Finite Group Databasebeta

← #137677

Subgroups — #137677

Order 512.

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

normal subgroups only

This group has 810 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#13870C211yesmaximal, characteristic
3C2×#1317C212yesmaximal
4#8300C211yesmaximal, characteristic
5#7187C212yesmaximal
6C22×#117C2211yescharacteristic
7C23×C4⋊C4C411yescharacteristic
8C2×#431—21no—
9C2×#381C411yescharacteristic
10C2×#381C411yescharacteristic
11C2×#381C2211yescharacteristic
12#1817—21no—
13#1458—24no—
14C2×#324—21no—
15C2×#324—21no—
16C2×#323C2212yes—
17C2×#323C2212yes—
18#1441—21no—
19#1441—21no—
20#1436—21no—
21#1436—21no—
22C22⋊#95C411yescharacteristic
23C22⋊#95C2211yescharacteristic
24#1317—24no—
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2×C411yescharacteristic
28C2×#117—22no—
29C2×#117—22no—
30C2×#117—41no—
31C2×#117—41no—
32C2×#117—42no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—41no—
37C22×C4⋊C4—41no—
38C22×C4⋊C4—42no—
39C22×C4⋊C4C814yescomplemented
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C22⋊C4C2×C411yescharacteristic
43C22×C22⋊C4C2×C411yescharacteristic
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4C2311yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48#431—24no—
49#431—42no—
50C2×C4⋊C8—41no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C4×C8—41no—
54#381—22no—
55#381—22no—
56#381—42no—
57#381—42no—
58C2×#95—21no—
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—24no—
71#324—42no—
72#323—22no—
73#323—22no—
74#323—22no—
75#323D414yes—
76C25C22×C411yescharacteristic
77C23×C4—21no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—41no—
91C23×C4—41no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4—41no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C2×C812yes—
105C23×C4C2×C812yes—
106C23×C4C2×C812yes—
107C23×C4M1612yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C22×C8—42no—
111C22×C8—42no—
112#117—44no—
113#117—44no—
114#117—44no—
115#117—44no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—44no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4C2×D412yes—
146C2×C22⋊C4#4112yes—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—22no—
155C2×C42—41no—
156C4⋊C8—44no—
157C22⋊C8—24no—
158C22⋊C8—42no—
159C22⋊C8—42no—
160C22⋊C8—42no—
161C4×C8—44no—
162#95—42no—
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—41no—
178C24—41no—
179C24—41no—
180C24—41no—
181C24C2×C22⋊C411yescharacteristic
182C24C4×D412yes—
183C24C22×C811yescharacteristic
184C24C2×M1611yescharacteristic
185C24#13112yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—48no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
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—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4C22⋊C812yes—
275C22×C4C22⋊C812yes—
276C22×C4#9911yescharacteristic
277C22×C4#9911yescharacteristic
278C22×C4#9912yes—
279C22×C4#10012yes—
280C22×C4#12011yescharacteristic
281C22×C4#12112yes—
282C22×C4#12311yescharacteristic
283C2×C8—44no—
284C2×C8—82no—
285C2×C8—82no—
286C2×C8—82no—
287C2×C8—82no—
288C2×C8—82no—
289C4⋊C4—44no—
290C4⋊C4—48no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—44no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—42no—
305C42—42no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—22no—
311C23—22no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—42no—
321C23—42no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#38611yescharacteristic
337C23C2×C22⋊C811yescharacteristic
338C23#40811yescharacteristic
339C23C2×#9911yescharacteristic
340C23C2×#9911yescharacteristic
341C23C2×#10011yescharacteristic
342C23#41311yescharacteristic
343C23C8×D412yes—
344C23#43512yes—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—84no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—84no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4#32312yes—
421C2×C4#32312yes—
422C8—84no—
423C8—162no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—22no—
437C22—22no—
438C22—42no—
439C22—42no—
440C22—44no—
441C22—44no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22C2×#32311yescharacteristic
453C22#145911yescharacteristic
454C22#146811yescharacteristic
455C22#147111yescharacteristic
456C22#186911yescharacteristic
457C22#189811yescharacteristic
458C22#190011yescharacteristic
459C22#212011yescharacteristic
460C22#212011yescharacteristic
461C22#212311yescharacteristic
462C22#212311yescharacteristic
463C4—22no—
464C4—22no—
465C4—42no—
466C4—42no—
467C4—41no—
468C4—41no—
469C4—42no—
470C4—44no—
471C4—81no—
472C4—82no—
473C4—161no—
474C4—161no—
475C2—21no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—42no—
480C2—42no—
481C2#907911yescharacteristic
482C2#1142511yescharacteristic
483C2#1142811yescharacteristic
484C2#1154411yescharacteristic
485C2#1154811yescharacteristic
486C2#1308111yescharacteristic
487C2#1308511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C22×C8
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
39. C22×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
C2×#14582
#138702
C2×#13172
#83002
#71872