Finite Group Databasebeta

← #224639

Subgroups — #224639

Order 512.

936 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36583C211yesmaximal, characteristic
2#21346C212yesmaximal, complemented
3#9689C212yesmaximal, complemented
4#9689C212yesmaximal, complemented
5#9685C212yesmaximal, complemented
6#8583C211yesmaximal, characteristic, complemented
7#8581C212yesmaximal, complemented
8#8226C212yesmaximal, complemented
9#8224C211yesmaximal, characteristic, complemented
10C2×#557C411yescharacteristic
11C2×#550C411yescharacteristic
12#2858C2212yes—
13#2858C2211yescharacteristic
14#2548C412yes—
15#2548C411yescharacteristic
16#2547C411yescharacteristic
17#2547C412yes—
18#2490—22no—
19#2490—22no—
20#2490—22no—
21#2341—22no—
22C4×#124C2211yescharacteristic
23C4×#124C2212yes—
24C42×Q8C2211yescharacteristic
25#1985—22no—
26#1940—22no—
27#1916—22no—
28#1916—22no—
29#1767C2212yes—
30#1767C2212yes—
31#1767C2212yes—
32#1745C2212yes—
33#1745C2212yes—
34C4×#97C2212yes—
35C2×#329C2212yes—
36C2×#329C2212yes—
37C2×#329C2212yes—
38C2×#329C2212yes—
39C2×#329C2212yes—
40C2×#329C2211yescharacteristic
41C2×#329C2211yescharacteristic
42#1295C2211yescharacteristic
43#1295C2211yescharacteristic
44#1295C2212yes—
45C22×#41—21no—
46#557—44no—
47#550—44no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#127C2×C412yes—
51C2×#127C2×C411yescharacteristic
52C2×#125—22no—
53C2×#125—21no—
54C2×#124C2×C412yes—
55C2×#124C2×C412yes—
56C2×#124C2×C412yes—
57C2×#124C2311yescharacteristic
58C2×#124C2312yes—
59C2×#121—22no—
60C2×#121—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D412yes—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C412yes—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×#97—22no—
72C2×#97C2312yes—
73C2×#97C2312yes—
74C2×#97C2312yes—
75C2×#97C2312yes—
76C2×C4×C8—22no—
77#397—21no—
78#397—21no—
79#397—22no—
80#397—22no—
81#394—21no—
82#394—21no—
83#394—22no—
84#394—22no—
85#391C2×C411yescharacteristic
86#391C2×C412yes—
87#390—22no—
88#390—22no—
89#390—22no—
90#390—22no—
91#390—22no—
92#390—22no—
93#390—22no—
94#390—22no—
95#390—22no—
96#390C2×C411yescharacteristic
97#390C2×C412yes—
98#388—22no—
99#388—22no—
100#388—22no—
101#382C2311yescharacteristic
102#380—22no—
103#380—22no—
104#380—22no—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2×C411yescharacteristic
107C4×C4⋊C4D412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C43C2312yes—
114C43C2311yescharacteristic
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#329—24no—
123#329—24no—
124#329—24no—
125#329—42no—
126#329—42no—
127#329—41no—
128#329—41no—
129C2×#41—44no—
130C2×#41—48no—
131C22×Q8C22⋊C411yescharacteristic
132C22×Q8C22×C412yes—
133C22×Q8C22×C411yescharacteristic
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4C22×C411yescharacteristic
139C22×D4C22×C412yes—
140C23×C4—22no—
141C23×C4—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—44no—
147C4⋊Q8—42no—
148#127—44no—
149#127—42no—
150#125—48no—
151#125—44no—
152#124—22no—
153#124—22no—
154#124—22no—
155#124—21no—
156#124—22no—
157#124—21no—
158#124—22no—
159#124—82no—
160#124—82no—
161#124—82no—
162#121—48no—
163#121—44no—
164C4×Q8—24no—
165C4×Q8—44no—
166C4×Q8—44no—
167C4×Q8—42no—
168C4×Q8—44no—
169C4×D4—42no—
170C4×D4—44no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C412yes—
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2411yescharacteristic
211C4⋊C8—44no—
212C4⋊C8—44no—
213C4⋊C8—44no—
214#97—22no—
215#97—22no—
216#97—22no—
217#97—22no—
218#97—22no—
219#97—22no—
220#97—22no—
221#97—44no—
222#97#4114yescomplemented
223C4×C8—44no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95C22⋊C411yescharacteristic
229#95C22⋊C411yescharacteristic
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232C24—21no—
233C24—22no—
234#41—816no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—22no—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—21no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—42no—
244C2×Q8—41no—
245C2×D4—21no—
246C2×D4—22no—
247C2×D4—21no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—82no—
261C2×D4—82no—
262C2×D4—82no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D412yes—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4#10014yes—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—44no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C42—24no—
324C42—22no—
325C42—24no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—41no—
331C42—42no—
332C42—42no—
333C42—44no—
334C42—42no—
335C42C2×#4112yes—
336C42H(5,𝔽2)11yescharacteristic
337C422−1+411yescharacteristic
338C23—41no—
339C23—42no—
340C23—41no—
341C23—42no—
342C23—41no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23C22×C22⋊C411yescharacteristic
351Q8—28no—
352Q8—82no—
353Q8—81no—
354Q8—82no—
355Q8—81no—
356D4—81no—
357D4—82no—
358D4—81no—
359D4—82no—
360D4—84no—
361D4—88no—
362D4—164no—
363D4—162no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—84no—
395C2×C4—88no—
396C2×C4C2×#10012yes—
397C2×C4C2×#10014yes—
398C2×C4#42114yes—
399C2×C4C4×#4111yescharacteristic
400C2×C4#52011yescharacteristic
401C2×C4#54812yes—
402C2×C4#55012yes—
403C8—44no—
404C8—44no—
405C8—82no—
406C8—82no—
407C22—81no—
408C22—82no—
409C22—81no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—81no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22#234111yescharacteristic
420C22C22×#10011yescharacteristic
421C22#288811yescharacteristic
422C22C2×#42112yes—
423C22#290112yes—
424C4—22no—
425C4—24no—
426C4—21no—
427C4—21no—
428C4—24no—
429C4—22no—
430C4—22no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—41no—
435C4—81no—
436C4—82no—
437C4—81no—
438C4—81no—
439C2—161no—
440C2—162no—
441C2#1766611yescharacteristic
442C2#1768412yes—
443C2#1768511yescharacteristic
444C2#2253611yescharacteristic
445C2#2255112yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#550C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21346C2semidirect
3. #9689C2semidirect
4. #9689C2semidirect
5. #9685C2semidirect
6. #8583C2semidirect
7. #8581C2semidirect
8. #8226C2semidirect
9. #8224C2semidirect
222. #97#41semidirect

Maximal subgroups

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

subgroupindex
#365832
#213462
#96892
#96852
#85832
#85812
#82262
#82242