Finite Group Databasebeta

← #104792

Subgroups — #104792

Order 512.

526 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 88 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#15011C211yesmaximal, characteristic
2#9909C211yesmaximal, characteristic
3#9884C211yesmaximal, characteristic
4#9714C211yesmaximal, characteristic
5#7039C211yesmaximal, characteristic
6#7028C211yesmaximal, characteristic
7#7024C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C2×#395—21no—
10C2×#391—21no—
11#1797C2211yescharacteristic
12#1797C2211yescharacteristic
13C4×C22⋊C8C2211yescharacteristic
14C2×#328—21no—
15C2×#328—21no—
16#1446—21no—
17#1442—21no—
18#1441—21no—
19#1436—21no—
20C4⋊#95C2211yescharacteristic
21#1434C2211yescharacteristic
22C4×#95C2211yescharacteristic
23#1294—21no—
24#1294—21no—
25#1294—21no—
26#1294—21no—
27C23×Q8—21no—
28C2×C4⋊Q8Q811yescharacteristic, complemented
29C2×C22⋊Q8—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C42—21no—
34C22×C42C2311yescharacteristic
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C4×C8—21no—
41#395—42no—
42#395—42no—
43#391—21no—
44#391—42no—
45#391—42no—
46#389—21no—
47#389—21no—
48#384Q811yescharacteristic, complemented
49#380—21no—
50#380—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95D411yescharacteristic
61C2×#95D411yescharacteristic
62C2×#95D411yescharacteristic
63C2×#95D411yescharacteristic
64C2×#95D411yescharacteristic
65C2×#95D411yescharacteristic
66#336—21no—
67#336—21no—
68#336—21no—
69#328—42no—
70#328—42no—
71C22×Q8—21no—
72C22×Q8—42no—
73C22×Q8—42no—
74C22×Q8—42no—
75C22×Q8—42no—
76C22×Q8—42no—
77C22×Q8—81no—
78C22×Q8—81no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—41no—
91C4⋊Q8—81no—
92C22⋊Q8—41no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—81no—
111C2×C4⋊C4—81no—
112C2×C22⋊C4—41no—
113C2×C42—42no—
114C2×C42—41no—
115C2×C42—41no—
116C2×C42—81no—
117C2×C42—81no—
118C2×C42C2×Q811yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122C4⋊C8—81no—
123C4⋊C8—81no—
124C22⋊C8—22no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130C4×C8—81no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—41no—
136#95—41no—
137#95—42no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—41no—
143#95—41no—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—81no—
149#95—81no—
150#95D812yes—
151#95SD1612yes—
152#95SD1612yes—
153#95Q1612yes—
154C24#12011yescharacteristic
155C2×Q8—48no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—84no—
159C2×Q8—81no—
160C2×Q8—81no—
161C2×Q8—84no—
162C2×Q8—84no—
163C2×Q8—84no—
164C2×Q8—84no—
165C2×Q8—84no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
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×C4C22⋊Q811yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4#12411yescharacteristic
222C22×C4#12411yescharacteristic
223C22×C4#12411yescharacteristic
224C22×C4C2×D811yescharacteristic
225C22×C4C2×SD1611yescharacteristic
226C22×C4C2×SD1611yescharacteristic
227C22×C4C2×Q1611yescharacteristic
228C22×C4AGL(1,ℤ/8)11yescharacteristic
229C22×C4AGL(1,ℤ/8)11yescharacteristic
230C22×C4#13711yescharacteristic
231C22×C4#13711yescharacteristic
232C2×C8—42no—
233C2×C8—81no—
234C2×C8—81no—
235C2×C8—81no—
236C2×C8—81no—
237C2×C8—81no—
238C2×C8—81no—
239C2×C8—81no—
240C2×C8—81no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—84no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—84no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C22⋊C4—81no—
254C22⋊C4—81no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42—82no—
260C42—82no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23#39311yescharacteristic
266C23#44711yescharacteristic
267C23#44811yescharacteristic
268C23#45011yescharacteristic
269C23#45111yescharacteristic
270C23#45511yescharacteristic
271C23#45611yescharacteristic
272Q8—816no—
273Q8—161no—
274Q8—161no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
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#47411yescharacteristic
348C2×C4#47511yescharacteristic
349C2×C4#47611yescharacteristic
350C2×C4#47711yescharacteristic
351C2×C4#48611yescharacteristic
352C2×C4#48711yescharacteristic
353C2×C4#48911yescharacteristic
354C2×C4#48911yescharacteristic
355C8—82no—
356C8—161no—
357C8—161no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22#163911yescharacteristic
371C22#164511yescharacteristic
372C22#164911yescharacteristic
373C22#165111yescharacteristic
374C22#202311yescharacteristic
375C22#202411yescharacteristic
376C22#202811yescharacteristic
377C22#203111yescharacteristic
378C22#203111yescharacteristic
379C22#203111yescharacteristic
380C22#203211yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—44no—
390C4—41no—
391C4—41no—
392C4—82no—
393C4—82no—
394C4—82no—
395C4—82no—
396C4—161no—
397C4—161no—
398C4—161no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2—21no—
403C2#1035011yescharacteristic
404C2#1036311yescharacteristic
405C2#1037011yescharacteristic
406C2#1037411yescharacteristic
407C2#1037511yescharacteristic
408C2#1037911yescharacteristic
409C2#1038311yescharacteristic
subgroupG/H
center, socleC23#393
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C22×C42 ▷ 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
28. C2×C4⋊Q8Q8semidirect
48. #384Q8semidirect

Maximal subgroups

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

subgroupindex
#150112
#99092
#98842
#97142
#70392
#70282
#70242