Finite Group Databasebeta

← #329436

Subgroups — #329436

Order 512.

742 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33043C211yesmaximal, characteristic
2#32957C211yesmaximal, characteristic
3#32954C211yesmaximal, characteristic
4C2×#1984C211yesmaximal, characteristic
5#10624C212yesmaximal, complemented
6#10487C212yesmaximal, complemented
7#10483C212yesmaximal, complemented
8#10277C212yesmaximal, complemented
9#9131C211yesmaximal, characteristic
10#9127C211yesmaximal, characteristic
11#8596C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#487C2211yescharacteristic
14C2×#477C2212yes—
15C2×#475C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#1990C2211yescharacteristic
21#1984—21no—
22#1984—21no—
23#1984C2212yes—
24#1984C2212yes—
25#1956C2211yescharacteristic
26#1933—21no—
27#1933—21no—
28#1892—21no—
29#1892—21no—
30#1865—22no—
31#1863—22no—
32#1853—21no—
33#1852—21no—
34#1834C2211yescharacteristic
35#1803C2212yes—
36#1803C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1787—21no—
40#1787—21no—
41C2×#336—21no—
42C2×#336—21no—
43C2×#335C2212yes—
44C2×#333C2212yes—
45C2×#332C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8D411yescharacteristic
50C23×C8—21no—
51C23×C8D411yescharacteristic
52C23×C8D411yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—42no—
60C2×C4×Q8—42no—
61C2×C4×Q8D412yes—
62C2×C4×Q8C2312yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#487—42no—
68#477—42no—
69#477—42no—
70#475—42no—
71#475—42no—
72C2×#107—22no—
73C2×#106—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—42no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—21no—
84C2×#103—21no—
85C2×#103—21no—
86C2×#103—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—42no—
90C2×C4×C8D411yescharacteristic
91C2×C4×C8D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#391—21no—
95#391—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#380—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106#340—21no—
107#340—21no—
108#340—21no—
109#340—21no—
110#340—22no—
111#340—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—21no—
117#336—21no—
118#335—42no—
119#333—42no—
120#332—42no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8Q1614yes—
128C22×Q8C2×D412yes—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4#4111yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C22⋊Q8—82no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—44no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4C2×D412yes—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42C2411yescharacteristic
196#107—44no—
197#106—44no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—42no—
205#103—42no—
206#103—42no—
207#103—42no—
208#103—44no—
209#103—44no—
210#103—44no—
211#103—44no—
212#103—42no—
213C22⋊C8—42no—
214C22⋊C8—42no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—41no—
218C4×C8—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223C24#12111yescharacteristic
224C2×Q8—22no—
225C2×Q8—24no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—48no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—48no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—42no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—81no—
242C2×Q8—81no—
243C2×Q8C2×Q1614yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4C2×Q1612yes—
269C22×C4#13512yes—
270C22×C4#13512yes—
271C22×C4#13712yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—84no—
298C2×C8#13712yes—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—81no—
317C4⋊C4—81no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C42—21no—
323C42—22no—
324C42—21no—
325C42—22no—
326C42—44no—
327C42—42no—
328C42—42no—
329C42H(5,𝔽2)12yes—
330C42H(5,𝔽2)12yes—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23#46412yes—
336C23#46712yes—
337C23#47112yes—
338C23C2×#12111yescharacteristic
339Q8—24no—
340Q8—24no—
341Q8—48no—
342Q8—42no—
343Q8—42no—
344Q8—81no—
345Q8—81no—
346Q8—81no—
347Q8—81no—
348Q8—162no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—24no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—84no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4#46412yes—
386C2×C4#46712yes—
387C2×C4#53412yes—
388C2×C4#53411yescharacteristic
389C2×C4#53511yescharacteristic
390C2×C4#53812yes—
391C2×C4C22×Q1611yescharacteristic
392C2×C4C2×#13511yescharacteristic
393C2×C4C2×#13511yescharacteristic
394C2×C4C2×#13711yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C2×C4#57512yes—
397C2×C4#57511yescharacteristic
398C8—22no—
399C8—41no—
400C8—41no—
401C8—41no—
402C8—82no—
403C8—82no—
404C8—82no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22#168414yes—
412C22#261711yescharacteristic
413C22C2×#46411yescharacteristic
414C22#303711yescharacteristic
415C22C2×#46711yescharacteristic
416C22#305411yescharacteristic
417C22C2×#47111yescharacteristic
418C22#306711yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—44no—
423C4—41no—
424C4—82no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4#318912yes—
430C4#319011yescharacteristic
431C4#319112yes—
432C4#319611yescharacteristic
433C4#321111yescharacteristic
434C4#321811yescharacteristic
435C2—24no—
436C2C2×#168411yescharacteristic
437C2#1915511yescharacteristic
438C2#1926211yescharacteristic
439C2#1928811yescharacteristic
440C2#2971911yescharacteristic
441C2#2973111yescharacteristic
442C2#2975011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10624C2semidirect
6. #10487C2semidirect
7. #10483C2semidirect
8. #10277C2semidirect

Maximal subgroups

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

subgroupindex
#330432
#329572
#329542
C2×#19842
#106242
#104872
#104832
#102772
#91312
#91272
#85962