Finite Group Databasebeta

← #140653

Subgroups — #140653

Order 512.

674 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 106 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
1#14001C211yesmaximal, characteristic
2C2×#1325C212yesmaximal, complemented
3C2×#1318C212yesmaximal
4#8560C211yesmaximal, characteristic, complemented
5C4×#324C211yesmaximal, characteristic, complemented
6C2×#390C411yescharacteristic
7C2×#390C411yescharacteristic
8C2×#390C2211yescharacteristic
9C2×#388—21no—
10C2×#388—21no—
11C2×#384C2211yescharacteristic
12C2×C4×C4⋊C4C2211yescharacteristic
13#1797—21no—
14C4×C22⋊C8—21no—
15C2×#324C2212yes—
16C2×#324C2212yes—
17#1439—21no—
18#1439—21no—
19C4⋊#95C411yescharacteristic
20C4×#95C411yescharacteristic
21#1325—24no—
22#1318—24no—
23#1280—21no—
24#1280—21no—
25C23×D4C2×C411yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2311yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42C2×C4⋊C8—41no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C4×C8—41no—
46#390—22no—
47#390—22no—
48#390—22no—
49#390—41no—
50#390—41no—
51#390C814yescomplemented
52#388—44no—
53#388—44no—
54#384—22no—
55#384—22no—
56C4×C4⋊C4—41no—
57C4×C4⋊C4—41no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95D411yescharacteristic
63C2×#95D411yescharacteristic
64#336—41no—
65#336—41no—
66#324—22no—
67#324—22no—
68#324—22no—
69#324D414yes—
70C25—21no—
71C22×D4—22no—
72C22×D4—21no—
73C22×D4—22no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4—41no—
77C22×D4—41no—
78C22×D4C2×C812yes—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22×C411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C22×C8—42no—
90C22×C8—42no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—81no—
104C2×C4⋊C4—81no—
105C2×C4⋊C4C2×C812yes—
106C2×C4⋊C4M1612yes—
107C2×C4⋊C4C2×D412yes—
108C2×C4⋊C4#4112yes—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—81no—
126C2×C22⋊C4—81no—
127C2×C22⋊C4—81no—
128C2×C22⋊C4—81no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—22no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—41no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42—41no—
144C2×C42—41no—
145C2×C42C2×C812yes—
146C4⋊C8—82no—
147C22⋊C8—24no—
148C22⋊C8—42no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151C4×C8—82no—
152#95—24no—
153#95—41no—
154#95—41no—
155#95—42no—
156#95—42no—
157#95—42no—
158C24—22no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24—41no—
169C24—41no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24C2×C22⋊C411yescharacteristic
174C2×D4—24no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—82no—
181C2×D4—82no—
182C2×D4—82no—
183C2×D4—82no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4C22⋊C812yes—
235C22×C4C22⋊C812yes—
236C22×C4#9911yescharacteristic
237C22×C4#9911yescharacteristic
238C22×C4C4×D412yes—
239C22×C4#12112yes—
240C22×C4#12411yescharacteristic
241C22×C4#12711yescharacteristic
242C22×C4C22×C811yescharacteristic
243C22×C4C2×M1611yescharacteristic
244C22×C4#13112yes—
245C2×C8—44no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—82no—
250C2×C8—82no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—82no—
256C4⋊C4—84no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C22⋊C4—48no—
260C22⋊C4—48no—
261C22⋊C4—82no—
262C22⋊C4—84no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C22⋊C4—84no—
266C22⋊C4—82no—
267C42—24no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42—81no—
274C42—81no—
275C42—82no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23#39011yescharacteristic
316C23C2×C22⋊C811yescharacteristic
317C23#40811yescharacteristic
318C23C2×#9911yescharacteristic
319C23#41011yescharacteristic
320C23#41311yescharacteristic
321C23#41311yescharacteristic
322D4—84no—
323D4—84no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4#35312yes—
388C2×C4C8×D412yes—
389C2×C4#43612yes—
390C8—84no—
391C8—162no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22—82no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22#145811yescharacteristic
420C22#145911yescharacteristic
421C22#146811yescharacteristic
422C22#147311yescharacteristic
423C22#195411yescharacteristic
424C22#196411yescharacteristic
425C22#196711yescharacteristic
426C22C2×#35311yescharacteristic
427C22#212811yescharacteristic
428C22#213211yescharacteristic
429C22#213311yescharacteristic
430C4—22no—
431C4—22no—
432C4—22no—
433C4—44no—
434C4—41no—
435C4—42no—
436C4—41no—
437C4—41no—
438C4—42no—
439C4—42no—
440C4—82no—
441C4—81no—
442C4—161no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2—21no—
447C2—162no—
448C2#965111yescharacteristic
449C2#1147511yescharacteristic
450C2#1148011yescharacteristic
451C2#1154911yescharacteristic
452C2#1155411yescharacteristic
453C2#1322311yescharacteristic
454C2#1322811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1325C2semidirect
4. #8560C2semidirect
5. C4×#324C2semidirect
51. #390C8semidirect

Maximal subgroups

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

subgroupindex
#140012
C2×#13252
C2×#13182
#85602
C4×#3242