Finite Group Databasebeta

← #415352

Subgroups — #415352

Order 512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35330C211yesmaximal, characteristic
2#21324C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic
5#11392C214yesmaximal
6#11276C214yesmaximal
7#10858C211yesmaximal, characteristic
8#10840C211yesmaximal, characteristic
9#10701C211yesmaximal, characteristic
10C2×#544C2212yes—
11C22×C4⋊Q8C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2862—21no—
15#2859—21no—
16#2660—22no—
17#2550—22no—
18#2550—22no—
19#2549—22no—
20#2549—22no—
21#2317C2212yes—
22#2285C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#1993—21no—
27#1993—21no—
28#1990—22no—
29#1988—22no—
30#1987—22no—
31C2×#343C2214yes—
32#1736C2212yes—
33#1736C2214yes—
34#1736C2212yes—
35C2×#339C2212yes—
36C2×#339C2214yes—
37C2×#339C2212yes—
38#1446—22no—
39#1445—22no—
40C4⋊#95C2214yes—
41C23×Q8—21no—
42C22×M16C2312yes—
43C22×M16C2312yes—
44#544—28no—
45#544—44no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8Q814yescomplemented
56C2×#125—22no—
57C2×#125—22no—
58C2×C22⋊Q8—24no—
59C2×#117C2312yes—
60C2×#117C2311yescharacteristic
61C2×#117C2311yescharacteristic
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67#423—22no—
68#423—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C4×M16—22no—
72C4×M16—22no—
73C2×#97—21no—
74C2×C4×C8—21no—
75#398—24no—
76#398—24no—
77#395—24no—
78#395—24no—
79#391—24no—
80#391—24no—
81#389—24no—
82#389—24no—
83#389—22no—
84#389—22no—
85#385—24no—
86#384—22no—
87#384D412yes—
88#382—22no—
89#382—21no—
90#382—21no—
91#382—21no—
92#382—21no—
93#382D411yescharacteristic
94#382D411yescharacteristic
95#380—21no—
96#380—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4D412yes—
100C4×C4⋊C4D412yes—
101C4×C22⋊C4—21no—
102C4×C22⋊C4—21no—
103#376—21no—
104#376—22no—
105#376—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—28no—
109C2×#95D412yes—
110C2×#95D412yes—
111C43—21no—
112#343—44no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#339—24no—
118#339—44no—
119#339Q818yescomplemented
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—41no—
123C22×Q8—41no—
124C22×Q8—42no—
125C22×Q8—84no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4C2×D414yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—24no—
140C2×M16—24no—
141C2×M16C2×Q814yes—
142C22×C8—22no—
143C22×C8—22no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—48no—
149C4⋊Q8—44no—
150C4⋊Q8—44no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155#125—48no—
156#125—44no—
157#125—44no—
158C22⋊Q8—416no—
159#117—22no—
160#117—22no—
161#117—24no—
162#117—24no—
163#117C2×Q814yes—
164#117#4114yes—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×Q814yes—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×Q814yes—
208C2×C42C2×Q812yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212C4⋊C8—42no—
213C4⋊C8—42no—
214#97—42no—
215C4×C8—42no—
216#95—24no—
217#95—24no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—42no—
229#95—42no—
230#95—88no—
231C24C22×D411yescharacteristic
232C2×Q8—48no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—84no—
236C2×Q8—84no—
237C2×Q8—88no—
238C2×Q8—84no—
239C2×Q8—84no—
240C2×Q8—88no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—48no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4C22⋊Q812yes—
265C22×C4C22⋊Q812yes—
266C22×C4#12412yes—
267C22×C4#12412yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×Q811yescharacteristic
271C22×C4C22×Q812yes—
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4H(5,𝔽2)12yes—
277C22×C42−1+412yes—
278M16—24no—
279M16—24no—
280M16—42no—
281M16—42no—
282M16—44no—
283M16—44no—
284C2×C8—24no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C4⋊C4#12414yes—
311C4⋊C4C4⋊Q814yes—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—44no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—82no—
324C42—82no—
325C42C22⋊Q812yes—
326C42C22⋊Q812yes—
327C42C22⋊Q812yes—
328C42C22⋊Q812yes—
329C42#12412yes—
330C42#12412yes—
331C42#12412yes—
332C42#12412yes—
333C23—22no—
334C23—22no—
335C23C2×#12011yescharacteristic
336C23C2×C22⋊Q811yescharacteristic
337C23C2×#12411yescharacteristic
338C23#54612yes—
339C23D4×Q812yes—
340Q8—88no—
341Q8—88no—
342Q8—168no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—84no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—88no—
376C2×C4#39312yes—
377C2×C4#39312yes—
378C2×C4#39312yes—
379C2×C4#39312yes—
380C2×C4C2×C22⋊Q811yescharacteristic
381C2×C4C2×C22⋊Q811yescharacteristic
382C2×C4C2×#12411yescharacteristic
383C2×C4C2×#12411yescharacteristic
384C2×C4C2×#12411yescharacteristic
385C2×C4C2×C4⋊Q811yescharacteristic
386C2×C4#53312yes—
387C2×C4#53312yes—
388C2×C4#53511yescharacteristic
389C2×C4#53611yescharacteristic
390C2×C4#54212yes—
391C2×C4#54212yes—
392C2×C4#54612yes—
393C2×C4D4×Q812yes—
394C2×C4#55412yes—
395C2×C4#55612yes—
396C2×C4#55712yes—
397C2×C4Q8212yes—
398C8—44no—
399C8—82no—
400C22—24no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—24no—
406C22C2×#39311yescharacteristic
407C22#243411yescharacteristic
408C22#243511yescharacteristic
409C22#245112yes—
410C22#249212yes—
411C22#302112yes—
412C22#302212yes—
413C4—22no—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—44no—
418C4—44no—
419C4—84no—
420C4—82no—
421C4—84no—
422C4—82no—
423C4—84no—
424C4#245112yes—
425C4#249212yes—
426C4#256711yescharacteristic
427C4#256811yescharacteristic
428C4#264011yescharacteristic
429C4#286211yescharacteristic
430C2—22no—
431C2—22no—
432C2#1519111yescharacteristic
433C2#2346311yescharacteristic
434C2#2346411yescharacteristic
435C2#2583312yes—
436C2#2588012yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
55. C2×C4⋊Q8Q8semidirect
119. #339Q8semidirect

Maximal subgroups

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

subgroupindex
#353302
#213242
#213132
#142822
#113922
#112762
#108582
#108402
#107012