Finite Group Databasebeta

← #99547

Subgroups — #99547

Order 512.

822 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21930C211yesmaximal, characteristic
2C4×#393C211yesmaximal, characteristic
3#8311C212yesmaximal, complemented
4C4×#323C211yesmaximal, characteristic
5#7028C212yesmaximal, complemented
6C22×C4×Q8C2211yescharacteristic
7C22×C22⋊C8—21no—
8C2×#393C411yescharacteristic
9C2×C4×C22⋊C4—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1850—24no—
12#1797—21no—
13#1797C2212yes—
14C4×C22⋊C8C2212yes—
15C2×Q8⋊C8—24no—
16C2×#323—21no—
17#1446C411yescharacteristic
18#1442—21no—
19#1442C411yescharacteristic
20#1439—21no—
21#1439C411yescharacteristic
22C4×#95—21no—
23C4×#95C2211yescharacteristic
24#1280—21no—
25C23×Q8C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C23×C8—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—22no—
30C2×C4×Q8—44no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4D411yescharacteristic
37C22×C42—21no—
38C22×C42—21no—
39C22×C42D411yescharacteristic
40C22×C42C2311yescharacteristic
41C2×C4⋊C8—22no—
42C2×C4⋊C8D414yes—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—42no—
47C2×C22⋊C8—42no—
48C2×C4×C8—22no—
49#393—22no—
50#393C814yescomplemented
51C4×C22⋊C4—22no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4—44no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95C2×C411yescharacteristic
62C2×#95C2×C411yescharacteristic
63C2×#95C2×C411yescharacteristic
64C2×#95D411yescharacteristic
65#336—22no—
66#336—22no—
67#336—22no—
68Q8⋊C8—48no—
69#323—24no—
70C25C22⋊C411yescharacteristic
71C22×Q8—21no—
72C22×Q8—22no—
73C22×Q8—22no—
74C22×Q8—42no—
75C22×Q8—42no—
76C22×Q8C2×C812yescomplemented
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4SD1612yes—
91C23×C4Q1612yes—
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C22×C8—24no—
96C22×C8—44no—
97C22×C8—42no—
98C22×C8—82no—
99C22×C8—82no—
100C4×Q8—28no—
101C4×Q8—44no—
102C4×Q8—44no—
103C4×Q8—44no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4C2×C812yes—
121C2×C22⋊C4M1612yes—
122C2×C42—24no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—42no—
126C2×C42—42no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42—42no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42C2×D412yes—
135C2×C42#4112yes—
136C4⋊C8—44no—
137C4⋊C8—82no—
138C22⋊C8—24no—
139C22⋊C8—42no—
140C22⋊C8—44no—
141C22⋊C8—42no—
142C22⋊C8—42no—
143C4×C8—44no—
144#95—24no—
145#95—22no—
146#95—22no—
147#95—21no—
148#95—22no—
149#95—42no—
150#95—42no—
151#95—42no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95C2×C812yescomplemented
156C24—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—41no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24C22⋊C812yes—
165C24#10312yes—
166C24#10412yes—
167C24C2×C22⋊C411yescharacteristic
168C2×Q8—24no—
169C2×Q8—24no—
170C2×Q8—24no—
171C2×Q8—44no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—44no—
175C2×Q8—48no—
176C2×Q8—42no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
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×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4C22⋊C812yes—
239C22×C4#10312yes—
240C22×C4C4×D412yes—
241C22×C4#12011yescharacteristic
242C22×C4#12112yes—
243C22×C4#12311yescharacteristic
244C22×C4C22×C811yescharacteristic
245C22×C4C2×M1611yescharacteristic
246C22×C4#13112yes—
247C22×C4C2×SD1611yescharacteristic
248C22×C4C2×Q1611yescharacteristic
249C22×C4AGL(1,ℤ/8)11yescharacteristic
250C22×C4#13711yescharacteristic
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—82no—
256C2×C8—82no—
257C2×C8—88no—
258C4⋊C4—24no—
259C4⋊C4—42no—
260C4⋊C4—48no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—44no—
264C22⋊C4—48no—
265C22⋊C4—48no—
266C42—24no—
267C42—42no—
268C42—44no—
269C42—42no—
270C42—44no—
271C42—81no—
272C42—82no—
273C42—82no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23Q8⋊C814yescomplemented
301C23#38611yescharacteristic
302C23C2×C22⋊C811yescharacteristic
303C23#40811yescharacteristic
304C23C2×#10311yescharacteristic
305C23#41711yescharacteristic
306C23C2×#10411yescharacteristic
307C23#42111yescharacteristic
308Q8—28no—
309Q8—44no—
310Q8—44no—
311Q8—44no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
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×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4C8×D412yes—
376C2×C4#43512yes—
377C2×C4#45011yescharacteristic
378C2×C4#45111yescharacteristic
379C2×C4#45512yes—
380C2×C4B(4,𝔽2)12yes—
381C2×C4#48311yescharacteristic
382C2×C4#48411yescharacteristic
383C8—84no—
384C8—84no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—24no—
398C22—44no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22C2×Q8⋊C811yescharacteristic
409C22#147711yescharacteristic
410C22#148111yescharacteristic
411C22#148311yescharacteristic
412C22#186911yescharacteristic
413C22#187111yescharacteristic
414C22#187211yescharacteristic
415C22#188411yescharacteristic
416C22#188511yescharacteristic
417C22#188911yescharacteristic
418C22#189111yescharacteristic
419C4—22no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—44no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4—81no—
431C4#160112yes—
432C4#160212yes—
433C2—21no—
434C2—21no—
435C2—21no—
436C2—21no—
437C2—44no—
438C2#893711yescharacteristic
439C2#895711yescharacteristic
440C2#896511yescharacteristic
441C2#900911yescharacteristic
442C2#901311yescharacteristic
443C2#902911yescharacteristic
444C2#903311yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8311C2semidirect
5. #7028C2semidirect
50. #393C8semidirect
76. C22×Q8C2×C8semidirect
155. #95C2×C8semidirect
300. C23Q8⋊C8semidirect

Maximal subgroups

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

subgroupindex
#219302
C4×#3932
#83112
C4×#3232
#70282