Finite Group Databasebeta

← #101347

Subgroups — #101347

Order 512.

444 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 90 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#27683C211yesmaximal, characteristic
2#22329C211yesmaximal, characteristic
3#8717C211yesmaximal, characteristic, complemented
4#8710C211yesmaximal, characteristic, complemented
5#8575C211yesmaximal, characteristic
6#7044C211yesmaximal, characteristic, complemented
7#7024C211yesmaximal, characteristic, complemented
8C4×#121C2211yescharacteristic
9#2285C2211yescharacteristic
10#2100C411yescharacteristic, complemented
11#2099C411yescharacteristic, complemented
12#2094C411yescharacteristic, complemented
13#2093C411yescharacteristic, complemented
14#1954—21no—
15#1949—21no—
16#1923—21no—
17#1916—21no—
18#1875—21no—
19#1869—21no—
20#1869—21no—
21#1854—21no—
22#1817C2211yescharacteristic
23#1804—21no—
24#1797C2211yescharacteristic
25#1797C2211yescharacteristic
26C8×C22⋊C4—21no—
27C4×C22⋊C8C2211yescharacteristic
28C2×#327—21no—
29#1434C2211yescharacteristic
30#1294—21no—
31#1294—21no—
32#1294—21no—
33#1279—21no—
34#1279—21no—
35C23×C8—21no—
36C2×#121D411yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4D411yescharacteristic
39C2×#117—21no—
40C22×C42C2311yescharacteristic
41C2×#107—21no—
42C2×#106—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×#102—21no—
47C2×#102—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8D411yescharacteristic
55C2×C22⋊C8D411yescharacteristic
56C2×C4×C8—21no—
57C2×C4×C8D411yescharacteristic
58C2×C4×C8D411yescharacteristic
59#390C2×C411yescharacteristic
60#389C2×C411yescharacteristic
61#388—21no—
62#386—21no—
63#384C2×C411yescharacteristic
64#384C2×C411yescharacteristic
65#382—21no—
66#380—21no—
67#380—21no—
68#380D411yescharacteristic
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C4×C4⋊C4D411yescharacteristic
72C4×C22⋊C4—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76#340—21no—
77#340—21no—
78#340—21no—
79#340—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336C2×C411yescharacteristic
87#336C2×C411yescharacteristic
88#327—42no—
89C22×D4—21no—
90C22×D4C22⋊C411yescharacteristic
91C23×C4—21no—
92C23×C4—41no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4#4111yescharacteristic
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—42no—
103C22×C8—41no—
104C22×C8—81no—
105C22×C8—81no—
106#121—21no—
107#121—21no—
108#121SD1612yes—
109#121Q1612yes—
110C4×D4—41no—
111C4×D4—41no—
112C4×D4—81no—
113C4×D4—81no—
114#117—81no—
115#117—81no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—41no—
132C2×C42—21no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42—41no—
136C2×C42C22×C411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42#4111yescharacteristic
140#107—81no—
141#106—81no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—81no—
147#102—81no—
148#102—81no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C22⋊C8—81no—
158C22⋊C8—81no—
159C4×C8—41no—
160C4×C8—41no—
161C4×C8—81no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—22no—
166#95—41no—
167#95—41no—
168#95—41no—
169#95—41no—
170#95—41no—
171#95—41no—
172#95—41no—
173C24—21no—
174C24#12111yescharacteristic
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—21no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—81no—
181C2×D4—81no—
182C2×D4#10312yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
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—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4#12011yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12311yescharacteristic
217C22×C4C2×SD1611yescharacteristic
218C22×C4C2×Q1611yescharacteristic
219C22×C4#13511yescharacteristic
220C22×C4#13511yescharacteristic
221C2×C8—22no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—84no—
240C2×C8—81no—
241C2×C8—81no—
242C2×C8—81no—
243C4⋊C4—21no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—81no—
254C4⋊C4—81no—
255C4⋊C4—81no—
256C4⋊C4#10312yes—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—81no—
262C22⋊C4—81no—
263C22⋊C4—81no—
264C22⋊C4—82no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—41no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—82no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—22no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23#38611yescharacteristic
288C23#46511yescharacteristic
289C23#46711yescharacteristic
290D4—42no—
291D4—81no—
292D4—82no—
293D4—82no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
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—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—161no—
339C2×C4—161no—
340C2×C4—161no—
341C2×C4—161no—
342C2×C4C2×#10311yescharacteristic
343C2×C4#41611yescharacteristic
344C2×C4#44311yescharacteristic
345C2×C4#44411yescharacteristic
346C2×C4#45811yescharacteristic
347C2×C4#45811yescharacteristic
348C2×C4#46511yescharacteristic
349C2×C4#46711yescharacteristic
350C8—42no—
351C8—81no—
352C8—81no—
353C8—81no—
354C8—81no—
355C8—81no—
356C8—161no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22#187611yescharacteristic
375C22#188111yescharacteristic
376C22#189011yescharacteristic
377C22#189211yescharacteristic
378C22#190411yescharacteristic
379C22#190611yescharacteristic
380C22#191011yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—42no—
392C4—81no—
393C4—81no—
394C4—162no—
395C4—161no—
396C4—161no—
397C2—41no—
398C2—41no—
399C2—82no—
400C2#903811yescharacteristic
401C2#920711yescharacteristic
402C2#928111yescharacteristic
403C2#930311yescharacteristic
404C2#937011yescharacteristic
405C2#946911yescharacteristic
406C2#952511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121D4
LCSG ▷ C2×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
3. #8717C2semidirect
4. #8710C2semidirect
6. #7044C2semidirect
7. #7024C2semidirect
10. #2100C4semidirect
11. #2099C4semidirect
12. #2094C4semidirect
13. #2093C4semidirect

Maximal subgroups

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

subgroupindex
#276832
#223292
#87172
#87102
#85752
#70442
#70242