Finite Group Databasebeta

← #99676

Subgroups — #99676

Order 512.

468 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 102 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#23815C211yesmaximal, characteristic
2#23152C211yesmaximal, characteristic, complemented
3#9678C211yesmaximal, characteristic
4#9678C211yesmaximal, characteristic
5C4×#332C211yesmaximal, characteristic, complemented
6#7027C211yesmaximal, characteristic, complemented
7#7027C211yesmaximal, characteristic, complemented
8C2×#446—21no—
9C4×C4⋊Q8C2211yescharacteristic
10C4×#127C2211yescharacteristic
11#2099C411yescharacteristic
12#2099C411yescharacteristic
13#2071—21no—
14#2041C411yescharacteristic
15#2041C411yescharacteristic
16#2023—21no—
17#1985—21no—
18#1940—21no—
19#1917—21no—
20#1916—21no—
21#1771—21no—
22#1767C2211yescharacteristic
23#1767C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C4×#102—21no—
27C4×#102—21no—
28#1744C2211yescharacteristic
29C2×#332—21no—
30C2×D4⋊C8—21no—
31C2×D4⋊C8—21no—
32C2×D4⋊C8—21no—
33C2×D4⋊C8—21no—
34#1296—21no—
35C2×C4⋊Q8D411yescharacteristic
36C2×#127C2×C411yescharacteristic
37C2×C4×Q8—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40C2×C4×D4—21no—
41#446—42no—
42#446—42no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8D411yescharacteristic
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D411yescharacteristic
51C2×#102—21no—
52C2×#102—21no—
53C2×#102—21no—
54C2×#102—21no—
55C2×#97—21no—
56C2×#97—21no—
57C2×#97—21no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61C2×C4×C8—21no—
62#391—21no—
63#390—21no—
64#390—21no—
65#390—21no—
66#389—21no—
67#389C2×C411yescharacteristic
68#389C2×C411yescharacteristic
69#384C2×C411yescharacteristic
70#384D411yescharacteristic
71C4×C4⋊C4—21no—
72C4×C4⋊C4D411yescharacteristic
73C4×C4⋊C4D411yescharacteristic
74C43C2311yescharacteristic
75#340—21no—
76#340—21no—
77#340—21no—
78#340—21no—
79#336—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#336C2×C411yescharacteristic
89#336C2×C411yescharacteristic
90#332—24no—
91D4⋊C8—42no—
92D4⋊C8—42no—
93D4⋊C8—42no—
94D4⋊C8—42no—
95C22×Q8—21no—
96C22×D4—21no—
97C22×D4—21no—
98C22×D4—21no—
99C23×C4—41no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—41no—
107C22×C8—41no—
108C4⋊Q8—22no—
109C4⋊Q8—41no—
110#127—42no—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×D4—41no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—41no—
119C4×D4—41no—
120C4×D4—41no—
121C4×D4—41no—
122C4×D4—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4C22⋊C411yescharacteristic
136C2×C4⋊C4C22⋊C411yescharacteristic
137C2×C4⋊C4C22⋊C411yescharacteristic
138C2×C4⋊C4C22⋊C411yescharacteristic
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42C22×C411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—41no—
160C4⋊C8—41no—
161C4⋊C8—42no—
162C4⋊C8—42no—
163C4⋊C8D812yes—
164C4⋊C8SD1612yes—
165#102—42no—
166#102—42no—
167#102—42no—
168#102—42no—
169#97—42no—
170#97—41no—
171#97—41no—
172#97—41no—
173#97—41no—
174C4×C8—41no—
175C4×C8—41no—
176C4×C8—42no—
177C4×C8—41no—
178C4×C8—41no—
179C4×C8—41no—
180C4×C8—41no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—21no—
185C24—41no—
186C2×Q8—42no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—81no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—41no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4#12011yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12311yescharacteristic
226C2×C8—22no—
227C2×C8—22no—
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—42no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—81no—
247C2×C8—81no—
248C2×C8—81no—
249C2×C8—81no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—81no—
269C4⋊C4#10412yes—
270C22⋊C4—81no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—41no—
281C42—41no—
282C42—41no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42—41no—
287C42C2×D811yescharacteristic
288C42C2×SD1611yescharacteristic
289C42#13511yescharacteristic
290C42#13511yescharacteristic
291C42AGL(1,ℤ/8)11yescharacteristic
292C42AGL(1,ℤ/8)11yescharacteristic
293C42AGL(1,ℤ/8)11yescharacteristic
294C42#13711yescharacteristic
295C23—42no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23#38611yescharacteristic
303Q8—82no—
304Q8—82no—
305D4—42no—
306D4—42no—
307D4—42no—
308D4—42no—
309D4—42no—
310D4—42no—
311D4—81no—
312D4—81no—
313D4—81no—
314D4—81no—
315D4—81no—
316D4—81no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—161no—
346C2×C4—161no—
347C2×C4—161no—
348C2×C4—161no—
349C2×C4C2×#10411yescharacteristic
350C2×C4#42111yescharacteristic
351C2×C4C4×D811yescharacteristic
352C2×C4C4×SD1611yescharacteristic
353C2×C4#44011yescharacteristic
354C2×C4#44211yescharacteristic
355C2×C4#44711yescharacteristic
356C2×C4#44811yescharacteristic
357C2×C4#44911yescharacteristic
358C2×C4#44911yescharacteristic
359C2×C4#45411yescharacteristic
360C2×C4#45411yescharacteristic
361C2×C4#45911yescharacteristic
362C2×C4#46011yescharacteristic
363C2×C4#46311yescharacteristic
364C2×C4#46411yescharacteristic
365C8—42no—
366C8—42no—
367C8—81no—
368C8—81no—
369C8—82no—
370C8—82no—
371C8—81no—
372C8—81no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—161no—
377C22—161no—
378C22—161no—
379C22—161no—
380C22#187111yescharacteristic
381C22#187911yescharacteristic
382C22#188011yescharacteristic
383C22#188111yescharacteristic
384C22#188111yescharacteristic
385C22#188811yescharacteristic
386C22#189111yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—41no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—161no—
403C4#162511yescharacteristic
404C4#162611yescharacteristic
405C4#162711yescharacteristic
406C4#162811yescharacteristic
407C2—162no—
408C2#899111yescharacteristic
409C2#899211yescharacteristic
410C2#900811yescharacteristic
411C2#903211yescharacteristic
412C2#904911yescharacteristic
413C2#905011yescharacteristic
414C2#906211yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socleC23#386
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #23152C2semidirect
5. C4×#332C2semidirect
6. #7027C2semidirect
7. #7027C2semidirect

Maximal subgroups

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

subgroupindex
#238152
#231522
#96782
C4×#3322
#70272