Finite Group Databasebeta

← #99559

Subgroups — #99559

Order 512.

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

normal subgroups only

This group has 710 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×#397C211yesmaximal, characteristic
3#8706C212yesmaximal, complemented
4C4×#323C211yesmaximal, characteristic
5#7028C212yesmaximal, complemented
6C22×C22⋊C8—21no—
7C2×#397C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×C4×C22⋊C4—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1850—24no—
12#1797—21no—
13#1797C2212yes—
14C4×C22⋊C8C2212yes—
15C2×#323—21no—
16#1445C411yescharacteristic
17#1443—21no—
18#1442C411yescharacteristic
19#1440—21no—
20#1439C411yescharacteristic
21C4×#95—21no—
22C4×#95C2211yescharacteristic
23#1280—21no—
24#1279—24no—
25C24×C4D411yescharacteristic
26C23×C8—22no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4D411yescharacteristic
33C22×C42—21no—
34C22×C42—21no—
35C22×C42D411yescharacteristic
36C22×C42C2311yescharacteristic
37C2×C4⋊C8—22no—
38C2×C4⋊C8D414yes—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—42no—
43C2×C22⋊C8—42no—
44C2×C4×C8—22no—
45#397—22no—
46#397C814yescomplemented
47C4×C4⋊C4—22no—
48C4×C4⋊C4—22no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—42no—
52C4×C22⋊C4—42no—
53C2×#95—21no—
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—
68#323—24no—
69C25C22⋊C411yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4D812yes—
84C23×C4SD1612yes—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C22×C8—24no—
89C22×C8—44no—
90C22×C8—42no—
91C22×C8—82no—
92C22×C8—82no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4C2×C812yescomplemented
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4C2×C812yes—
111C2×C22⋊C4M1612yes—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42—41no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42C2×D412yes—
127C2×C42#4112yes—
128C4⋊C8—44no—
129C4⋊C8—82no—
130C22⋊C8—24no—
131C22⋊C8—42no—
132C22⋊C8—44no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135C4×C8—44no—
136#95—24no—
137#95—22no—
138#95—22no—
139#95—21no—
140#95—22no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95C2×C812yescomplemented
149C24—21no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—41no—
154C24—41no—
155C24—41no—
156C24—41no—
157C24C22⋊C812yes—
158C24#10212yes—
159C24#10412yes—
160C24C2×C22⋊C411yescharacteristic
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—42no—
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×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4C22⋊C812yes—
223C22×C4#10212yes—
224C22×C4C4×D412yes—
225C22×C4#12011yescharacteristic
226C22×C4#12112yes—
227C22×C4#12311yescharacteristic
228C22×C4C22×C811yescharacteristic
229C22×C4C2×M1611yescharacteristic
230C22×C4#13112yes—
231C22×C4C2×D811yescharacteristic
232C22×C4C2×SD1611yescharacteristic
233C22×C4AGL(1,ℤ/8)11yescharacteristic
234C22×C4#13711yescharacteristic
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—82no—
240C2×C8—82no—
241C2×C8—88no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—44no—
249C22⋊C4—48no—
250C22⋊C4—48no—
251C42—24no—
252C42—42no—
253C42—42no—
254C42—81no—
255C42—82no—
256C42—82no—
257C42—81no—
258C42—81no—
259C42—81no—
260C42—81no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23D4⋊C814yes—
288C23#38611yescharacteristic
289C23C2×C22⋊C811yescharacteristic
290C23#40811yescharacteristic
291C23C2×#10211yescharacteristic
292C23#41711yescharacteristic
293C23C2×#10411yescharacteristic
294C23#42111yescharacteristic
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
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×C4C8×D412yes—
359C2×C4#43512yes—
360C2×C4#44711yescharacteristic
361C2×C4#44811yescharacteristic
362C2×C4#45512yes—
363C2×C4#45812yes—
364C2×C4#48011yescharacteristic
365C2×C4#48111yescharacteristic
366C8—84no—
367C8—84no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—24no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22C2×D4⋊C811yescharacteristic
392C22#147711yescharacteristic
393C22#148011yescharacteristic
394C22#148311yescharacteristic
395C22#186911yescharacteristic
396C22#187111yescharacteristic
397C22#187411yescharacteristic
398C22#188411yescharacteristic
399C22#188511yescharacteristic
400C22#189011yescharacteristic
401C22#189111yescharacteristic
402C4—22no—
403C4—22no—
404C4—21no—
405C4—21no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—82no—
410C4—82no—
411C4—81no—
412C4—82no—
413C4—82no—
414C4#160712yes—
415C4#160812yes—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—44no—
421C2#893911yescharacteristic
422C2#895711yescharacteristic
423C2#896611yescharacteristic
424C2#901111yescharacteristic
425C2#902511yescharacteristic
426C2#902911yescharacteristic
427C2#903511yescharacteristic
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. #8706C2semidirect
5. #7028C2semidirect
46. #397C8semidirect
98. C2×C4⋊C4C2×C8semidirect
148. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
#219302
C4×#3972
#87062
C4×#3232
#70282