Finite Group Databasebeta

← #99995

Subgroups — #99995

Order 512.

474 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 96 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#23326C211yesmaximal, characteristic
2#22047C211yesmaximal, characteristic
3#9878C211yesmaximal, characteristic, complemented
4#7595C211yesmaximal, characteristic, complemented
5#7582C211yesmaximal, characteristic
6#7033C211yesmaximal, characteristic, complemented
7#7021C211yesmaximal, characteristic, complemented
8C2×#448C411yescharacteristic
9C4×#121C2211yescharacteristic
10C2×#391C2211yescharacteristic
11#1973—21no—
12#1954—21no—
13#1935—21no—
14#1928—21no—
15#1923—21no—
16#1916—21no—
17#1877—21no—
18#1874C411yescharacteristic
19#1869—21no—
20#1869—21no—
21#1852C411yescharacteristic
22#1850—21no—
23#1817C2211yescharacteristic
24#1817C2211yescharacteristic
25#1797C2211yescharacteristic
26#1787C411yescharacteristic
27#1749C2211yescharacteristic
28C4×C22⋊C8C2211yescharacteristic
29#1280—21no—
30#1280—21no—
31#1272—21no—
32#1272—21no—
33#1271—21no—
34#1270—21no—
35C23×Q8C2×C411yescharacteristic
36C22×SD16—21no—
37C23×C8—21no—
38C2×#121C2×C411yescharacteristic
39C2×C4×D4—21no—
40C2×C4×D4D411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C42C2311yescharacteristic
43#448—22no—
44#448C814yescomplemented
45C2×#107—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×#103—21no—
49C2×#103—21no—
50C2×#103—21no—
51C2×#102—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8C2×C411yescharacteristic
59C2×C22⋊C8C2×C411yescharacteristic
60C2×C22⋊C8D411yescharacteristic
61C2×C22⋊C8D411yescharacteristic
62C2×#97—21no—
63C2×C4×C8—21no—
64C2×C4×C8—21no—
65#391—21no—
66#391—21no—
67#391—21no—
68#391—21no—
69#390D411yescharacteristic
70#389D411yescharacteristic
71#388—21no—
72#386—21no—
73#380C2×C411yescharacteristic
74C4×C4⋊C4D411yescharacteristic
75C4×C22⋊C4—21no—
76C2×#95—21no—
77C2×#95—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
90C22×Q8—21no—
91C22×Q8—21no—
92C22×Q8—21no—
93C22×Q8—21no—
94C22×Q8—21no—
95C22×Q8—81no—
96C22×Q8—81no—
97C22×Q8C2×C812yescomplemented
98C22×D4—21no—
99C22×D4C22⋊C411yescharacteristic
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4#4111yescharacteristic
106C2×SD16—24no—
107C2×SD16—81no—
108C2×SD16—81no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—81no—
118C22×C8—81no—
119C22×C8—81no—
120#121—21no—
121#121—21no—
122#121C2×C812yes—
123#121M1612yes—
124C4×D4—81no—
125C4×D4—81no—
126C4×D4—81no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—81no—
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C4⋊C4C22⋊C411yescharacteristic
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C42—21no—
140C2×C42—81no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42#4111yescharacteristic
145#107—81no—
146C4⋊C8—81no—
147C4⋊C8—81no—
148#103—24no—
149#103—81no—
150#103—81no—
151#102—81no—
152C22⋊C8—21no—
153C22⋊C8—22no—
154C22⋊C8—21no—
155C22⋊C8—21no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—81no—
160C22⋊C8—81no—
161C22⋊C8—81no—
162C22⋊C8—81no—
163C22⋊C8—81no—
164C22⋊C8C2×C812yescomplemented
165#97—81no—
166C4×C8—81no—
167C4×C8—81no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—41no—
172#95—41no—
173#95—41no—
174#95—41no—
175C24—21no—
176C24C4×D411yescharacteristic
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—81no—
186C2×Q8—81no—
187C2×Q8—82no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×Q8—82no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—81no—
197C2×D4—81no—
198C2×D4C22⋊C812yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
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—42no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4C22×C811yescharacteristic
233C22×C4C2×M1611yescharacteristic
234C22×C4#13111yescharacteristic
235C22×C4#13111yescharacteristic
236SD16—84no—
237C2×C8—22no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—81no—
257C2×C8—81no—
258C2×C8—84no—
259C2×C8—81no—
260C2×C8—81no—
261C2×C8—81no—
262C4⋊C4—21no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—81no—
268C4⋊C4—81no—
269C4⋊C4C22⋊C812yes—
270C22⋊C4—22no—
271C22⋊C4—22no—
272C22⋊C4—81no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C42—41no—
277C42—41no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42—41no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23#38611yescharacteristic
296C23C8×D411yescharacteristic
297C23#43511yescharacteristic
298Q8—42no—
299Q8—42no—
300Q8—82no—
301Q8—81no—
302Q8—81no—
303Q8—82no—
304D4—42no—
305D4—81no—
306D4—82no—
307D4—82no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
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—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—162no—
352C2×C4C2×C22⋊C811yescharacteristic
353C2×C4#40811yescharacteristic
354C2×C4C8×D411yescharacteristic
355C2×C4#43511yescharacteristic
356C2×C4#45511yescharacteristic
357C2×C4#45611yescharacteristic
358C8—42no—
359C8—81no—
360C8—81no—
361C8—81no—
362C8—81no—
363C8—81no—
364C8—82no—
365C8—161no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22#186911yescharacteristic
384C22#188111yescharacteristic
385C22#188211yescharacteristic
386C22#190211yescharacteristic
387C22#190411yescharacteristic
388C22#191311yescharacteristic
389C22#191411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—42no—
400C4—81no—
401C4—81no—
402C4—82no—
403C4—161no—
404C2—41no—
405C2—41no—
406C2—82no—
407C2#895311yescharacteristic
408C2#908711yescharacteristic
409C2#910711yescharacteristic
410C2#933111yescharacteristic
411C2#933311yescharacteristic
412C2#933711yescharacteristic
413C2#934111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC2×#121C2×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
3. #9878C2semidirect
4. #7595C2semidirect
6. #7033C2semidirect
7. #7021C2semidirect
44. #448C8semidirect
97. C22×Q8C2×C8semidirect
164. C22⋊C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#233262
#220472
#98782
#75952
#75822
#70332
#70212