Finite Group Databasebeta

← #141985

Subgroups — #141985

Order 512.

676 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 96 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#13923C211yesmaximal, characteristic
2#8559C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7187C212yesmaximal
5#7187C212yesmaximal
6C23×C4⋊C4C411yescharacteristic
7C2×#380C411yescharacteristic
8C2×C4×C22⋊C4C2211yescharacteristic
9#1797—21no—
10#1749—21no—
11C2×#323C2212yes—
12C2×#323C2212yes—
13#1441—21no—
14#1441—21no—
15#1436—21no—
16#1436—21no—
17C22⋊#95C411yescharacteristic
18C22⋊#95C411yescharacteristic
19C22⋊#95C2211yescharacteristic
20C22⋊#95C2211yescharacteristic
21#1280—21no—
22#1280—21no—
23C24×C4C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—41no—
32C22×C4⋊C4—41no—
33C22×C4⋊C4—42no—
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4C2×C411yescharacteristic
37C22×C22⋊C4C2311yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40C2×C4⋊C8—41no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×#97—41no—
44#380—42no—
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×#95D411yescharacteristic
57C2×#95D411yescharacteristic
58#336—41no—
59#336—41no—
60#323—22no—
61#323—24no—
62#323Q814yes—
63C25C22×C411yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—41no—
78C23×C4—41no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—81no—
82C23×C4—81no—
83C23×C4—81no—
84C23×C4—81no—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4M1612yes—
90C23×C4M1612yes—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—42no—
94C22×C8—42no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—44no—
101C2×C4⋊C4—44no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—81no—
108C2×C4⋊C4—81no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4C2×Q812yes—
121C2×C22⋊C4#4112yes—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—81no—
127C2×C42—81no—
128C4⋊C8—82no—
129C22⋊C8—24no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133#97—82no—
134#95—41no—
135#95—41no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—81no—
142#95—81no—
143#95—81no—
144#95—81no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24C2×C22⋊C411yescharacteristic
154C24C4×Q812yes—
155C24C2×M1611yescharacteristic
156C24C2×M1611yescharacteristic
157C24#13112yes—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—48no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
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—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
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—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—84no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—84no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4#9911yescharacteristic
239C22×C4#9911yescharacteristic
240C22×C4#10012yes—
241C22×C4#10112yes—
242C22×C4C22⋊Q812yes—
243C22×C4#12411yescharacteristic
244C22×C4C4⋊Q811yescharacteristic
245C2×C8—44no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—82no—
250C2×C8—82no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—48no—
254C4⋊C4—84no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—44no—
258C22⋊C4—82no—
259C22⋊C4—82no—
260C42—42no—
261C42—42no—
262C42—81no—
263C42—81no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#39111yescharacteristic
295C23#40711yescharacteristic
296C23#40811yescharacteristic
297C23C2×#9911yescharacteristic
298C23#41011yescharacteristic
299C23C2×#10011yescharacteristic
300C23C2×#10111yescharacteristic
301C23#44612yes—
302C23#44612yes—
303C2×C4—24no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—84no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—84no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C8—84no—
377C8—162no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—22no—
392C22—42no—
393C22—42no—
394C22—44no—
395C22—42no—
396C22—42no—
397C22—44no—
398C22—42no—
399C22—42no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—42no—
405C22—42no—
406C22#146211yescharacteristic
407C22#146711yescharacteristic
408C22#146911yescharacteristic
409C22#147211yescharacteristic
410C22#198511yescharacteristic
411C22#199611yescharacteristic
412C22#199711yescharacteristic
413C22#212011yescharacteristic
414C22#212011yescharacteristic
415C22#212111yescharacteristic
416C22#212111yescharacteristic
417C4—42no—
418C4—44no—
419C4—41no—
420C4—41no—
421C4—42no—
422C4—41no—
423C4—41no—
424C4—42no—
425C4—81no—
426C4—82no—
427C4—161no—
428C4—161no—
429C2—21no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—42no—
434C2—42no—
435C2#988211yescharacteristic
436C2#1150111yescharacteristic
437C2#1150211yescharacteristic
438C2#1157711yescharacteristic
439C2#1157811yescharacteristic
440C2#1329411yescharacteristic
441C2#1329711yescharacteristic
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
#139232
#85592
#81522
#71872