Finite Group Databasebeta

← #6435199

Subgroups — #6435199

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34642C211yesmaximal, characteristic
2C2×#2320C211yesmaximal, characteristic
3#29993C218yesmaximal, complemented
4#29992C218yesmaximal, complemented
5#23747C212yesmaximal
6#23732C212yesmaximal
7#23554C212yesmaximal
8#23455C212yesmaximal
9#21170C212yesmaximal
10#20753C211yesmaximal, characteristic
11#20647C212yesmaximal
12C2×#533C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C2×#428C2214yes—
15C2×#423C2212yes—
16C2×#419C2214yes—
17#2660C2214yes—
18#2660C2214yes—
19#2659—21no—
20#2659—21no—
21#2659—21no—
22#2659—21no—
23#2399—24no—
24#2320—24no—
25#2320C2218yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#387—21no—
31C2×#387—21no—
32#2285C2212yes—
33#2285C2212yes—
34C2×#383—21no—
35C2×#382—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×#376C2211yescharacteristic
38#2101C2218yes—
39#2100C2214yes—
40#2099C2214yes—
41#2097C2218yes—
42#2096C2214yes—
43#2095C2214yes—
44#2059C2218yes—
45#2058C2214yes—
46#2057C2214yes—
47#2026C2218yes—
48#2024C2214yes—
49#2024C2214yes—
50#1949C2218yes—
51#1948C2214yes—
52#1947C2214yes—
53#1832C2218yes—
54#1805C2218yes—
55#1803C2214yes—
56#1803C2214yes—
57#1736C2214yes—
58#1736C2214yes—
59#1728C2212yes—
60#1728C2212yes—
61C23×Q8C2311yescharacteristic
62C22×M16C2314yes—
63#533—48no—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×#125C2314yes—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—24no—
73C2×C4×Q8Q818yescomplemented
74C2×C4×Q8C2314yes—
75C2×#117C2314yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#428—48no—
88C2×#107C2318yes—
89C2×#106C2318yes—
90#423—28no—
91C2×C4⋊C8C2314yes—
92C2×C4⋊C8C2314yes—
93#419—48no—
94C2×#103C2318yes—
95C2×#103C2318yes—
96#398—24no—
97#398—24no—
98#398—24no—
99#398—24no—
100#395—24no—
101#395—24no—
102#395—24no—
103#395—24no—
104#391—22no—
105#391—22no—
106#391C2314yes—
107#391C2314yes—
108#389—22no—
109#389—22no—
110#389C2314yes—
111#389C2314yes—
112#387—28no—
113#387—28no—
114#384C2312yes—
115#384C2312yes—
116#384C2312yes—
117#384C2312yes—
118#383—28no—
119#382—28no—
120#382C2314yes—
121#382C2314yes—
122#380—22no—
123#380—22no—
124#380—22no—
125#380—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2314yes—
132#376D414yes—
133#376C2314yes—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95D411yescharacteristic
142C2×#95D411yescharacteristic
143C2×#95D411yescharacteristic
144C2×#95D411yescharacteristic
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151#336C2314yes—
152#336C2314yes—
153C22×Q8—21no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8C2×Q814yes—
160C22×Q8C2412yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C2×M16—28no—
173C2×M16—28no—
174C2×M16—28no—
175C22×C8C2418yes—
176C4⋊Q8—44no—
177C4⋊Q8—44no—
178#125—44no—
179#125—44no—
180C22⋊Q8—48no—
181C22⋊Q8—48no—
182C4×Q8—216no—
183C4×Q8—28no—
184C4×Q8—48no—
185C4×Q8—48no—
186C4×Q8C2×Q8116yes—
187#117—28no—
188#117—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×Q814yes—
214C2×C4⋊C4C2412yes—
215C2×C4⋊C4C2412yes—
216C2×C4⋊C4C2414yes—
217C2×C4⋊C4C2414yes—
218C2×C4⋊C4C2412yes—
219C2×C22⋊C4—24no—
220C2×C42—28no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—82no—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×Q814yes—
231C2×C42#4114yes—
232C2×C42C2412yes—
233C2×C42C2412yes—
234C2×C42C2412yes—
235#107—48no—
236#106—48no—
237C4⋊C8—28no—
238C4⋊C8—28no—
239#103—48no—
240#103—48no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95C2×D414yes—
253#95C2×D414yes—
254#95C2×D414yes—
255#95C2×D414yes—
256C24C22×D411yescharacteristic
257C2×Q8—24no—
258C2×Q8—22no—
259C2×Q8—216no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—48no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8C22×Q814yes—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4C22⋊Q814yes—
307C22×C4C22⋊Q814yes—
308C22×C4C22×D412yes—
309C22×C4C22×D412yes—
310C22×C4C22×D412yes—
311C22×C4C22×D412yes—
312C22×C4C22×D412yes—
313C22×C4C22×D412yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×Q812yes—
316C22×C4C2×#4112yes—
317C22×C4H(5,𝔽2)14yes—
318C22×C42−1+414yes—
319C22×C4C2511yescharacteristic
320M16—416no—
321C2×C8—28no—
322C2×C8—28no—
323C2×C8—28no—
324C4⋊C4—216no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C4⋊C4C22×Q814yes—
350C22⋊C4—48no—
351C42—216no—
352C42—24no—
353C42—24no—
354C42—24no—
355C42—24no—
356C42—48no—
357C42—84no—
358C42C22⋊Q818yes—
359C42C22×Q814yes—
360C42C2×#4114yes—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23C2×C22⋊Q812yes—
366C23#53512yes—
367C23#53612yes—
368C23C23×D411yescharacteristic
369Q8—28no—
370Q8—48no—
371Q8—44no—
372Q8—88no—
373Q8—88no—
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—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—48no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4C2×C22⋊Q814yes—
419C2×C4C2×C22⋊Q814yes—
420C2×C4C2×C22⋊Q812yes—
421C2×C4#53512yes—
422C2×C4#53612yes—
423C2×C4#54218yes—
424C2×C4#54418yes—
425C2×C4C23×Q811yescharacteristic
426C2×C4C22×#4111yescharacteristic
427C2×C4C2×H(5,𝔽2)12yes—
428C2×C4C2×2−1+412yes—
429C8—48no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22#262814yes—
437C22C22×C22⋊Q811yescharacteristic
438C22C2×#53511yescharacteristic
439C22C2×#53611yescharacteristic
440C22#358511yescharacteristic
441C22#358611yescharacteristic
442C22#358611yescharacteristic
443C22#358611yescharacteristic
444C4—24no—
445C4—24no—
446C4—44no—
447C4—44no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—84no—
453C4—82no—
454C4—84no—
455C4—82no—
456C4#262814yes—
457C4C2×#54212yes—
458C4C2×#54412yes—
459C2—24no—
460C2C2×#262811yescharacteristic
461C2#6049111yescharacteristic
462C2#6049211yescharacteristic
463C2#6080911yescharacteristic
464C2#6081011yescharacteristic
465C2#6086911yescharacteristic
466C2#6087011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29993C2semidirect
4. #29992C2semidirect
73. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#346422
C2×#23202
#299932
#299922
#237472
#237322
#235542
#234552
#211702
#207532
#206472