Finite Group Databasebeta

← #480440

Subgroups — #480440

Order 512.

685 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60207C211yesmaximal, characteristic
2#33426C212yesmaximal, complemented
3#11540C212yesmaximal, complemented
4#11537C212yesmaximal, complemented
5#11523C212yesmaximal, complemented
6#11521C212yesmaximal, complemented
7#11486C212yesmaximal, complemented
8#11483C212yesmaximal, complemented
9#3444C411yescharacteristic
10#2970—22no—
11#2970—22no—
12#2968—22no—
13#2968—22no—
14#2961C411yescharacteristic
15#2961C2211yescharacteristic
16#2932C411yescharacteristic
17#2932C2211yescharacteristic
18#2929C411yescharacteristic
19#2929C411yescharacteristic
20#2929C2211yescharacteristic
21#2929C2211yescharacteristic
22#2923C411yescharacteristic
23#2923C2211yescharacteristic
24#2920C2211yescharacteristic
25#2875C411yescharacteristic
26#2875C2211yescharacteristic
27C22×#97C411yescharacteristic
28#2166—22no—
29#2166C2214yes—
30#2165—22no—
31#2165C2214yes—
32#2153—22no—
33#2115C2214yes—
34C2×#347—22no—
35#1328C2214yes—
36#1327C2214yes—
37#1321C2214yes—
38#1320C2214yes—
39C2×#131C2×C411yescharacteristic
40C2×#131C2×C411yescharacteristic
41C2×#131C2×C411yescharacteristic
42C2×#131C2311yescharacteristic
43C23×C8C2×C411yescharacteristic
44C23×C8C2×C411yescharacteristic
45#533—21no—
46#533C2×C411yescharacteristic
47#532—21no—
48#532C2×C411yescharacteristic
49#529—21no—
50#529—21no—
51#529—22no—
52#529—22no—
53#525—22no—
54#525C2×C411yescharacteristic
55#525C2×C411yescharacteristic
56C4×#41—22no—
57C4×#41C2×C411yescharacteristic
58C4×#41C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C2×M32—24no—
61#446—22no—
62#446—22no—
63#446—22no—
64#446—22no—
65#435—22no—
66#435—22no—
67#435—22no—
68#435—22no—
69#435—22no—
70#435—22no—
71#435—22no—
72#435—22no—
73C8×D4—22no—
74C8×D4—22no—
75C8×D4—22no—
76C8×D4—22no—
77#433—22no—
78#433—22no—
79#432—22no—
80#432—22no—
81#432—22no—
82#431—22no—
83#424C2×C411yescharacteristic
84#424C2×C411yescharacteristic
85#424C2×C411yescharacteristic
86#424C2311yescharacteristic
87#423C2×C411yescharacteristic
88#423C2311yescharacteristic
89#408C2×C411yescharacteristic
90#408C2×C411yescharacteristic
91#408C2×C411yescharacteristic
92#408C2311yescharacteristic
93#407C2×C411yescharacteristic
94#407C2311yescharacteristic
95#405—21no—
96#405—21no—
97#405C2×C411yescharacteristic
98#405C2×C411yescharacteristic
99#405C2×C411yescharacteristic
100#405D412yes—
101#405D412yes—
102#405C2311yescharacteristic
103C4×M16—21no—
104C4×M16C2×C411yescharacteristic
105C4×M16D412yes—
106C4×M16C2311yescharacteristic
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97C2×C411yescharacteristic
113C2×#97C2×C411yescharacteristic
114C2×#97C2×C411yescharacteristic
115C2×#97C2×C411yescharacteristic
116C2×#97D412yes—
117#364—24no—
118#364—24no—
119#349—24no—
120#349C2318yes—
121#347—24no—
122#347—24no—
123C2×#41—22no—
124C2×#41C22×C411yescharacteristic
125C2×#41C22×C411yescharacteristic
126C23×C4—21no—
127C23×C4C22×C411yescharacteristic
128#131—42no—
129#131—42no—
130#131—42no—
131#131—42no—
132C2×M16—21no—
133C2×M16—21no—
134C2×M16C22⋊C412yes—
135C2×M16C22×C411yescharacteristic
136C2×M16C22×C411yescharacteristic
137C2×M16C22×C411yescharacteristic
138C2×M16C2×D412yes—
139C2×M16C2411yescharacteristic
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8C22⋊C412yes—
149C22×C8C22×C411yescharacteristic
150C22×C8C22×C411yescharacteristic
151C22×C8C22×C411yescharacteristic
152C22×C8C22×C411yescharacteristic
153C22×C8C2×D412yes—
154C4⋊Q8—21no—
155C4⋊Q8M1611yescharacteristic
156#127M1611yescharacteristic
157#126—22no—
158#126—22no—
159#125—22no—
160#125—21no—
161#124—22no—
162#124M1611yescharacteristic
163#124M1611yescharacteristic
164#123—22no—
165#123—22no—
166#123—21no—
167#123M1612yes—
168C22⋊Q8—22no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—22no—
172#121—22no—
173#121—22no—
174#121—21no—
175#121—21no—
176#120—21no—
177#120M1612yes—
178C4×Q8—22no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192#117—22no—
193#117—22no—
194#117—22no—
195#117—21no—
196#117C22⋊C412yes—
197#117C22×C411yescharacteristic
198#117C22×C411yescharacteristic
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42C22⋊C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208M32—28no—
209M32—44no—
210C2×C16—44no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223#97—21no—
224#97—21no—
225#97—21no—
226#97—21no—
227#97—21no—
228#97—21no—
229#97—21no—
230#97—21no—
231#97—21no—
232#97C22⋊C411yescharacteristic
233#97C22⋊C411yescharacteristic
234#97C22⋊C411yescharacteristic
235#97C22⋊C412yes—
236#97C22⋊C411yescharacteristic
237#97C2×D412yes—
238#97C2×D412yes—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8C22⋊C412yes—
243C4×C8C2×D412yes—
244C4×C8C2×D412yes—
245C24C2×M1611yescharacteristic
246#41—41no—
247#41—41no—
248#41—41no—
249#41—41no—
250#41—84no—
251C2×Q8—22no—
252C2×Q8C2×M1611yescharacteristic
253C2×Q8C2×M1611yescharacteristic
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4C2×M1611yescharacteristic
259C2×D4C2×M1611yescharacteristic
260C2×D4C2×M1612yes—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—82no—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×M1611yescharacteristic
277C22×C4C23×C411yescharacteristic
278M16—24no—
279M16—41no—
280M16—42no—
281M16—41no—
282M16—42no—
283M16—42no—
284M16—41no—
285M16—41no—
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8C2×C22⋊C411yescharacteristic
303C2×C8C2×C22⋊C411yescharacteristic
304C2×C8C2×C22⋊C412yes—
305C2×C8C2×C22⋊C411yescharacteristic
306C2×C8C2×C22⋊C411yescharacteristic
307C2×C8C22×D412yes—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—21no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4C2×M1612yes—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—22no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—22no—
337C42C2×C22⋊C411yescharacteristic
338C42C2×C22⋊C412yes—
339C42C2×C22⋊C411yescharacteristic
340C42C2×M1611yescharacteristic
341C42C2×M1611yescharacteristic
342C16—48no—
343C23—21no—
344C23—21no—
345C23—41no—
346C23—42no—
347C23#40712yes—
348C23C22×M1611yescharacteristic
349Q8—41no—
350Q8—41no—
351Q8—82no—
352D4—41no—
353D4—42no—
354D4—41no—
355D4—42no—
356D4—44no—
357D4—44no—
358D4—82no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4#40712yes—
386C2×C4C22×C22⋊C411yescharacteristic
387C2×C4C22×M1611yescharacteristic
388C8—22no—
389C8—21no—
390C8—21no—
391C8—22no—
392C8—41no—
393C8—41no—
394C8—41no—
395C8—41no—
396C8#41012yes—
397C8#41312yes—
398C22—22no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—42no—
403C22—41no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—41no—
408C22—41no—
409C22C2×#40711yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—42no—
417C4—42no—
418C4—41no—
419C4—41no—
420C4C2×#41011yescharacteristic
421C4C2×#41311yescharacteristic
422C2—21no—
423C2—42no—
424C2—42no—
425C2—41no—
426C2—81no—
427C2#1736311yescharacteristic
subgroupG/H
centerC4C2×#410
commutatorC22×C4C23×C4
Frattini, cosocleC2×M16C24
socleC2#17363
prime radical#3444C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ #97 ◁ #2875 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33426C2semidirect
3. #11540C2semidirect
4. #11537C2semidirect
5. #11523C2semidirect
6. #11521C2semidirect
7. #11486C2semidirect
8. #11483C2semidirect

Maximal subgroups

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

subgroupindex
#602072
#334262
#115402
#115372
#115232
#115212
#114862
#114832