Finite Group Databasebeta

← #100007

Subgroups — #100007

Order 512.

439 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 84 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#23334C211yesmaximal, characteristic
2#22082C211yesmaximal, characteristic
3#9650C211yesmaximal, characteristic, complemented
4#7593C211yesmaximal, characteristic, complemented
5#7582C211yesmaximal, characteristic
6#7033C211yesmaximal, characteristic, complemented
7#7021C211yesmaximal, characteristic, complemented
8C2×#481C411yescharacteristic
9C4×#121C2211yescharacteristic
10C2×#389C2211yescharacteristic
11#1955—21no—
12#1944—21no—
13#1936—21no—
14#1924—21no—
15#1919—21no—
16#1917—21no—
17#1874C411yescharacteristic
18#1870—21no—
19#1870—21no—
20#1862—21no—
21#1852C411yescharacteristic
22#1851—21no—
23#1824C411yescharacteristic
24#1817C2211yescharacteristic
25#1817C2211yescharacteristic
26#1797C2211yescharacteristic
27#1749C2211yescharacteristic
28C4×C22⋊C8C2211yescharacteristic
29C2×#324—21no—
30C2×#324—21no—
31#1271—21no—
32#1271—21no—
33#1271—21no—
34#1270—21no—
35C22×M16—21no—
36C2×#121C2×C411yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4D411yescharacteristic
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C42C2311yescharacteristic
43#481—22no—
44#481—22no—
45C2×#107—21no—
46C2×#106—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×#102—21no—
50C2×#102—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8C2×C411yescharacteristic
58C2×C22⋊C8C2×C411yescharacteristic
59C2×C22⋊C8D411yescharacteristic
60C2×C22⋊C8D411yescharacteristic
61C2×#97—21no—
62C2×#97—21no—
63C2×C4×C8—21no—
64#390D411yescharacteristic
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389D411yescharacteristic
70#388—21no—
71#386—21no—
72#380C2×C411yescharacteristic
73C4×C4⋊C4D411yescharacteristic
74C4×C22⋊C4—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#336—21no—
87#336—21no—
88#336D411yescharacteristic
89#336D411yescharacteristic
90#324—42no—
91#324—42no—
92C22×D4—21no—
93C22×D4C22⋊C411yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—81no—
101C2×M16—81no—
102C2×M16—81no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—41no—
110C22×C8—41no—
111#121—21no—
112#121—21no—
113#121—21no—
114#121—21no—
115C4×D4—81no—
116C4×D4—81no—
117C4×D4—81no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—81no—
133C2×C4⋊C4C22⋊C411yescharacteristic
134C2×C4⋊C4C22⋊C411yescharacteristic
135C2×C4⋊C4C22⋊C411yescharacteristic
136C2×C4⋊C4M1612yescomplemented
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C42—21no—
143C2×C42—81no—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42#4111yescharacteristic
148#107—81no—
149#106—24no—
150C4⋊C8—81no—
151C4⋊C8—81no—
152#102—24no—
153#102—81no—
154C22⋊C8—21no—
155C22⋊C8—22no—
156C22⋊C8—21no—
157C22⋊C8—21no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C22⋊C8—81no—
162C22⋊C8—81no—
163C22⋊C8—81no—
164C22⋊C8—81no—
165C22⋊C8—81no—
166C22⋊C8M1612yescomplemented
167#97—81no—
168#97—81no—
169C4×C8—81no—
170#95—21no—
171#95—21no—
172#95—21no—
173#95—41no—
174#95—41no—
175C24—21no—
176C24C4×D411yescharacteristic
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—81no—
184C2×D4—81no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
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—81no—
214C22×C4—81no—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4#12011yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12311yescharacteristic
221C22×C4C2×M1611yescharacteristic
222C22×C4C2×M1611yescharacteristic
223C22×C4#13111yescharacteristic
224C22×C4#13111yescharacteristic
225M16—84no—
226C2×C8—22no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—42no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—81no—
246C2×C8—81no—
247C2×C8—81no—
248C2×C8—81no—
249C2×C8—81no—
250C4⋊C4—22no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C22⋊C4—81no—
272C42—41no—
273C42—41no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42—41no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23#38611yescharacteristic
292C23#43511yescharacteristic
293C23#43611yescharacteristic
294D4—42no—
295D4—81no—
296D4—82no—
297D4—82no—
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—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—161no—
344C2×C4—161no—
345C2×C4#40711yescharacteristic
346C2×C4#40811yescharacteristic
347C2×C4#43511yescharacteristic
348C2×C4#43611yescharacteristic
349C2×C4#45311yescharacteristic
350C2×C4#45411yescharacteristic
351C8—42no—
352C8—81no—
353C8—81no—
354C8—81no—
355C8—81no—
356C8—81no—
357C8—82no—
358C8—161no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22#187011yescharacteristic
377C22#188111yescharacteristic
378C22#188311yescharacteristic
379C22#190211yescharacteristic
380C22#190411yescharacteristic
381C22#191311yescharacteristic
382C22#191511yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—82no—
396C4—161no—
397C2—41no—
398C2—41no—
399C2—161no—
400C2#895611yescharacteristic
401C2#909111yescharacteristic
402C2#911011yescharacteristic
403C2#933111yescharacteristic
404C2#933511yescharacteristic
405C2#933911yescharacteristic
406C2#934311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121C2×C4
LCSG ▷ C23×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
3. #9650C2semidirect
4. #7593C2semidirect
6. #7033C2semidirect
7. #7021C2semidirect
136. C2×C4⋊C4M16semidirect
166. C22⋊C8M16semidirect

Maximal subgroups

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

subgroupindex
#233342
#220822
#96502
#75932
#75822
#70332
#70212