Finite Group Databasebeta

← #137683

Subgroups — #137683

Order 512.

756 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 112 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×#1458C211yesmaximal, characteristic
2#14138C211yesmaximal, characteristic
3C2×#1319C212yesmaximal, complemented
4#8300C211yesmaximal, characteristic
5#7189C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C22×#117C2211yescharacteristic
8C2×#431—21no—
9C2×#391C411yescharacteristic
10C2×#391C2211yescharacteristic
11C2×C4×C22⋊C4C411yescharacteristic
12#1817—21no—
13#1458—24no—
14C2×#324C2212yes—
15C2×#324C2212yes—
16C2×#323—21no—
17C2×#323—21no—
18#1440—21no—
19#1440—21no—
20#1436—21no—
21#1436—21no—
22C4⋊#95C411yescharacteristic
23C4⋊#95C2211yescharacteristic
24#1319—24no—
25C23×Q8C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C2×C22⋊Q8—22no—
28C2×C22⋊Q8—41no—
29C2×C22⋊Q8—41no—
30C2×C22⋊Q8—42no—
31C2×C22⋊Q8C814yescomplemented
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—41no—
35C2×#117—41no—
36C2×#117—42no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2311yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4D411yescharacteristic
44C22×C22⋊C4D411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49#431—24no—
50#431—42no—
51C2×C4⋊C8—41no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C4×C8—41no—
55#391—22no—
56#391—22no—
57#391—42no—
58C4×C22⋊C4—42no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68#336—41no—
69#336—41no—
70#324—22no—
71#324—22no—
72#324—22no—
73#324D414yes—
74#323—24no—
75#323—42no—
76C25C22⋊C411yescharacteristic
77C22×Q8—21no—
78C22×Q8—22no—
79C22×Q8—22no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8—41no—
83C22×Q8—41no—
84C22×Q8C2×C812yes—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C2×C812yes—
100C23×C4M1612yes—
101C23×C4C22×C411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C22×C8—42no—
105C22×C8—42no—
106C22⋊Q8—44no—
107C22⋊Q8—44no—
108C22⋊Q8—44no—
109C22⋊Q8—44no—
110#117—44no—
111#117—44no—
112#117—44no—
113#117—44no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4C2×C812yes—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4#4112yes—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—22no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42—41no—
157C4⋊C8—44no—
158C22⋊C8—24no—
159C22⋊C8—42no—
160C22⋊C8—42no—
161C22⋊C8—42no—
162C4×C8—44no—
163#95—42no—
164#95—42no—
165#95—42no—
166#95—42no—
167#95—42no—
168#95—42no—
169#95—42no—
170#95—41no—
171#95—42no—
172#95—41no—
173C24—21no—
174C24—21no—
175C24—21no—
176C24—21no—
177C24—21no—
178C24—21no—
179C24—41no—
180C24—41no—
181C24—41no—
182C24—41no—
183C24C22⋊C812yes—
184C24C2×C22⋊C411yescharacteristic
185C2×Q8—24no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—82no—
192C2×Q8—82no—
193C2×Q8—82no—
194C2×Q8—82no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
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—41no—
243C22×C4—41no—
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—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4C22⋊C812yes—
263C22×C4#9911yescharacteristic
264C22×C4#9911yescharacteristic
265C22×C4#9912yes—
266C22×C4#10012yes—
267C22×C4C4×D412yes—
268C22×C4#12011yescharacteristic
269C22×C4#12112yes—
270C22×C4#12311yescharacteristic
271C22×C4C22×C811yescharacteristic
272C22×C4C2×M1611yescharacteristic
273C22×C4#13112yes—
274C2×C8—44no—
275C2×C8—82no—
276C2×C8—82no—
277C2×C8—82no—
278C2×C8—82no—
279C2×C8—82no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—48no—
291C22⋊C4—48no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—42no—
297C42—42no—
298C42—82no—
299C42—82no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23#38611yescharacteristic
328C23C2×C22⋊C811yescharacteristic
329C23#40811yescharacteristic
330C23C2×#9911yescharacteristic
331C23C2×#9911yescharacteristic
332C23C2×#10011yescharacteristic
333C23#41311yescharacteristic
334Q8—84no—
335Q8—84no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4#32312yes—
407C2×C4#32312yes—
408C2×C4C8×D412yes—
409C2×C4#43512yes—
410C8—84no—
411C8—162no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22C2×#32311yescharacteristic
436C22#145911yescharacteristic
437C22#146811yescharacteristic
438C22#147111yescharacteristic
439C22#186911yescharacteristic
440C22#189811yescharacteristic
441C22#190011yescharacteristic
442C22#212611yescharacteristic
443C22#212911yescharacteristic
444C22#213211yescharacteristic
445C22#213511yescharacteristic
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—41no—
451C4—42no—
452C4—41no—
453C4—42no—
454C4—82no—
455C4—82no—
456C4—81no—
457C4—82no—
458C4—82no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—21no—
463C2—82no—
464C2#907911yescharacteristic
465C2#1143111yescharacteristic
466C2#1143411yescharacteristic
467C2#1155211yescharacteristic
468C2#1155611yescharacteristic
469C2#1309111yescharacteristic
470C2#1309511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1319C2semidirect
5. #7189C2semidirect
31. C2×C22⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#14582
#141382
C2×#13192
#83002
#71892