Finite Group Databasebeta

← #98544

Subgroups — #98544

Order 512.

976 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2429C211yesmaximal, characteristic
2#8559C211yesmaximal, characteristic
3#8559C211yesmaximal, characteristic
4#8342C212yesmaximal, complemented
5#7641C212yesmaximal, complemented
6C22×C4⋊Q8C2211yescharacteristic
7C22×C22⋊Q8C411yescharacteristic
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×Q8C411yescharacteristic
10#2429—24no—
11#2429—24no—
12C2×#398—21no—
13C2×#396—21no—
14C2×#391C411yescharacteristic
15C2×#389—21no—
16C2×#389C2211yescharacteristic
17C2×#388—21no—
18C2×C4×C22⋊C4C2211yescharacteristic
19#1797C2212yes—
20#1797C2212yes—
21C2×#333—24no—
22C2×#323—21no—
23C2×#323—21no—
24#1280—21no—
25#1280—21no—
26#1271—24no—
27C23×Q8C2×C411yescharacteristic
28C23×Q8C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—44no—
32C2×C4⋊Q8D414yes—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—44no—
38C2×C22⋊Q8—44no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—44no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4D411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42—21no—
55C22×C42C2×C411yescharacteristic
56C22×C42C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61#398—28no—
62#396—28no—
63#391—22no—
64#391—22no—
65#389—28no—
66#389—22no—
67#389D414yes—
68#388—28no—
69C4×C22⋊C4—22no—
70C4×C22⋊C4—22no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95D411yescharacteristic
75C2×#95D411yescharacteristic
76#336—22no—
77#336—22no—
78#333—48no—
79#323—42no—
80#323—42no—
81C25C22⋊C411yescharacteristic
82C22×Q8—24no—
83C22×Q8—21no—
84C22×Q8—21no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8—42no—
88C22×Q8—41no—
89C22×Q8—41no—
90C22×Q8—41no—
91C22×Q8—41no—
92C22×Q8C22⋊C412yes—
93C22×Q8C22⋊C412yes—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—41no—
101C23×C4—41no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C22×C8—42no—
111C22×C8—42no—
112C4⋊Q8—28no—
113C4⋊Q8—44no—
114C4⋊Q8—48no—
115C22⋊Q8—48no—
116C22⋊Q8—44no—
117C22⋊Q8—48no—
118C22⋊Q8—44no—
119C22⋊Q8—44no—
120C22⋊Q8—44no—
121C4×Q8—48no—
122C4×Q8—44no—
123C4×Q8—44no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C412yes—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4SD1612yes—
155C2×C22⋊C4Q1612yes—
156C2×C42—24no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—42no—
160C2×C42—44no—
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—21no—
180C24—21no—
181C24—41no—
182C24—41no—
183C24—41no—
184C24—41no—
185C24#10312yes—
186C24C2×C22⋊C411yescharacteristic
187C2×Q8—24no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—48no—
191C2×Q8—48no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—84no—
202C2×Q8—84no—
203C2×Q8#9914yes—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—84no—
253C22×C4—84no—
254C22×C4#10312yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4#12012yes—
258C22×C4#12012yes—
259C22×C4C2×SD1611yescharacteristic
260C22×C4C2×Q1611yescharacteristic
261C22×C4#13512yes—
262C22×C4AGL(1,ℤ/8)11yescharacteristic
263C22×C4#13712yes—
264C22×C4#13711yescharacteristic
265C2×C8—82no—
266C2×C8—82no—
267C2×C8—82no—
268C2×C8—82no—
269C2×C8—82no—
270C2×C8—82no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—48no—
274C4⋊C4—48no—
275C4⋊C4—42no—
276C4⋊C4—48no—
277C4⋊C4—48no—
278C4⋊C4—48no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4#10014yes—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—48no—
286C22⋊C4—48no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C42—24no—
290C42—42no—
291C42—42no—
292C42—48no—
293C42—42no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23#37911yescharacteristic
322C23#41311yescharacteristic
323C23#41311yescharacteristic
324C23C2×#10311yescharacteristic
325C23#41611yescharacteristic
326C23#41711yescharacteristic
327C23#41911yescharacteristic
328Q8—44no—
329Q8—44no—
330Q8—48no—
331Q8—84no—
332Q8—84no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4C2×#9912yes—
387C2×C4C2×#10012yes—
388C2×C4#44812yes—
389C2×C4#44912yes—
390C2×C4#45112yes—
391C2×C4#45212yes—
392C8—162no—
393C8—162no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#154211yescharacteristic
418C22#155011yescharacteristic
419C22#155411yescharacteristic
420C22#155911yescharacteristic
421C22#178111yescharacteristic
422C22#178311yescharacteristic
423C22#178511yescharacteristic
424C22#178711yescharacteristic
425C22#178811yescharacteristic
426C22#179611yescharacteristic
427C22#179611yescharacteristic
428C4—22no—
429C4—22no—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—44no—
435C4—42no—
436C4—44no—
437C4—81no—
438C4—82no—
439C4—82no—
440C4—81no—
441C2—21no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—82no—
446C2#834311yescharacteristic
447C2#837711yescharacteristic
448C2#837911yescharacteristic
449C2#839111yescharacteristic
450C2#839311yescharacteristic
451C2#852711yescharacteristic
452C2#854911yescharacteristic
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
4. #8342C2semidirect
5. #7641C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24292
#85592
#83422
#76412