Finite Group Databasebeta

← #100155

Subgroups — #100155

Order 512.

822 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 128 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#20523C211yesmaximal, characteristic
2#14172C211yesmaximal, characteristic
3#8559C211yesmaximal, characteristic
4#8311C212yesmaximal, complemented
5#7024C212yesmaximal, complemented
6C22×C4×Q8C2211yescharacteristic
7C22×C22⋊C8—21no—
8C2×#393C411yescharacteristic
9C2×#385—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1797C2212yes—
12C8×C22⋊C4—24no—
13C4×C22⋊C8—21no—
14C4×C22⋊C8C2212yes—
15C2×Q8⋊C8—24no—
16C2×#323—21no—
17#1446C411yescharacteristic
18#1444C411yescharacteristic
19#1439C411yescharacteristic
20#1437—21no—
21#1436—21no—
22#1436—21no—
23#1434C2211yescharacteristic
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⋊C4—21no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4C2×C411yescharacteristic
37C22×C22⋊C4D411yescharacteristic
38C22×C42—21no—
39C22×C42D411yescharacteristic
40C22×C42C2311yescharacteristic
41C2×C4⋊C8—22no—
42C2×C22⋊C8—24no—
43C2×C22⋊C8—24no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C4×C8—22no—
48C2×C4×C8D414yes—
49#393—22no—
50#393C814yescomplemented
51#385—44no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4—22no—
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—42no—
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—22no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—42no—
86C23×C4—81no—
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—24no—
94C22×C8—28no—
95C22×C8—42no—
96C22×C8—82no—
97C22×C8—82no—
98C4×Q8—28no—
99C4×Q8—44no—
100C4×Q8—44no—
101C4×Q8—44no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—81no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—81no—
121C2×C22⋊C4C2×C812yes—
122C2×C22⋊C4M1612yes—
123C2×C22⋊C4SD1612yes—
124C2×C22⋊C4Q1612yes—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—42no—
128C2×C42—42no—
129C2×C42—42no—
130C2×C42—42no—
131C2×C42—81no—
132C2×C42C2×D412yes—
133C2×C42#4112yes—
134C4⋊C8—44no—
135C22⋊C8—28no—
136C22⋊C8—24no—
137C22⋊C8—24no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140C4×C8—44no—
141C4×C8—82no—
142#95—24no—
143#95—22no—
144#95—22no—
145#95—21no—
146#95—22no—
147#95—42no—
148#95—42no—
149#95—42no—
150#95—42no—
151#95—81no—
152#95—81no—
153#95C2×C812yescomplemented
154C24—21no—
155C24—21no—
156C24—21no—
157C24—21no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24C22⋊C812yes—
163C24#10312yes—
164C24#10412yes—
165C24C2×C22⋊C411yescharacteristic
166C2×Q8—24no—
167C2×Q8—24no—
168C2×Q8—24no—
169C2×Q8—44no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—48no—
174C2×Q8—42no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—24no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
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×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—84no—
231C22×C4—84no—
232C22×C4C22⋊C812yes—
233C22×C4#10312yes—
234C22×C4#10412yes—
235C22×C4C4×D412yes—
236C22×C4#12011yescharacteristic
237C22×C4#12112yes—
238C22×C4#12311yescharacteristic
239C22×C4C22×C811yescharacteristic
240C22×C4C2×M1611yescharacteristic
241C22×C4#13112yes—
242C22×C4C2×SD1611yescharacteristic
243C22×C4C2×Q1611yescharacteristic
244C22×C4#13512yes—
245C2×C8—28no—
246C2×C8—44no—
247C2×C8—44no—
248C2×C8—88no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C4⋊C4—24no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—84no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—48no—
262C22⋊C4—84no—
263C42—24no—
264C42—42no—
265C42—42no—
266C42—48no—
267C42—81no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23Q8⋊C814yescomplemented
295C23#38611yescharacteristic
296C23C2×C22⋊C811yescharacteristic
297C23#40811yescharacteristic
298C23C2×#10311yescharacteristic
299C23#41611yescharacteristic
300C23C2×#10411yescharacteristic
301C23C2×#10411yescharacteristic
302Q8—28no—
303Q8—44no—
304Q8—44no—
305Q8—44no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
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—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—84no—
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×C4C8×D412yes—
370C2×C4#43512yes—
371C2×C4B(4,𝔽2)12yes—
372C2×C4#46512yes—
373C2×C4#46712yes—
374C8—48no—
375C8—162no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—24no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22C2×Q8⋊C811yescharacteristic
400C22#147611yescharacteristic
401C22#149311yescharacteristic
402C22#149411yescharacteristic
403C22#186911yescharacteristic
404C22#188911yescharacteristic
405C22#189111yescharacteristic
406C22#189211yescharacteristic
407C22#189611yescharacteristic
408C22#190911yescharacteristic
409C22#191111yescharacteristic
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—22no—
415C4—41no—
416C4—42no—
417C4—44no—
418C4—42no—
419C4—42no—
420C4—82no—
421C4—82no—
422C4—161no—
423C2—21no—
424C2—21no—
425C2—21no—
426C2—21no—
427C2—44no—
428C2#896511yescharacteristic
429C2#907111yescharacteristic
430C2#909911yescharacteristic
431C2#946711yescharacteristic
432C2#947511yescharacteristic
433C2#948311yescharacteristic
434C2#949111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8311C2semidirect
5. #7024C2semidirect
50. #393C8semidirect
76. C22×Q8C2×C8semidirect
153. #95C2×C8semidirect
294. C23Q8⋊C8semidirect

Maximal subgroups

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

subgroupindex
#205232
#141722
#85592
#83112
#70242