Finite Group Databasebeta

← #382003

Subgroups — #382003

Order 512.

924 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2087C211yesmaximal, characteristic
2C2×#1943C211yesmaximal, characteristic
3C2×#1768C211yesmaximal, characteristic
4#14266C212yesmaximal
5#11379C211yesmaximal, characteristic
6#11316C212yesmaximal
7#11257C211yesmaximal, characteristic
8#11231C211yesmaximal, characteristic
9#11227C212yesmaximal
10#10880C212yesmaximal
11#10821C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#376C2211yescharacteristic
21#2087C418yescomplemented
22#1943—24no—
23#1768—24no—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1445—22no—
35#1443—22no—
36#1442—22no—
37#1441—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C2×#107—22no—
56C2×#107—22no—
57C2×#107C2×C414yes—
58C2×#106—22no—
59C2×#106—42no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8C2×C414yes—
66C2×#97—42no—
67C2×#97D414yes—
68C2×C4×C8—22no—
69C2×C4×C8—42no—
70#389—24no—
71#389—24no—
72#389—24no—
73#389C2×C418yes—
74#376—22no—
75#376—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95D412yes—
87C2×#95D412yes—
88#340—24no—
89#340—24no—
90#340—22no—
91#340—42no—
92#340—42no—
93#340—42no—
94#340C2×C414yes—
95#340C2×C418yes—
96#336—22no—
97#336—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D412yes—
106C23×C4C2411yescharacteristic
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8C22×C412yes—
122C22×C8C22×C412yes—
123C22×C8C2×D412yes—
124C22×C8C2×D412yes—
125C22×C8#4112yes—
126C22×C8#4112yes—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4C22×C414yes—
156C2×C4⋊C4C22×C414yes—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42C22×C412yes—
168C2×C42C2×D412yes—
169C2×C42#4112yes—
170C2×C42#4112yes—
171#107—28no—
172#106—44no—
173C4⋊C8—28no—
174C4⋊C8—28no—
175#97—44no—
176C4×C8—44no—
177#95—28no—
178#95—24no—
179#95—24no—
180#95—44no—
181#95—82no—
182#95—82no—
183#95—82no—
184#95—82no—
185#95—82no—
186#95—82no—
187#95—82no—
188#95—82no—
189C24C22×D411yescharacteristic
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C4×D412yes—
240C22×C4C4×D412yes—
241C22×C4#12114yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C2×#4111yescharacteristic
245C22×C4C2×#4111yescharacteristic
246C22×C4C2×#4111yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4H(5,𝔽2)12yes—
249C22×C42−1+412yes—
250C22×C42−1+412yes—
251C22×C42−1+412yes—
252C2×C8—24no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—24no—
256C2×C8—22no—
257C2×C8—24no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—24no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8C4×D414yes—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—88no—
289C4⋊C4—88no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—24no—
297C42—41no—
298C42—41no—
299C42—42no—
300C23C2×C4×D411yescharacteristic
301C23C2×#12111yescharacteristic
302C23#53211yescharacteristic
303C23#53612yes—
304C23#53712yes—
305C23#57612yes—
306C23#57712yes—
307C23#57712yes—
308C23#57812yes—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4C2×C4×D411yescharacteristic
362C2×C4C4×#4111yescharacteristic
363C2×C4#51912yes—
364C2×C4#52012yes—
365C2×C4#52511yescharacteristic
366C2×C4#52911yescharacteristic
367C2×C4#53411yescharacteristic
368C2×C4#53611yescharacteristic
369C2×C4#53611yescharacteristic
370C2×C4#53611yescharacteristic
371C2×C4#53912yes—
372C2×C4#53912yes—
373C2×C4#53912yes—
374C2×C4#54112yes—
375C2×C4#54112yes—
376C2×C4#54112yes—
377C2×C4#54112yes—
378C2×C4#54212yes—
379C8—24no—
380C8—42no—
381C8—42no—
382C8—42no—
383C22C4×#12111yescharacteristic
384C22#232312yes—
385C22#261811yescharacteristic
386C22#262111yescharacteristic
387C22#266412yes—
388C22#294612yes—
389C22#295011yescharacteristic
390C22#295211yescharacteristic
391C22#306212yes—
392C22#306312yes—
393C22#317311yescharacteristic
394C22#317611yescharacteristic
395C22#317812yes—
396C22#324112yes—
397C22#324212yes—
398C22#324312yes—
399C22#324412yes—
400C22#325712yes—
401C22#325812yes—
402C22#326012yes—
403C22#326212yes—
404C4—42no—
405C4—42no—
406C4—44no—
407C4—84no—
408C4—84no—
409C4—82no—
410C4—162no—
411C4—162no—
412C4#232211yescharacteristic
413C4#232711yescharacteristic
414C4#233612yes—
415C4#263411yescharacteristic
416C4#263811yescharacteristic
417C4#265512yes—
418C2#1431411yescharacteristic
419C2#2153011yescharacteristic
420C2#2154111yescharacteristic
421C2#2212212yes—
422C2#2212512yes—
423C2#2978712yes—
424C2#2988512yes—
425C2#3038912yes—
426C2#3039412yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #2087C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20872
C2×#19432
C2×#17682
#142662
#113792
#113162
#112572
#112312
#112272
#108802
#108212