Finite Group Databasebeta

← #96758

Subgroups — #96758

Order 512.

656 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14703C211yesmaximal, characteristic
2#10795C211yesmaximal, characteristic
3#10795C211yesmaximal, characteristic
4C2×#1272C212yesmaximal
5#7025C212yesmaximal
6C2×#391C411yescharacteristic
7C2×#391C411yescharacteristic
8C2×#391C2211yescharacteristic
9C2×#389C411yescharacteristic
10C2×C43C2211yescharacteristic
11C2×#336—21no—
12C2×#336—21no—
13C2×#336—21no—
14C2×#336—21no—
15C2×#336C2212yes—
16C2×#336C2212yes—
17#1436—21no—
18#1436—21no—
19C4⋊#95C411yescharacteristic
20C4⋊#95C2211yescharacteristic
21#1434—21no—
22#1434—21no—
23#1272C418yescomplemented
24C23×Q8C2×C411yescharacteristic
25C23×C8—22no—
26C23×C8—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42C2311yescharacteristic
38#391—22no—
39#391—42no—
40#391—42no—
41#391C2×C414yes—
42#389—42no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C43—42no—
55#336—22no—
56#336—22no—
57#336—42no—
58#336—42no—
59#336—42no—
60#336—42no—
61#336C2×C414yes—
62#336C2×C414yes—
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8—41no—
70C22×Q8C4212yes—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C22×C8—24no—
83C22×C8—24no—
84C22×C8—42no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8—42no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8—42no—
95C22×C8—42no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4C4212yes—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—21no—
117C2×C42—22no—
118C2×C42—21no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42SD1612yes—
130C2×C42Q1612yes—
131C2×C42C22×C412yes—
132C2×C42#4112yes—
133#95—24no—
134#95—24no—
135#95—42no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—41no—
144#95—41no—
145C24C2×C22⋊C411yescharacteristic
146C2×Q8—24no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—82no—
153C2×Q8—82no—
154C2×Q8—82no—
155C2×Q8—82no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
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—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
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—22no—
193C22×C4—22no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
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#9912yes—
221C22×C4#10012yes—
222C22×C4#10312yes—
223C22×C4#10312yes—
224C22×C4C2×C4211yescharacteristic
225C22×C4#11711yescharacteristic
226C22×C4C4×D412yes—
227C22×C4C4×D412yes—
228C22×C4#13112yes—
229C22×C4#13112yes—
230C22×C4C2×SD1611yescharacteristic
231C22×C4C2×Q1611yescharacteristic
232C22×C4#13512yes—
233C22×C4AGL(1,ℤ/8)11yescharacteristic
234C22×C4#13712yes—
235C22×C4#13711yescharacteristic
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—82no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—84no—
255C42—42no—
256C42—42no—
257C42—42no—
258C42—42no—
259C42—44no—
260C42—44no—
261C42—44no—
262C42—44no—
263C42—42no—
264C42—42no—
265C23C4×C22⋊C411yescharacteristic
266C23#40811yescharacteristic
267C23#40811yescharacteristic
268C23C2×#9911yescharacteristic
269C23#41011yescharacteristic
270C23C2×#10011yescharacteristic
271C23#41311yescharacteristic
272C23C2×#10311yescharacteristic
273C23#41611yescharacteristic
274C23#41711yescharacteristic
275C23#41911yescharacteristic
276C23#42111yescharacteristic
277C23#42111yescharacteristic
278C23#42111yescharacteristic
279C23#42111yescharacteristic
280Q8—84no—
281Q8—84no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4#40512yes—
363C2×C4#43312yes—
364C2×C4C4×SD1612yes—
365C2×C4C4×Q1612yes—
366C2×C4#44012yes—
367C2×C4#44112yes—
368C8—84no—
369C8—84no—
370C22#146811yescharacteristic
371C22#146911yescharacteristic
372C22#147111yescharacteristic
373C22#147311yescharacteristic
374C22#148111yescharacteristic
375C22#148111yescharacteristic
376C22#148211yescharacteristic
377C22#148211yescharacteristic
378C22#149511yescharacteristic
379C22#149511yescharacteristic
380C22#149711yescharacteristic
381C22#149711yescharacteristic
382C22#151111yescharacteristic
383C22#151211yescharacteristic
384C22#151311yescharacteristic
385C22#151511yescharacteristic
386C22#151611yescharacteristic
387C22#151811yescharacteristic
388C22#152011yescharacteristic
389C22#152111yescharacteristic
390C22#153311yescharacteristic
391C22#153511yescharacteristic
392C22#153511yescharacteristic
393C22#153611yescharacteristic
394C22#154811yescharacteristic
395C22#155011yescharacteristic
396C22#155111yescharacteristic
397C22#155411yescharacteristic
398C22#175011yescharacteristic
399C22C4×#9911yescharacteristic
400C22C4×#10011yescharacteristic
401C22#175911yescharacteristic
402C22#175911yescharacteristic
403C22C4×#10311yescharacteristic
404C22#176311yescharacteristic
405C4—22no—
406C4—22no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—82no—
416C4—82no—
417C4—161no—
418C4—161no—
419C2#771511yescharacteristic
420C2#771611yescharacteristic
421C2#772111yescharacteristic
422C2#772211yescharacteristic
423C2#772711yescharacteristic
424C2#772911yescharacteristic
425C2#773011yescharacteristic
426C2#773311yescharacteristic
427C2#815511yescharacteristic
428C2#817811yescharacteristic
429C2#818211yescharacteristic
430C2#821711yescharacteristic
431C2#822111yescharacteristic
432C2#822411yescharacteristic
433C2#823511yescharacteristic
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. #1272C4semidirect

Maximal subgroups

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

subgroupindex
#147032
#107952
C2×#12722
#70252