Finite Group Databasebeta

← #100220

Subgroups — #100220

Order 512.

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

normal subgroups only

This group has 570 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#22317C211yesmaximal, characteristic
2#21455C211yesmaximal, characteristic
3#8311C211yesmaximal, characteristic
4#8303C211yesmaximal, characteristic
5#7600C211yesmaximal, characteristic
6#7033C211yesmaximal, characteristic
7#7021C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#451C411yescharacteristic
10C2×#441—21no—
11C4×C22⋊Q8C2211yescharacteristic
12#1984—21no—
13#1923—21no—
14#1923—21no—
15#1916—21no—
16#1916—21no—
17#1878—21no—
18#1872C411yescharacteristic
19#1865—21no—
20#1853C411yescharacteristic
21#1817C2211yescharacteristic
22#1817C2211yescharacteristic
23#1797C2211yescharacteristic
24#1787C411yescharacteristic
25C8×C4⋊C4—21no—
26#1763—21no—
27#1749C2211yescharacteristic
28C4×C22⋊C8C2211yescharacteristic
29C2×Q8⋊C8—21no—
30C2×Q8⋊C8—21no—
31C2×#324—21no—
32C2×#324—21no—
33#1272—21no—
34#1271—21no—
35C23×Q8C2×C411yescharacteristic
36C22×Q16—21no—
37C2×C22⋊Q8C2×C411yescharacteristic
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—42no—
43C2×C4×Q8—42no—
44C2×C4×Q8D411yescharacteristic
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C42—21no—
48C22×C42C2311yescharacteristic
49#451—22no—
50#451C814yescomplemented
51#441—42no—
52#441—42no—
53C2×#106—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×#103—21no—
58C2×#103—21no—
59C2×#103—21no—
60C2×#103—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8C2×C411yescharacteristic
65C2×C22⋊C8C2×C411yescharacteristic
66C2×C22⋊C8D411yescharacteristic
67C2×C22⋊C8D411yescharacteristic
68C2×#97—21no—
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73#391D411yescharacteristic
74#389D411yescharacteristic
75#387—21no—
76#385—21no—
77#380C2×C411yescharacteristic
78C4×C4⋊C4—21no—
79C4×C4⋊C4D411yescharacteristic
80C4×C22⋊C4—21no—
81#340—21no—
82#340—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#336—21no—
91#336—21no—
92#336—21no—
93#336—21no—
94Q8⋊C8—42no—
95Q8⋊C8—42no—
96#324—42no—
97#324—42no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—42no—
104C22×Q8—42no—
105C22×Q8C22⋊C411yescharacteristic
106C22×Q8C2×C812yes—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C2×Q16—24no—
113C2×Q16—42no—
114C2×Q16—42no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—41no—
122C22×C8—41no—
123C22⋊Q8—21no—
124C22⋊Q8—21no—
125C22⋊Q8C2×C812yes—
126C22⋊Q8M1612yes—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—42no—
138C4×Q8—42no—
139C4×Q8—42no—
140C4×Q8—42no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4C22⋊C411yescharacteristic
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C22⋊C411yescharacteristic
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—42no—
163C2×C42—42no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170#106—42no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—42no—
174C4⋊C8—81no—
175#103—24no—
176#103—42no—
177#103—42no—
178#103—42no—
179C22⋊C8—21no—
180C22⋊C8—21no—
181C22⋊C8—21no—
182C22⋊C8—42no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—81no—
188C22⋊C8C2×C812yes—
189#97—41no—
190#97—41no—
191C4×C8—41no—
192C4×C8—41no—
193C4×C8—42no—
194C4×C8—81no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200C24C4×D411yescharacteristic
201C2×Q8—21no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—44no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—44no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—44no—
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8C22⋊C812yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4C22×C811yescharacteristic
254C22×C4C2×M1611yescharacteristic
255C22×C4#13111yescharacteristic
256C22×C4#13111yescharacteristic
257Q16—48no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—81no—
277C2×C8—81no—
278C2×C8—81no—
279C2×C8—81no—
280C2×C8—81no—
281C2×C8—81no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—81no—
298C4⋊C4C22⋊C812yes—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—81no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—41no—
307C42—41no—
308C42—41no—
309C42—41no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—41no—
315C42—41no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23#38611yescharacteristic
323C23C8×D411yescharacteristic
324C23#43511yescharacteristic
325Q8—24no—
326Q8—24no—
327Q8—42no—
328Q8—42no—
329Q8—42no—
330Q8—42no—
331Q8—42no—
332Q8—42no—
333Q8—42no—
334Q8—81no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4C2×C22⋊C811yescharacteristic
382C2×C4#40811yescharacteristic
383C2×C4C8×D411yescharacteristic
384C2×C4#43511yescharacteristic
385C2×C4B(4,𝔽2)11yescharacteristic
386C2×C4B(4,𝔽2)11yescharacteristic
387C2×C4#47111yescharacteristic
388C2×C4#47311yescharacteristic
389C8—42no—
390C8—42no—
391C8—81no—
392C8—81no—
393C8—81no—
394C8—81no—
395C8—161no—
396C8—161no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22#186911yescharacteristic
408C22#188911yescharacteristic
409C22#189111yescharacteristic
410C22#190211yescharacteristic
411C22#190511yescharacteristic
412C22#190611yescharacteristic
413C22#190811yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—44no—
423C4—42no—
424C4—41no—
425C4—41no—
426C4—42no—
427C4—81no—
428C4—82no—
429C4—81no—
430C2—41no—
431C2—41no—
432C2#896511yescharacteristic
433C2#908811yescharacteristic
434C2#909611yescharacteristic
435C2#951911yescharacteristic
436C2#952411yescharacteristic
437C2#952811yescharacteristic
438C2#953111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
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
50. #451C8semidirect

Maximal subgroups

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

subgroupindex
#223172
#214552
#83112
#83032
#76002
#70332
#70212