Finite Group Databasebeta

← #101235

Subgroups — #101235

Order 512.

704 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 98 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#20627C211yesmaximal, characteristic
2#14907C211yesmaximal, characteristic
3#8300C211yesmaximal, characteristic
4#7610C212yesmaximal, complemented
5#7037C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7C2×#423—21no—
8C2×#407—21no—
9C2×#398C411yescharacteristic
10C2×#397C411yescharacteristic
11C2×#389C2211yescharacteristic
12C2×#385—21no—
13#1817C2212yes—
14#1798—24no—
15#1728C2212yes—
16C2×#324—21no—
17C2×#323—21no—
18#1441—21no—
19#1441C411yescharacteristic
20#1440—21no—
21#1439C411yescharacteristic
22#1436—21no—
23#1434C2211yescharacteristic
24#1271—24no—
25C24×C4D411yescharacteristic
26C22×M16—22no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—42no—
30C2×#117—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4D411yescharacteristic
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C22⋊C4C2×C411yescharacteristic
40C22×C22⋊C4D411yescharacteristic
41C22×C42D411yescharacteristic
42C22×C42C2311yescharacteristic
43#423—42no—
44#423—42no—
45C2×C4⋊C8—24no—
46#407—42no—
47#407—42no—
48C2×C22⋊C8—24no—
49C2×C22⋊C8—22no—
50#398—42no—
51#397—42no—
52#389—22no—
53#389—22no—
54#385—42no—
55#385—42no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95C2×C411yescharacteristic
65C2×#95C2×C411yescharacteristic
66#336—22no—
67#336—22no—
68#336—22no—
69#336D414yes—
70#324—42no—
71#323—42no—
72C25C22⋊C411yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—41no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—41no—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4D812yes—
89C23×C4SD1612yes—
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C2×M16—42no—
94C2×M16—42no—
95C2×M16—42no—
96C2×M16—42no—
97C2×M16—42no—
98C2×M16—42no—
99C22×C8—24no—
100C22×C8—42no—
101#117—48no—
102#117—48no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42C2×D412yes—
142C2×C42#4112yes—
143C4⋊C8—48no—
144C22⋊C8—48no—
145C22⋊C8—42no—
146C22⋊C8—42no—
147#95—41no—
148#95—41no—
149#95—42no—
150#95—42no—
151#95—42no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95—41no—
157#95—41no—
158C24—21no—
159C24—21no—
160C24—21no—
161C24—21no—
162C24—21no—
163C24—21no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24#10212yes—
169C24C2×C22⋊C411yescharacteristic
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
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—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
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—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4#9912yes—
241C22×C4#9912yes—
242C22×C4#10212yes—
243C22×C4C4×D412yes—
244C22×C4#12011yescharacteristic
245C22×C4#12112yes—
246C22×C4#12311yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×SD1611yescharacteristic
249C22×C4AGL(1,ℤ/8)11yescharacteristic
250C22×C4#13711yescharacteristic
251M16—82no—
252M16—82no—
253M16—82no—
254M16—82no—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—82no—
259C2×C8—82no—
260C2×C8—82no—
261C4⋊C4—24no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—42no—
265C4⋊C4—82no—
266C4⋊C4—84no—
267C4⋊C4—81no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C4⋊C4—81no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—48no—
274C22⋊C4—84no—
275C42—24no—
276C42—24no—
277C42—42no—
278C42—42no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23#38611yescharacteristic
307C23C2×#9911yescharacteristic
308C23C2×#9911yescharacteristic
309C23C2×#10211yescharacteristic
310C23#41711yescharacteristic
311C23#42111yescharacteristic
312C23#42111yescharacteristic
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
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×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4#44711yescharacteristic
385C2×C4#44811yescharacteristic
386C2×C4#48011yescharacteristic
387C2×C4#48111yescharacteristic
388C8—84no—
389C8—162no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—44no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22#151311yescharacteristic
414C22#151311yescharacteristic
415C22#151911yescharacteristic
416C22#151911yescharacteristic
417C22#187411yescharacteristic
418C22#188411yescharacteristic
419C22#189111yescharacteristic
420C22#189711yescharacteristic
421C22#189811yescharacteristic
422C22#190911yescharacteristic
423C22#191511yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—82no—
434C4—82no—
435C4—161no—
436C4—161no—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—82no—
442C2#903511yescharacteristic
443C2#915511yescharacteristic
444C2#919511yescharacteristic
445C2#939011yescharacteristic
446C2#944311yescharacteristic
447C2#949311yescharacteristic
448C2#951311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ 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. #7610C2semidirect
5. #7037C2semidirect

Maximal subgroups

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

subgroupindex
#206272
#149072
#83002
#76102
#70372