Finite Group Databasebeta

← #99725

Subgroups — #99725

Order 512.

748 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1744C211yesmaximal, characteristic
2#14703C211yesmaximal, characteristic
3#10795C211yesmaximal, characteristic
4C2×#1272C212yesmaximal
5#7025C212yesmaximal
6C22×C4×C8—21no—
7C2×#391C411yescharacteristic
8C2×#391C411yescharacteristic
9C2×#391C2211yescharacteristic
10C2×#389C411yescharacteristic
11C2×C43C2211yescharacteristic
12#1744—24no—
13C2×#336—21no—
14C2×#336—21no—
15C2×#336—21no—
16C2×#336C2212yes—
17C2×#336C2212yes—
18#1436—21no—
19#1436—21no—
20C4⋊#95C411yescharacteristic
21C4⋊#95C2211yescharacteristic
22#1434—21no—
23#1434—21no—
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C23×C8—22no—
27C23×C8—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42C2311yescharacteristic
39C2×C4×C8—24no—
40C2×C4×C8—24no—
41C2×C4×C8—24no—
42#391—22no—
43#391—22no—
44#391—22no—
45#391—42no—
46#391C814yescomplemented
47#389—42no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C43—22no—
60C43—22no—
61#336—24no—
62#336—24no—
63#336—22no—
64#336—22no—
65#336—22no—
66#336—42no—
67#336D414yes—
68C22×Q8—21no—
69C22×Q8—22no—
70C22×Q8—22no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×Q8C2×C812yes—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—24no—
88C22×C8—24no—
89C22×C8—24no—
90C22×C8—24no—
91C22×C8—42no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8—42no—
95C22×C8—42no—
96C22×C8—42no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8—42no—
100C22×C8—42no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4C2×C812yes—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—21no—
118C2×C42—22no—
119C2×C42—21no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—21no—
126C2×C42—22no—
127C2×C42—21no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42C2×C812yes—
135C2×C42M1612yes—
136C2×C42SD1612yes—
137C2×C42Q1612yes—
138C2×C42C2×D412yes—
139C2×C42#4112yes—
140C4×C8—216no—
141#95—24no—
142#95—24no—
143#95—42no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—41no—
149#95—42no—
150#95—41no—
151#95—42no—
152#95—42no—
153C24C2×C22⋊C411yescharacteristic
154C2×Q8—24no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—82no—
161C2×Q8—82no—
162C2×Q8—82no—
163C2×Q8—82no—
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—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
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—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4C22⋊C812yes—
225C22×C4C22⋊C812yes—
226C22×C4#10012yes—
227C22×C4#10312yes—
228C22×C4#10312yes—
229C22×C4#10412yes—
230C22×C4#10412yes—
231C22×C4#10412yes—
232C22×C4C4×D412yes—
233C22×C4#12011yescharacteristic
234C22×C4#12112yes—
235C22×C4#12311yescharacteristic
236C22×C4C22×C811yescharacteristic
237C22×C4C2×M1611yescharacteristic
238C22×C4#13112yes—
239C22×C4C2×SD1611yescharacteristic
240C22×C4C2×Q1611yescharacteristic
241C22×C4#13512yes—
242C22×C4#13512yes—
243C22×C4#13712yes—
244C2×C8—28no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—82no—
255C2×C8—82no—
256C2×C8—82no—
257C2×C8—82no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—84no—
263C42—24no—
264C42—24no—
265C42—24no—
266C42—24no—
267C42—24no—
268C42—24no—
269C42—42no—
270C42—42no—
271C42—44no—
272C42—44no—
273C42—42no—
274C42—42no—
275C23#38611yescharacteristic
276C23C2×C22⋊C811yescharacteristic
277C23#40811yescharacteristic
278C23#41011yescharacteristic
279C23#41011yescharacteristic
280C23C2×#10011yescharacteristic
281C23#41311yescharacteristic
282C23C2×#10311yescharacteristic
283C23#41611yescharacteristic
284C23#41611yescharacteristic
285C23#41911yescharacteristic
286C23C2×#10411yescharacteristic
287C23C2×#10411yescharacteristic
288C23C2×#10411yescharacteristic
289C23#42111yescharacteristic
290Q8—84no—
291Q8—84no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
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—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
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—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4Q8⋊C814yes—
371C2×C4#32914yes—
372C2×C4C8×D412yes—
373C2×C4#43512yes—
374C2×C4#45212yes—
375C2×C4#46512yes—
376C2×C4#46712yes—
377C2×C4#48512yes—
378C8—48no—
379C8—84no—
380C22#145811yescharacteristic
381C22#146011yescharacteristic
382C22#146911yescharacteristic
383C22#147311yescharacteristic
384C22C2×Q8⋊C811yescharacteristic
385C22#147611yescharacteristic
386C22#147611yescharacteristic
387C22#147911yescharacteristic
388C22#148211yescharacteristic
389C22#148511yescharacteristic
390C22#149311yescharacteristic
391C22#149411yescharacteristic
392C22#150011yescharacteristic
393C22#150011yescharacteristic
394C22#150011yescharacteristic
395C22#150311yescharacteristic
396C22#150611yescharacteristic
397C22#150611yescharacteristic
398C22#151811yescharacteristic
399C22#152111yescharacteristic
400C22C2×#32911yescharacteristic
401C22#152411yescharacteristic
402C22#153011yescharacteristic
403C22#153211yescharacteristic
404C22#154211yescharacteristic
405C22#154311yescharacteristic
406C22#154811yescharacteristic
407C22#155411yescharacteristic
408C22#186911yescharacteristic
409C22#187111yescharacteristic
410C22#187311yescharacteristic
411C22#189211yescharacteristic
412C22#189611yescharacteristic
413C22#189911yescharacteristic
414C22#190011yescharacteristic
415C4—24no—
416C4—22no—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—24no—
421C4—22no—
422C4—22no—
423C4—42no—
424C4—42no—
425C4—82no—
426C4—82no—
427C4—161no—
428C4—161no—
429C2#756211yescharacteristic
430C2#756711yescharacteristic
431C2#758111yescharacteristic
432C2#759011yescharacteristic
433C2#765611yescharacteristic
434C2#766911yescharacteristic
435C2#768211yescharacteristic
436C2#769311yescharacteristic
437C2#893811yescharacteristic
438C2#907111yescharacteristic
439C2#908011yescharacteristic
440C2#911211yescharacteristic
441C2#911911yescharacteristic
442C2#912811yescharacteristic
443C2#914511yescharacteristic
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
46. #391C8semidirect

Maximal subgroups

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

subgroupindex
C2×#17442
#147032
#107952
C2×#12722
#70252