Finite Group Databasebeta

← #140663

Subgroups — #140663

Order 512.

634 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 94 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#13992C211yesmaximal, characteristic
2C2×#1326C212yesmaximal
3C2×#1317C212yesmaximal
4#8557C211yesmaximal, characteristic
5#8152C211yesmaximal, characteristic
6C2×#385C2211yescharacteristic
7C2×#381C411yescharacteristic
8C2×#381C2211yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10C22×#95C411yescharacteristic
11#1797—21no—
12#1749—21no—
13C2×#323C2212yes—
14C2×#323C2212yes—
15#1443—21no—
16#1442—21no—
17#1436—21no—
18#1434—21no—
19C22⋊#95C411yescharacteristic
20C22⋊#95C411yescharacteristic
21#1326—24no—
22#1317—24no—
23#1280—21no—
24#1280—21no—
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2×C411yescharacteristic
28C22×C4⋊C4—21no—
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4C2311yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C4⋊C8—41no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×#97—41no—
39#385—22no—
40#385—22no—
41#381—22no—
42#381—22no—
43#381—22no—
44#381—22no—
45C4×C22⋊C4—41no—
46C4×C22⋊C4—41no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—41no—
59C2×#95—41no—
60C2×#95D411yescharacteristic
61C2×#95D411yescharacteristic
62#336—41no—
63#336—41no—
64#323—22no—
65#323—22no—
66#323—22no—
67#323D414yes—
68C25C22×C411yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—22no—
80C23×C4—41no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—41no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4—41no—
87C23×C4—41no—
88C23×C4—41no—
89C23×C4—81no—
90C23×C4—81no—
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4M1612yes—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C22×C8—42no—
99C22×C8—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—81no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4M1612yes—
116C2×C22⋊C4C2×D412yes—
117C2×C22⋊C4#4112yes—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—42no—
123C2×C42—81no—
124C4⋊C8—82no—
125C22⋊C8—24no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#97—82no—
130#95—24no—
131#95—24no—
132#95—41no—
133#95—41no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—41no—
138#95—41no—
139#95—42no—
140#95—41no—
141#95—41no—
142#95—41no—
143#95—41no—
144#95—81no—
145#95—81no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24C2×C22⋊C411yescharacteristic
155C24C4×D412yes—
156C24C2×M1611yescharacteristic
157C24C2×M1611yescharacteristic
158C24#13112yes—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—24no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
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—42no—
199C22×C4—42no—
200C22×C4—42no—
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—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4—84no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4#9911yescharacteristic
239C22×C4#9911yescharacteristic
240C22×C4#12112yes—
241C22×C4#12411yescharacteristic
242C22×C4#12711yescharacteristic
243C2×C8—44no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C4⋊C4—84no—
250C22⋊C4—24no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—82no—
257C42—42no—
258C42—42no—
259C42—81no—
260C42—81no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—22no—
266C23—22no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—22no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23#39011yescharacteristic
292C23#40711yescharacteristic
293C23#40811yescharacteristic
294C23C2×#9911yescharacteristic
295C23#41011yescharacteristic
296C23#41311yescharacteristic
297C23#41311yescharacteristic
298C23#43512yes—
299C23#43512yes—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—84no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—84no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4#35212yes—
369C8—84no—
370C8—162no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—42no—
386C22—42no—
387C22—44no—
388C22—42no—
389C22—42no—
390C22—44no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22#146411yescharacteristic
400C22#146511yescharacteristic
401C22#146811yescharacteristic
402C22#147311yescharacteristic
403C22#195511yescharacteristic
404C22#196411yescharacteristic
405C22#196711yescharacteristic
406C22C2×#35211yescharacteristic
407C22#212011yescharacteristic
408C22#212111yescharacteristic
409C22#212311yescharacteristic
410C4—22no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—42no—
415C4—82no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4—82no—
420C4—82no—
421C4—161no—
422C2—21no—
423C2—21no—
424C2—21no—
425C2—21no—
426C2—42no—
427C2—42no—
428C2#965211yescharacteristic
429C2#1152611yescharacteristic
430C2#1152811yescharacteristic
431C2#1154311yescharacteristic
432C2#1154811yescharacteristic
433C2#1321711yescharacteristic
434C2#1322211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutator, prime radicalC25C22×C4
Frattini, cosocleC22×C22⋊C4C23
LCSG ▷ C25 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#139922
C2×#13262
C2×#13172
#85572
#81522