Finite Group Databasebeta

← #378942

Subgroups — #378942

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2089C211yesmaximal, characteristic
2#21159C212yesmaximal
3C2×#1942C211yesmaximal, characteristic
4#20435C211yesmaximal, characteristic
5#11365C211yesmaximal, characteristic
6#11331C211yesmaximal, characteristic
7#11239C211yesmaximal, characteristic
8#10913C212yesmaximal
9#10896C212yesmaximal
10#10896C212yesmaximal
11C4×#341C211yesmaximal, characteristic
12C2×#428C2212yes—
13C2×#428C2212yes—
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C2×#424C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C2×C4×M16C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2296C2211yescharacteristic
22C2×#389C2212yes—
23#2089C418yescomplemented
24#1944—22no—
25#1943—22no—
26#1942—21no—
27#1942—21no—
28#1942—21no—
29#1942—21no—
30#1942—22no—
31#1941—22no—
32#1771—21no—
33#1771—21no—
34C8×C4⋊C4—21no—
35C8×C4⋊C4—21no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#341C2211yescharacteristic
41C2×#341C2211yescharacteristic
42C2×#341C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45C2×#340C2212yes—
46#1729C2211yescharacteristic
47#1728C2211yescharacteristic
48C22×M16C2311yescharacteristic
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C22×M16C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C42C2311yescharacteristic
60#428—24no—
61#428—24no—
62#428—42no—
63#428C2×C418yes—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—22no—
71C2×#107—42no—
72C2×#107C2×C414yes—
73C2×#106—22no—
74C2×#106—22no—
75#424—22no—
76#424C2×C414yes—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79#405—22no—
80#405—22no—
81#405—42no—
82C4×M16—24no—
83C4×M16—42no—
84C2×#97—21no—
85C2×#97—21no—
86C2×#97—22no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—22no—
90C2×C4×C8—42no—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389D411yescharacteristic
100#389D411yescharacteristic
101#389D411yescharacteristic
102#389D411yescharacteristic
103#385—21no—
104#385—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C2×#95—22no—
108#343—24no—
109#343—24no—
110#343—42no—
111#343C2×C418yes—
112#341—22no—
113#341C2×C414yes—
114#341Q818yes—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#336—21no—
122#336—21no—
123#336—21no—
124#336—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4#4111yescharacteristic
128C23×C4#4111yescharacteristic
129C23×C4C2411yescharacteristic
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16C22×C412yes—
145C2×M16C22×C414yes—
146C2×M16C2×Q814yes—
147C2×M16C2×Q814yes—
148C2×M16#4112yes—
149C2×M16#4114yes—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8C22×C412yes—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×Q814yes—
165C22×C8#4112yes—
166#117—21no—
167#117—21no—
168#117—22no—
169#117—22no—
170#117C22×C412yes—
171#117#4112yes—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—84no—
190C2×C4⋊C4C22×C414yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205#107—24no—
206#107—24no—
207#107—28no—
208#107—44no—
209#107—42no—
210#107—42no—
211#106—28no—
212#106—44no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215#97—24no—
216#97—44no—
217#97—42no—
218C4×C8—24no—
219C4×C8—41no—
220C4×C8—41no—
221C4×C8—41no—
222C4×C8—41no—
223C4×C8—44no—
224C4×C8—42no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—42no—
230#95—42no—
231C24C2×#4111yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4C23×C411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×Q812yes—
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4112yes—
263C22×C4C2×#4111yescharacteristic
264C22×C4H(5,𝔽2)12yes—
265C22×C42−1+412yes—
266M16—24no—
267M16—24no—
268M16—24no—
269M16—24no—
270M16—24no—
271M16—24no—
272M16—24no—
273M16—42no—
274M16—42no—
275M16—42no—
276M16—42no—
277M16C4×Q818yes—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8C4×D412yes—
303C2×C8C4×D412yes—
304C2×C8C4⋊Q812yes—
305C2×C8C4⋊Q812yes—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C4⋊C4C4×D412yes—
325C4⋊C4C4×D412yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—42no—
329C42—22no—
330C42—22no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—22no—
336C42—22no—
337C42—41no—
338C42—41no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23C4×#4111yescharacteristic
344C23C2×#12111yescharacteristic
345C23#52911yescharacteristic
346C23#55312yes—
347C23#55712yes—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—84no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—162no—
381C2×C4—162no—
382C2×C4#47212yes—
383C2×C4#47312yes—
384C2×C4C2×C4×D411yescharacteristic
385C2×C4C2×C4×D411yescharacteristic
386C2×C4C2×C4×Q812yes—
387C2×C4#52012yes—
388C2×C4#52511yescharacteristic
389C2×C4C2×C4⋊Q811yescharacteristic
390C2×C4#53211yescharacteristic
391C2×C4#53211yescharacteristic
392C2×C4#53312yes—
393C2×C4#53411yescharacteristic
394C2×C4#53611yescharacteristic
395C2×C4#53912yes—
396C2×C4#54612yes—
397C2×C4D4×Q812yes—
398C2×C4#55212yes—
399C2×C4#55212yes—
400C2×C4#55412yes—
401C2×C4#55412yes—
402C8—22no—
403C8—22no—
404C8—22no—
405C8—22no—
406C8—28no—
407C8—42no—
408C8—42no—
409C8—44no—
410C22—22no—
411C22—22no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22C4×#12111yescharacteristic
419C22#237012yes—
420C22#254311yescharacteristic
421C22#254611yescharacteristic
422C22#258512yes—
423C22C2×#47211yescharacteristic
424C22C2×#47311yescharacteristic
425C22#306711yescharacteristic
426C22#306711yescharacteristic
427C4—22no—
428C4—22no—
429C4—42no—
430C4—42no—
431C4—41no—
432C4—41no—
433C4—84no—
434C4—81no—
435C4—81no—
436C4—162no—
437C4C4×C4⋊Q811yescharacteristic
438C4#232211yescharacteristic
439C4#236012yes—
440C4#255612yes—
441C4#258311yescharacteristic
442C4#261111yescharacteristic
443C2—22no—
444C2—22no—
445C2#1422611yescharacteristic
446C2C4×#47211yescharacteristic
447C2C4×#47311yescharacteristic
448C2#2859111yescharacteristic
449C2#2859211yescharacteristic
450C2#2869911yescharacteristic
451C2#2870111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×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
23. #2089C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20892
#211592
C2×#19422
#204352
#113652
#113312
#112392
#109132
#108962
C4×#3412