Finite Group Databasebeta

← #99550

Subgroups — #99550

Order 512.

776 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 120 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#21902C211yesmaximal, characteristic
2C4×#393C211yesmaximal, characteristic
3#8311C212yesmaximal, complemented
4C4×#323C211yesmaximal, characteristic
5#7028C212yesmaximal, complemented
6C22×C4×Q8C2211yescharacteristic
7C2×#393C411yescharacteristic
8C2×C4×C22⋊C4—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1923—24no—
11#1797C2212yes—
12#1779—21no—
13C4×C22⋊C8—21no—
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
27C2×C4×Q8—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—44no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4D411yescharacteristic
36C22×C42—21no—
37C22×C42—21no—
38C22×C42D411yescharacteristic
39C22×C42C2311yescharacteristic
40C2×C4⋊C8D414yes—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C4×C8—22no—
46C2×C4×C8—22no—
47#393—22no—
48#393C814yescomplemented
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—44no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95D411yescharacteristic
63#336—22no—
64#336—22no—
65#336—22no—
66Q8⋊C8—48no—
67#323—24no—
68C25C22⋊C411yescharacteristic
69C22×Q8—21no—
70C22×Q8—22no—
71C22×Q8—22no—
72C22×Q8—42no—
73C22×Q8—42no—
74C22×Q8C2×C812yes—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C22×C8—24no—
92C22×C8—42no—
93C4×Q8—28no—
94C4×Q8—44no—
95C4×Q8—44no—
96C4×Q8—44no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4C2×C812yes—
114C2×C22⋊C4M1612yes—
115C2×C42—24no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—42no—
119C2×C42—42no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—42no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42C2×D412yes—
127C2×C42#4112yes—
128C4⋊C8—44no—
129C22⋊C8—24no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133C22⋊C8—82no—
134C22⋊C8—82no—
135C4×C8—44no—
136C4×C8—82no—
137#95—24no—
138#95—22no—
139#95—22no—
140#95—21no—
141#95—22no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95C2×C812yes—
149C24—21no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—41no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24C22⋊C812yes—
160C24C2×C22⋊C411yescharacteristic
161C2×Q8—24no—
162C2×Q8—24no—
163C2×Q8—24no—
164C2×Q8—44no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—44no—
168C2×Q8—48no—
169C2×Q8—42no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—48no—
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—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—84no—
227C22×C4C22⋊C812yes—
228C22×C4#10412yes—
229C22×C4C4×D412yes—
230C22×C4#12011yescharacteristic
231C22×C4#12112yes—
232C22×C4#12311yescharacteristic
233C22×C4C22×C811yescharacteristic
234C22×C4C2×M1611yescharacteristic
235C22×C4#13112yes—
236C22×C4#13512yes—
237C22×C4#13712yes—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—82no—
243C2×C8—82no—
244C4⋊C4—24no—
245C4⋊C4—42no—
246C4⋊C4—48no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—44no—
250C22⋊C4—48no—
251C22⋊C4—48no—
252C42—24no—
253C42—42no—
254C42—44no—
255C42—42no—
256C42—44no—
257C42—81no—
258C42—84no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23—81no—
286C23#38611yescharacteristic
287C23C2×C22⋊C811yescharacteristic
288C23#40811yescharacteristic
289C23#41611yescharacteristic
290C23#41911yescharacteristic
291C23C2×#10411yescharacteristic
292C23#42111yescharacteristic
293Q8—28no—
294Q8—44no—
295Q8—44no—
296Q8—44no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—84no—
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—84no—
355C2×C4—81no—
356C2×C4C8×D412yes—
357C2×C4#43512yes—
358C2×C4#45212yes—
359C2×C4#45512yes—
360C2×C4B(4,𝔽2)12yes—
361C2×C4#48512yes—
362C8—84no—
363C8—84no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—44no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22#147611yescharacteristic
388C22#147911yescharacteristic
389C22#148211yescharacteristic
390C22#148511yescharacteristic
391C22#186911yescharacteristic
392C22#187111yescharacteristic
393C22#187311yescharacteristic
394C22#188411yescharacteristic
395C22#188511yescharacteristic
396C22#188911yescharacteristic
397C22#189111yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—44no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—82no—
406C4—82no—
407C4—82no—
408C4—81no—
409C4#160614yes—
410C2—21no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2—82no—
415C2#893811yescharacteristic
416C2#895711yescharacteristic
417C2#896511yescharacteristic
418C2#901011yescharacteristic
419C2#901611yescharacteristic
420C2#902911yescharacteristic
421C2#903411yescharacteristic
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
48. #393C8semidirect

Maximal subgroups

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

subgroupindex
#219022
C4×#3932
#83112
C4×#3232
#70282