Finite Group Databasebeta

← #214841

Subgroups — #214841

Order 512.

920 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35748C211yesmaximal, characteristic
2#20731C211yesmaximal, characteristic
3#17351C211yesmaximal, characteristic
4#9483C212yesmaximal, complemented
5#9395C212yesmaximal, complemented
6#9194C212yesmaximal, complemented
7#9034C212yesmaximal, complemented
8#8766C212yesmaximal, complemented
9#8396C212yesmaximal, complemented
10C2×Q82C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2972C2212yes—
13#2921C2212yes—
14#2628C412yes—
15#2628C2211yescharacteristic
16#2580C412yes—
17#2580C2211yescharacteristic
18#2542C412yes—
19#2542C2211yescharacteristic
20#2452—22no—
21#2452—21no—
22#2440—21no—
23#2440—22no—
24#2396C412yes—
25#2396C2211yescharacteristic
26#2351C2211yescharacteristic
27Q8×C4⋊C4—21no—
28#2341—21no—
29#1821—21no—
30#1801C2212yes—
31#1800C2212yes—
32#1785—21no—
33C4×#101—21no—
34C4×#100—21no—
35#1549C2214yes—
36#1487C2214yes—
37#1461C2212yes—
38#1459—22no—
39#1458C2212yes—
40C2×#324—22no—
41#1303C2214yes—
42#1286C2214yes—
43C22×#41D411yescharacteristic
44Q82—44no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52#520—44no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×D4C2×C412yes—
60C2×C4×D4D412yes—
61C2×C4×D4D411yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×#117C2×C412yes—
64C2×#117C2311yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D412yes—
67C8×D4—44no—
68#433—44no—
69#431C2314yes—
70#423C2314yes—
71C2×#101—22no—
72C2×#100—22no—
73#407—22no—
74C2×C22⋊C8—24no—
75C4×M16—22no—
76#400C2×C411yescharacteristic
77#400C2×C412yes—
78#399C2×C412yes—
79#399C2×C412yes—
80#399C2×C412yes—
81#399C2×C412yes—
82#399C2×C411yescharacteristic
83#398—22no—
84#398—21no—
85#396—22no—
86#396—21no—
87#395—22no—
88#395—21no—
89#395C2×C411yescharacteristic
90#393—22no—
91#393—21no—
92#393C2×C411yescharacteristic
93#393C2×C412yes—
94#390—22no—
95#390—22no—
96#390—21no—
97#390—21no—
98#389—22no—
99#389—21no—
100#389—21no—
101#389—22no—
102#389—22no—
103#389—21no—
104#385—22no—
105#385—22no—
106#385—22no—
107#384—22no—
108#384—21no—
109#382C2×C412yes—
110#382C2311yescharacteristic
111#380—22no—
112#380—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—22no—
118C4×C22⋊C4—21no—
119#343—22no—
120#343—22no—
121#332—44no—
122Q8⋊C8—44no—
123#324—22no—
124#324—24no—
125#323—22no—
126C2×#41—24no—
127C2×#41—24no—
128C2×#41—42no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8C22×C411yescharacteristic
132C22×Q8C2×D411yescharacteristic
133C22×D4C22⋊C412yes—
134C22×D4C2×D411yescharacteristic
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C2×M16—24no—
141C22×C8—44no—
142C4⋊Q8—24no—
143C4⋊Q8—44no—
144C4⋊Q8—48no—
145C4⋊Q8—44no—
146C22⋊Q8—84no—
147C22⋊Q8—84no—
148C22⋊Q8—84no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—44no—
152C4×Q8—42no—
153C4×Q8—44no—
154C4×Q8—44no—
155C4×D4—24no—
156C4×D4—44no—
157C4×D4—42no—
158C4×D4—44no—
159#117—24no—
160#117—44no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22×C412yes—
195C2×C22⋊C4C22×C411yescharacteristic
196C2×C22⋊C4C2×D411yescharacteristic
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42C22⋊C412yes—
202C2×C42C22×C412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2411yescharacteristic
205C4⋊C8—24no—
206C4⋊C8—24no—
207C4⋊C8—24no—
208#101—44no—
209#100—44no—
210C22⋊C8—24no—
211C22⋊C8—44no—
212C22⋊C8—44no—
213#97—42no—
214C4×C8—24no—
215C4×C8—42no—
216#95—21no—
217#95—22no—
218#95—21no—
219#95—22no—
220#95—21no—
221#95—22no—
222#95—21no—
223#95—22no—
224#95—21no—
225#95C22×C411yescharacteristic
226#95C22×C412yes—
227#95C22×C411yescharacteristic
228C24C2×C22⋊C411yescharacteristic
229C24C2×C22⋊C412yes—
230#41—416no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—42no—
235C2×Q8—44no—
236C2×Q8—42no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8#13514yes—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—42no—
246C2×D4#13714yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274M16—44no—
275M16—44no—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—164no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C22⋊C4—22no—
303C22⋊C4—24no—
304C22⋊C4—22no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—42no—
308C22⋊C4—44no—
309C22⋊C4—42no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C42—24no—
314C42—22no—
315C42—42no—
316C42—44no—
317C42—42no—
318C42—42no—
319C42—44no—
320C42H(5,𝔽2)14yes—
321C23—22no—
322C23—22no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—42no—
327C23—41no—
328C23—42no—
329C23—41no—
330C23#41914yes—
331C23C22×C22⋊C411yescharacteristic
332Q8—28no—
333Q8—24no—
334Q8—42no—
335Q8—162no—
336Q8—164no—
337D4—28no—
338D4—48no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—48no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—84no—
375C2×C4—162no—
376C2×C4—162no—
377C2×C4#41614yes—
378C2×C4#51812yes—
379C2×C4#53412yes—
380C2×C4#53412yes—
381C2×C4C2×#13512yes—
382C2×C4C2×#13712yes—
383C8—84no—
384C8—84no—
385C22—21no—
386C22—21no—
387C22—24no—
388C22—44no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—82no—
396C22—81no—
397C22#230811yescharacteristic
398C22C2×#41611yescharacteristic
399C22C2×#41911yescharacteristic
400C4—24no—
401C4—22no—
402C4—22no—
403C4—42no—
404C4—41no—
405C4—41no—
406C4—42no—
407C4—42no—
408C4—41no—
409C4—42no—
410C4—81no—
411C4—82no—
412C4—82no—
413C4—81no—
414C4#319014yes—
415C2—21no—
416C2—21no—
417C2—44no—
418C2—81no—
419C2#2176211yescharacteristic
420C2#2178011yescharacteristic
421C2#2178211yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9483C2semidirect
5. #9395C2semidirect
6. #9194C2semidirect
7. #9034C2semidirect
8. #8766C2semidirect
9. #8396C2semidirect

Maximal subgroups

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

subgroupindex
#357482
#207312
#173512
#94832
#93952
#91942
#90342
#87662
#83962