Finite Group Databasebeta

← #102406

Subgroups — #102406

Order 512.

680 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14709C211yesmaximal, characteristic
2#10832C211yesmaximal, characteristic
3#10795C211yesmaximal, characteristic
4#7025C212yesmaximal
5#7025C212yesmaximal
6C22×#125C411yescharacteristic
7C22×C4⋊C8—21no—
8C2×#382C411yescharacteristic
9C2×C43C2211yescharacteristic
10C2×#336—21no—
11C2×#336—21no—
12C2×#336—21no—
13C2×#336C2212yes—
14C2×#336C2212yes—
15#1436—21no—
16#1436—21no—
17#1436—21no—
18#1436—21no—
19C4⋊#95C411yescharacteristic
20C4⋊#95C411yescharacteristic
21C4⋊#95C2211yescharacteristic
22C4⋊#95C2211yescharacteristic
23C23×C8—22no—
24C23×C8—22no—
25C2×#125—22no—
26C2×#125—42no—
27C2×#125—42no—
28C2×#125C814yescomplemented
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C42C2×C411yescharacteristic
36C22×C42C2×C411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42C2311yescharacteristic
42C2×C4⋊C8—24no—
43C2×C4⋊C8—42no—
44C2×C4⋊C8—42no—
45#382—42no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C43—42no—
57#336—22no—
58#336—24no—
59#336—42no—
60#336—42no—
61#336—42no—
62#336Q814yes—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C22×C8—24no—
75C22×C8—24no—
76C22×C8—42no—
77C22×C8—42no—
78C22×C8—42no—
79C22×C8—42no—
80C22×C8—42no—
81C22×C8—42no—
82C22×C8—42no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—42no—
88#125—48no—
89#125—48no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4C2×C812yes—
109C2×C4⋊C4C2×C812yes—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42C2×C812yes—
133C2×C42M1612yes—
134C2×C42C2×Q812yes—
135C2×C42#4112yes—
136C4⋊C8—48no—
137#95—42no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—41no—
149#95—41no—
150C24C2×C22⋊C411yescharacteristic
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
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—22no—
186C22×C4—22no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
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×C4C22⋊C812yes—
214C22×C4C22⋊C812yes—
215C22×C4#9912yes—
216C22×C4#10012yes—
217C22×C4#10112yes—
218C22×C4#10412yes—
219C22×C4C4×Q812yes—
220C22×C4C22⋊Q812yes—
221C22×C4#12411yescharacteristic
222C22×C4C4⋊Q811yescharacteristic
223C22×C4C22×C811yescharacteristic
224C22×C4C2×M1611yescharacteristic
225C22×C4#13112yes—
226C22×C4#13512yes—
227C22×C4AGL(1,ℤ/8)11yescharacteristic
228C22×C4AGL(1,ℤ/8)11yescharacteristic
229C22×C4AGL(1,ℤ/8)11yescharacteristic
230C22×C4#13711yescharacteristic
231C22×C4#13711yescharacteristic
232C22×C4#13711yescharacteristic
233C2×C8—44no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—82no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—82no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—48no—
252C4⋊C4—48no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C42—24no—
256C42—24no—
257C42—24no—
258C42—24no—
259C42—42no—
260C42—42no—
261C42—44no—
262C42—44no—
263C42—42no—
264C42—42no—
265C42—42no—
266C42—42no—
267C23#39111yescharacteristic
268C23C2×C22⋊C811yescharacteristic
269C23#40811yescharacteristic
270C23C2×#9911yescharacteristic
271C23#41011yescharacteristic
272C23C2×#10011yescharacteristic
273C23C2×#10111yescharacteristic
274C23#41611yescharacteristic
275C23#41711yescharacteristic
276C23#41711yescharacteristic
277C23#41711yescharacteristic
278C23C2×#10411yescharacteristic
279C23#42111yescharacteristic
280C23#42111yescharacteristic
281C23#42111yescharacteristic
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
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—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
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—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4#32314yes—
366C2×C4C8×Q812yes—
367C2×C4#44612yes—
368C2×C4#47812yes—
369C2×C4#47912yes—
370C2×C4#49011yescharacteristic
371C2×C4#49111yescharacteristic
372C2×C4#50111yescharacteristic
373C2×C4#50111yescharacteristic
374C8—84no—
375C8—84no—
376C22C2×#32311yescharacteristic
377C22#146111yescharacteristic
378C22#146911yescharacteristic
379C22#147211yescharacteristic
380C22#147611yescharacteristic
381C22#147711yescharacteristic
382C22#147711yescharacteristic
383C22#147711yescharacteristic
384C22#148211yescharacteristic
385C22#148311yescharacteristic
386C22#149511yescharacteristic
387C22#149511yescharacteristic
388C22#150411yescharacteristic
389C22#150911yescharacteristic
390C22#151211yescharacteristic
391C22#151211yescharacteristic
392C22#151311yescharacteristic
393C22#151311yescharacteristic
394C22#151611yescharacteristic
395C22#152111yescharacteristic
396C22#152811yescharacteristic
397C22#153011yescharacteristic
398C22#153411yescharacteristic
399C22#153611yescharacteristic
400C22#155111yescharacteristic
401C22#155211yescharacteristic
402C22#156011yescharacteristic
403C22#156111yescharacteristic
404C22#198411yescharacteristic
405C22#198611yescharacteristic
406C22#198911yescharacteristic
407C22#199211yescharacteristic
408C22#199511yescharacteristic
409C22#199611yescharacteristic
410C22#199711yescharacteristic
411C4—22no—
412C4—22no—
413C4—24no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—82no—
424C4—82no—
425C4—161no—
426C4—161no—
427C2#757811yescharacteristic
428C2#758511yescharacteristic
429C2#759711yescharacteristic
430C2#760211yescharacteristic
431C2#769611yescharacteristic
432C2#771111yescharacteristic
433C2#771611yescharacteristic
434C2#773111yescharacteristic
435C2#986611yescharacteristic
436C2#987211yescharacteristic
437C2#987811yescharacteristic
438C2#988711yescharacteristic
439C2#989411yescharacteristic
440C2#990611yescharacteristic
441C2#991311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C22×C8
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
28. C2×#125C8semidirect

Maximal subgroups

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

subgroupindex
#147092
#108322
#107952
#70252