Finite Group Databasebeta

← #97422

Subgroups — #97422

Order 512.

1070 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 120 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×#2452C211yesmaximal, characteristic
2#10802C211yesmaximal, characteristic, complemented
3#8701C211yesmaximal, characteristic, complemented
4#7610C212yesmaximal, complemented
5#7606C212yesmaximal
6C22×C4⋊Q8C411yescharacteristic
7C22×C22⋊Q8—21no—
8C22×C4×D4C411yescharacteristic
9#2452—24no—
10#2452—24no—
11C2×#396—21no—
12C2×#393—21no—
13C2×#390C411yescharacteristic
14C2×#390C2211yescharacteristic
15C2×#389C411yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#385—21no—
18C2×C4×C4⋊C4C2211yescharacteristic
19#1817C2212yes—
20C2×#343—21no—
21C2×#343—21no—
22#1728C2212yes—
23#1280—21no—
24#1280—21no—
25#1271—24no—
26#1270—24no—
27C23×Q8—21no—
28C23×D4C2×C411yescharacteristic
29C24×C4—21no—
30C22×M16—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—44no—
34C2×C22⋊Q8—28no—
35C2×C22⋊Q8—44no—
36C2×C22⋊Q8—42no—
37C2×C22⋊Q8—42no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—44no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4D411yescharacteristic
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42C2311yescharacteristic
57C2×C22⋊C8—22no—
58#396—28no—
59#393—28no—
60#390—22no—
61#390—22no—
62#390—22no—
63#390D414yes—
64#389—22no—
65#389—22no—
66#389—22no—
67#389D414yes—
68#385—28no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95D411yescharacteristic
74C2×#95D411yescharacteristic
75#343—24no—
76#343—42no—
77#336—22no—
78#336—22no—
79#336—22no—
80#336—22no—
81C25—21no—
82C22×Q8—24no—
83C22×Q8—42no—
84C22×Q8—44no—
85C22×Q8—42no—
86C22×Q8—82no—
87C22×D4—24no—
88C22×D4—21no—
89C22×D4—41no—
90C22×D4—41no—
91C22×D4—42no—
92C22×D4C22⋊C412yes—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—24no—
97C23×C4—44no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—82no—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C2×M16—24no—
109C2×M16—42no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C2×M16—42no—
114C22×C8—42no—
115C22×C8—42no—
116C4⋊Q8—48no—
117C4⋊Q8—44no—
118C4⋊Q8—44no—
119C22⋊Q8—416no—
120C22⋊Q8—416no—
121C4×D4—48no—
122C4×D4—44no—
123C4×D4—44no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4D812yes—
149C2×C4⋊C4SD1612yes—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—44no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—82no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C22⋊C8—42no—
175C22⋊C8—42no—
176#95—24no—
177#95—24no—
178#95—22no—
179#95—22no—
180C24—24no—
181C24—22no—
182C24—42no—
183C24—42no—
184C24—42no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24—41no—
189C24—41no—
190C24—41no—
191C24C2×C22⋊C411yescharacteristic
192C2×Q8—48no—
193C2×Q8—48no—
194C2×Q8—84no—
195C2×Q8—88no—
196C2×Q8—88no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—48no—
202C2×D4—48no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4#10114yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—48no—
232C22×C4—48no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—84no—
255C22×C4—88no—
256C22×C4—88no—
257C22×C4#10212yes—
258C22×C4#10212yes—
259C22×C4#10412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4#12012yes—
263C22×C4#12012yes—
264C22×C4C2×D811yescharacteristic
265C22×C4C2×SD1611yescharacteristic
266C22×C4AGL(1,ℤ/8)12yes—
267M16—44no—
268M16—44no—
269M16—82no—
270M16—82no—
271C2×C8—44no—
272C2×C8—82no—
273C2×C8—82no—
274C2×C8—82no—
275C2×C8—82no—
276C2×C8—82no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—48no—
286C4⋊C4—48no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4#10014yes—
291C22⋊C4—48no—
292C22⋊C4—48no—
293C22⋊C4—48no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—84no—
297C42—42no—
298C42—42no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—42no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—24no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—82no—
321C23—82no—
322C23—82no—
323C23—82no—
324C23—82no—
325C23—82no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23#37911yescharacteristic
331C23#41011yescharacteristic
332C23#41011yescharacteristic
333C23C2×#10211yescharacteristic
334C23#41811yescharacteristic
335C23C2×#10411yescharacteristic
336C23#42111yescharacteristic
337Q8—816no—
338D4—44no—
339D4—48no—
340D4—44no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—816no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4C2×#10012yes—
399C2×C4C2×#10112yes—
400C2×C4#44712yes—
401C2×C4#45012yes—
402C8—84no—
403C8—162no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—84no—
420C22—84no—
421C22—84no—
422C22—84no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22#150211yescharacteristic
428C22#150511yescharacteristic
429C22#151411yescharacteristic
430C22#151711yescharacteristic
431C22#178211yescharacteristic
432C22#178311yescharacteristic
433C22#178411yescharacteristic
434C22#178611yescharacteristic
435C22#178911yescharacteristic
436C22#179511yescharacteristic
437C22#179611yescharacteristic
438C4—22no—
439C4—22no—
440C4—22no—
441C4—22no—
442C4—42no—
443C4—42no—
444C4—44no—
445C4—42no—
446C4—42no—
447C4—82no—
448C4—81no—
449C4—82no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2—84no—
455C2#833211yescharacteristic
456C2#836911yescharacteristic
457C2#837211yescharacteristic
458C2#838311yescharacteristic
459C2#838811yescharacteristic
460C2#851911yescharacteristic
461C2#855511yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×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. #10802C2semidirect
3. #8701C2semidirect
4. #7610C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24522
#108022
#87012
#76102
#76062