no—
249C42—22no—
250C42—42no—
251C42—44no—
252C42—44no—
253C42—44no—
254C42—42no—
255C42—44no—
256C42—81no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23#38611yescharacteristic
262C23#40711yescharacteristic
263C23#40811yescharacteristic
264C23C2×#10311yescharacteristic
265C23#41711yescharacteristic
266C23C2×#10411yescharacteristic
267C23#42111yescharacteristic
268Q8—28no—
269Q8—44no—
270Q8—44no—
271Q8—44no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
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#43512yes—
339C2×C4#43612yes—
340C2×C4#45011yescharacteristic
341C2×C4#45111yescharacteristic
342C2×C4#45612yes—
343C2×C4B(4,𝔽2)12yes—
344C2×C4#48311yescharacteristic
345C2×C4#48411yescharacteristic
346C8—84no—
347C8—84no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22#148711yescharacteristic
361C22#148911yescharacteristic
362C22#149311yescharacteristic
363C22#149511yescharacteristic
364C22#187011yescharacteristic
365C22#187111yescharacteristic
366C22#187211yescharacteristic
367C22#188411yescharacteristic
368C22#188611yescharacteristic
369C22#188911yescharacteristic
370C22#189111yescharacteristic
371C4—22no—
372C4—22no—
373C4—21no—
374C4—21no—
375C4—44no—
376C4—44no—
377C4—42no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—82no—
382C4—81no—
383C4—82no—
384C4#160412yes—
385C4#160512yes—
386C2—21no—
387C2—21no—
388C2—21no—
389C2—21no—
390C2#894211yescharacteristic
391C2#896211yescharacteristic
392C2#896711yescharacteristic
393C2#900911yescharacteristic
394C2#901511yescharacteristic
395C2#903011yescharacteristic
396C2#903311yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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
#219552
C4×#3952
#83112
C4×#3242
#70282