Finite Group Databasebeta

← #200586

Subgroups — #200586

Order 512.

1077 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#35179C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic
4#17747C212yesmaximal
5#9404C212yesmaximal, complemented
6#9400C212yesmaximal, complemented
7#9396C212yesmaximal, complemented
8#9392C212yesmaximal, complemented
9#8094C212yesmaximal
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2984—24no—
13#2983—22no—
14#2983C2212yes—
15#2982C2212yes—
16Q8×M16—24no—
17#2920—22no—
18#2920C2212yes—
19#2920C2212yes—
20#2855—22no—
21#2627C414yes—
22#2582—22no—
23#2581C414yes—
24#2568C2211yescharacteristic
25#2568C2211yescharacteristic
26#2567C2211yescharacteristic
27#2567C2211yescharacteristic
28#2565—21no—
29#2565—21no—
30#2317—21no—
31#2310—21no—
32#2285C2211yescharacteristic
33#1821—21no—
34#1801C2212yes—
35#1801C2212yes—
36#1784—21no—
37C4×#101—21no—
38C4×#101—21no—
39#1559—24no—
40#1554—24no—
41#1549C2214yes—
42#1548C2214yes—
43#1303C2214yes—
44#1303C2214yes—
45C23×Q8D411yescharacteristic
46#533—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2×C412yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—21no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D412yes—
65C2×C4×Q8C2311yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×#117—21no—
68C2×#117—21no—
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4D411yescharacteristic
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73#446—216no—
74#446—28no—
75C8×Q8—28no—
76#432—24no—
77#432—24no—
78#432—22no—
79#432—22no—
80#432—22no—
81#432D414yes—
82#431—24no—
83#431—24no—
84#431—22no—
85#431—22no—
86#431—22no—
87#431C2314yes—
88#423—24no—
89#423D414yes—
90#423C2314yes—
91C2×#101—22no—
92C2×#101—22no—
93#407—22no—
94#407—22no—
95C4×M16—24no—
96C4×M16—22no—
97C4×M16—22no—
98#400C2×C414yes—
99#400C2×C414yes—
100#398—24no—
101#398—24no—
102#395C2×C414yes—
103#395C2×C414yes—
104#391—24no—
105#391—21no—
106#391—21no—
107#391—21no—
108#391—21no—
109#391—22no—
110#391—22no—
111#389—24no—
112#389—22no—
113#389—22no—
114#387—24no—
115#387—24no—
116#385—24no—
117#384—21no—
118#384—21no—
119#384—21no—
120#382—21no—
121#382—21no—
122#382C2×C412yes—
123#381—24no—
124#380—21no—
125#380C2×C412yes—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129#343—22no—
130#343—22no—
131#333—28no—
132#332—28no—
133C22×Q8—24no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—42no—
138C22×Q8C22⋊C412yes—
139C22×Q8C22×C412yes—
140C22×Q8C2×D411yescharacteristic
141C22×Q8C2×D411yescharacteristic
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—24no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153#125—42no—
154#125—42no—
155C22⋊Q8—24no—
156C22⋊Q8—24no—
157C22⋊Q8—41no—
158C22⋊Q8—41no—
159C22⋊Q8—41no—
160C22⋊Q8—41no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—24no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—42no—
176C4×Q8—42no—
177#117—44no—
178#117—42no—
179#117—42no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42C22⋊C412yes—
213C2×C42C22⋊C412yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42C2411yescharacteristic
224C4⋊C8—28no—
225C4⋊C8—28no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228C4⋊C8—24no—
229C4⋊C8C2×D418yes—
230#101—44no—
231#101—44no—
232C22⋊C8—24no—
233C22⋊C8—24no—
234C22⋊C8—42no—
235C22⋊C8—42no—
236#97—28no—
237#97—24no—
238#97—42no—
239C4×C8—28no—
240C4×C8—24no—
241C4×C8—42no—
242#95—22no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95C22×C412yes—
250#95C22×C412yes—
251C24C2×C22⋊C411yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—42no—
261C2×Q8—48no—
262C2×Q8—48no—
263C2×Q8—44no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—82no—
267C2×Q8#13714yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—81no—
282C22×C4—82no—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12112yes—
295C22×C4#12112yes—
296C22×C4#12311yescharacteristic
297C22×C4#12311yescharacteristic
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301M16—48no—
302M16—44no—
303M16—44no—
304C2×C8—28no—
305C2×C8—44no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—82no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—42no—
339C42—42no—
340C42—44no—
341C42C4×D412yes—
342C42C4×D412yes—
343C42#12012yes—
344C42#12112yes—
345C42#12112yes—
346C42#12312yes—
347C42C22×D412yes—
348C42C2×#4112yes—
349C23—21no—
350C23—21no—
351C23—41no—
352C23—41no—
353C23#38611yescharacteristic
354C23#38611yescharacteristic
355C23C22×C22⋊C411yescharacteristic
356Q8—28no—
357Q8—48no—
358Q8—44no—
359Q8—84no—
360Q8—82no—
361Q8—82no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—82no—
399C2×C4#38611yescharacteristic
400C2×C4#38611yescharacteristic
401C2×C4#38612yes—
402C2×C4#38612yes—
403C2×C4#41914yes—
404C2×C4C2×C4×D412yes—
405C2×C4C2×#12011yescharacteristic
406C2×C4C2×#12112yes—
407C2×C4C2×#12311yescharacteristic
408C2×C4C2×#13712yes—
409C2×C4#57512yes—
410C8—48no—
411C8—84no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—42no—
421C22—42no—
422C22C2×#38611yescharacteristic
423C22C2×#41911yescharacteristic
424C22#289611yescharacteristic
425C4—22no—
426C4—22no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—22no—
431C4—24no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—82no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4#302012yes—
440C4#305614yes—
441C2—21no—
442C2—21no—
443C2—41no—
444C2—41no—
445C2#2095411yescharacteristic
446C2#2097211yescharacteristic
447C2#2101711yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9404C2semidirect
6. #9400C2semidirect
7. #9396C2semidirect
8. #9392C2semidirect

Maximal subgroups

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

subgroupindex
#602482
#351792
#207272
#177472
#94042
#94002
#93962
#93922
#80942