Finite Group Databasebeta

← #98552

Subgroups — #98552

Order 512.

1036 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×#2440C211yesmaximal, characteristic
2#8724C212yesmaximal, complemented
3#8559C211yesmaximal, characteristic
4#8559C211yesmaximal, characteristic
5#7641C212yesmaximal, complemented
6C22×C4⋊Q8C411yescharacteristic
7C22×C22⋊Q8—21no—
8C23×C4⋊C4C411yescharacteristic
9#2440—24no—
10#2440—24no—
11C2×#398—21no—
12C2×#395—21no—
13C2×#389C411yescharacteristic
14C2×#389C2211yescharacteristic
15C2×#385—21no—
16C2×#384C2211yescharacteristic
17C2×#380C411yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19#1797C2212yes—
20#1797C2212yes—
21C2×#323—21no—
22C2×#323—21no—
23#1296—24no—
24#1280—21no—
25#1280—21no—
26#1271—24no—
27C23×Q8—21no—
28C24×C4—21no—
29C24×C4C2×C411yescharacteristic
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—44no—
33C2×C22⋊Q8—28no—
34C2×C22⋊Q8—44no—
35C2×C22⋊Q8—42no—
36C2×C22⋊Q8—42no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—44no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C22⋊C4—21no—
51C22×C22⋊C4D411yescharacteristic
52C22×C22⋊C4D411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42C2311yescharacteristic
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59#398—28no—
60#395—28no—
61#389—22no—
62#389—22no—
63#389—22no—
64#389D414yes—
65#385—28no—
66#384—22no—
67#384D414yes—
68#380—22no—
69#380—22no—
70C4×C22⋊C4—22no—
71C4×C22⋊C4—22no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95D411yescharacteristic
76C2×#95D411yescharacteristic
77#336—22no—
78#336—22no—
79#323—42no—
80#323—42no—
81C25C22⋊C411yescharacteristic
82C22×Q8—24no—
83C22×Q8—42no—
84C22×Q8—44no—
85C22×Q8—42no—
86C22×Q8—82no—
87C23×C4—22no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—24no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4—44no—
99C23×C4—42no—
100C23×C4—42no—
101C23×C4—82no—
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C22×C8—42no—
109C22×C8—42no—
110C4⋊Q8—48no—
111C4⋊Q8—44no—
112C4⋊Q8—44no—
113C22⋊Q8—416no—
114C22⋊Q8—416no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—48no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—82no—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4C22⋊C412yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4D812yes—
155C2×C22⋊C4SD1612yes—
156C2×C22⋊C4SD1612yes—
157C2×C22⋊C4Q1612yes—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C4⋊C8—44no—
165C4⋊C8—44no—
166C22⋊C8—42no—
167C22⋊C8—42no—
168C22⋊C8—42no—
169C22⋊C8—42no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—22no—
174#95—22no—
175C24—21no—
176C24—21no—
177C24—21no—
178C24—21no—
179C24—41no—
180C24—41no—
181C24—41no—
182C24—41no—
183C24#10112yes—
184C24#10212yes—
185C24#10312yes—
186C24C2×C22⋊C411yescharacteristic
187C2×Q8—48no—
188C2×Q8—48no—
189C2×Q8—84no—
190C2×Q8—88no—
191C2×Q8—88no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—48no—
215C22×C4—48no—
216C22×C4—42no—
217C22×C4—48no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—48no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—84no—
243C22×C4—88no—
244C22×C4—88no—
245C22×C4#10112yes—
246C22×C4#10212yes—
247C22×C4#10312yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4#12012yes—
251C22×C4#12012yes—
252C22×C4C2×D811yescharacteristic
253C22×C4C2×SD1611yescharacteristic
254C22×C4C2×SD1611yescharacteristic
255C22×C4C2×Q1611yescharacteristic
256C22×C4AGL(1,ℤ/8)12yes—
257C22×C4#13712yes—
258C2×C8—82no—
259C2×C8—82no—
260C2×C8—82no—
261C2×C8—82no—
262C2×C8—82no—
263C2×C8—82no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—48no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—84no—
280C4⋊C4#10014yes—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—48no—
286C42—24no—
287C42—42no—
288C42—42no—
289C42—42no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23#33214yes—
317C23#37911yescharacteristic
318C23C2×#10111yescharacteristic
319C23C2×#10111yescharacteristic
320C23C2×#10211yescharacteristic
321C23C2×#10311yescharacteristic
322C23#41811yescharacteristic
323C23#41911yescharacteristic
324Q8—816no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—816no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4#41012yes—
380C2×C4C2×#10012yes—
381C2×C4#44712yes—
382C2×C4#44812yes—
383C2×C4#45012yes—
384C2×C4#45112yes—
385C8—162no—
386C8—162no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—24no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—44no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22C2×#33211yescharacteristic
411C22#154611yescharacteristic
412C22#154911yescharacteristic
413C22#155611yescharacteristic
414C22#178211yescharacteristic
415C22#178311yescharacteristic
416C22#178411yescharacteristic
417C22#178611yescharacteristic
418C22#178711yescharacteristic
419C22#179511yescharacteristic
420C22#179611yescharacteristic
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—42no—
428C4—44no—
429C4—42no—
430C4—42no—
431C4—82no—
432C4—82no—
433C4—82no—
434C2—21no—
435C2—21no—
436C2—21no—
437C2—21no—
438C2—44no—
439C2#834011yescharacteristic
440C2#837611yescharacteristic
441C2#838011yescharacteristic
442C2#838311yescharacteristic
443C2#839111yescharacteristic
444C2#852511yescharacteristic
445C2#855511yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #8724C2semidirect
5. #7641C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
#87242
#85592
#76412