Finite Group Databasebeta

← #96743

Subgroups — #96743

Order 512.

728 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#389C211yesmaximal, characteristic
2#10795C211yesmaximal, characteristic
3#10795C211yesmaximal, characteristic
4C2×#1271C212yesmaximal
5#7025C212yesmaximal
6C2×#389C411yescharacteristic
7C2×#389C2211yescharacteristic
8C2×C4×C4⋊C4—21no—
9C2×C4×C4⋊C4C411yescharacteristic
10C2×C4×C4⋊C4C411yescharacteristic
11C2×C43C2211yescharacteristic
12C2×#336—21no—
13C2×#336—21no—
14C2×#336—21no—
15C2×#336—21no—
16C2×#336C2212yes—
17C2×#336C2212yes—
18#1436—21no—
19#1436—21no—
20C4⋊#95C411yescharacteristic
21C4⋊#95C2211yescharacteristic
22C4×#95—21no—
23#1271C418yescomplemented
24C23×C8—22no—
25C23×C8—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C42—21no—
33C22×C42—21no—
34C22×C42C2×C411yescharacteristic
35C22×C42C2×C411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42C2311yescharacteristic
41#389—22no—
42#389—22no—
43#389—22no—
44#389C2×C414yes—
45C4×C4⋊C4—22no—
46C4×C4⋊C4—22no—
47C4×C4⋊C4—42no—
48C4×C4⋊C4—42no—
49C4×C4⋊C4—42no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C43—42no—
59#336—22no—
60#336—22no—
61#336—42no—
62#336—42no—
63#336—42no—
64#336—42no—
65#336C2×C414yes—
66#336C2×C414yes—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C22×C8—24no—
79C22×C8—24no—
80C22×C8—42no—
81C22×C8—42no—
82C22×C8—42no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8—42no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4C4212yes—
112C2×C4⋊C4C4212yes—
113C2×C4⋊C4M1612yes—
114C2×C4⋊C4M1612yes—
115C2×C4⋊C4M1612yes—
116C2×C4⋊C4M1612yes—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—24no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42—42no—
139C2×C42—42no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42D812yes—
144C2×C42SD1612yes—
145C2×C42SD1612yes—
146C2×C42Q1612yes—
147C2×C42C22×C412yes—
148C2×C42#4112yes—
149#95—24no—
150#95—24no—
151#95—24no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—41no—
156#95—41no—
157#95—42no—
158C24C2×C22⋊C411yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
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#9912yes—
222C22×C4#10112yes—
223C22×C4#10212yes—
224C22×C4#10212yes—
225C22×C4#10312yes—
226C22×C4#10312yes—
227C22×C4C2×C4211yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4C4×D412yes—
230C22×C4C4×D412yes—
231C22×C4C2×M1611yescharacteristic
232C22×C4C2×M1611yescharacteristic
233C22×C4C2×M1611yescharacteristic
234C22×C4C2×M1611yescharacteristic
235C22×C4C2×D811yescharacteristic
236C22×C4C2×SD1611yescharacteristic
237C22×C4C2×SD1611yescharacteristic
238C22×C4C2×Q1611yescharacteristic
239C22×C4#13512yes—
240C22×C4#13512yes—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—48no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4#9714yes—
268C42—24no—
269C42—24no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42—42no—
274C42—42no—
275C42—42no—
276C42—44no—
277C42—44no—
278C42—42no—
279C42—42no—
280C42—48no—
281C42—84no—
282C23C4×C22⋊C411yescharacteristic
283C23#40711yescharacteristic
284C23#40711yescharacteristic
285C23C2×#9911yescharacteristic
286C23#41011yescharacteristic
287C23C2×#10111yescharacteristic
288C23#41311yescharacteristic
289C23C2×#10211yescharacteristic
290C23C2×#10311yescharacteristic
291C23#41611yescharacteristic
292C23#41611yescharacteristic
293C23#42111yescharacteristic
294C23#42111yescharacteristic
295C23#42111yescharacteristic
296C23#42111yescharacteristic
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—81no—
369C2×C4—81no—
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#33214yes—
381C2×C4C2×#9711yescharacteristic
382C2×C4C4×M1611yescharacteristic
383C2×C4#43211yescharacteristic
384C2×C4#43211yescharacteristic
385C2×C4C4×D812yes—
386C2×C4C4×SD1612yes—
387C2×C4C4×SD1612yes—
388C2×C4C4×Q1612yes—
389C8—84no—
390C8—84no—
391C22#146311yescharacteristic
392C22#146411yescharacteristic
393C22#146511yescharacteristic
394C22#146711yescharacteristic
395C22#148611yescharacteristic
396C22#148611yescharacteristic
397C22#148711yescharacteristic
398C22#148711yescharacteristic
399C22#148811yescharacteristic
400C22#148811yescharacteristic
401C22#148811yescharacteristic
402C22#148811yescharacteristic
403C22#151611yescharacteristic
404C22#151611yescharacteristic
405C22#151711yescharacteristic
406C22#151811yescharacteristic
407C22#151911yescharacteristic
408C22#152011yescharacteristic
409C22#152111yescharacteristic
410C22#152111yescharacteristic
411C22#153411yescharacteristic
412C22#153511yescharacteristic
413C22#153511yescharacteristic
414C22#153711yescharacteristic
415C22C2×#33211yescharacteristic
416C22#154011yescharacteristic
417C22#154211yescharacteristic
418C22#154411yescharacteristic
419C22#174911yescharacteristic
420C22C4×#9911yescharacteristic
421C22C4×#10111yescharacteristic
422C22#175911yescharacteristic
423C22#175911yescharacteristic
424C22C4×#10211yescharacteristic
425C22C4×#10311yescharacteristic
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—24no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—44no—
439C4—82no—
440C4—82no—
441C4—82no—
442C2#764111yescharacteristic
443C2#764211yescharacteristic
444C2#764411yescharacteristic
445C2#764611yescharacteristic
446C2#764711yescharacteristic
447C2#764811yescharacteristic
448C2#765011yescharacteristic
449C2#765211yescharacteristic
450C2#815311yescharacteristic
451C2#817411yescharacteristic
452C2#817511yescharacteristic
453C2#822011yescharacteristic
454C2#822111yescharacteristic
455C2#822511yescharacteristic
456C2C4×#33211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #1271C4semidirect

Maximal subgroups

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

subgroupindex
C4×#3892
#107952
C2×#12712
#70252