Finite Group Databasebeta

← #378945

Subgroups — #378945

Order 512.

846 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2089C211yesmaximal, characteristic
2#21159C212yesmaximal
3C2×#1943C211yesmaximal, characteristic
4#20436C211yesmaximal, characteristic
5#11365C211yesmaximal, characteristic
6#11330C211yesmaximal, characteristic
7#11238C211yesmaximal, characteristic
8#10913C212yesmaximal
9#10896C212yesmaximal
10#10896C212yesmaximal
11#10705C211yesmaximal, characteristic
12C2×#428C2212yes—
13C2×#428C2212yes—
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#424C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C2×C4×M16C2212yes—
20C22×#97C2211yescharacteristic
21#2296C2211yescharacteristic
22C2×#389C2212yes—
23#2089C418yescomplemented
24#1944—22no—
25#1943—21no—
26#1943—21no—
27#1943—21no—
28#1943—21no—
29#1943—22no—
30#1942—22no—
31#1941—22no—
32#1771—21no—
33#1771—21no—
34#1770—21no—
35#1770—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—22no—
69C2×#107C2×C414yes—
70C2×#106—22no—
71C2×#106—21no—
72C2×#106—21no—
73C2×#106—22no—
74C2×#106—42no—
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×#97—42no—
88C2×#97D411yescharacteristic
89C2×#97D411yescharacteristic
90C2×#97D411yescharacteristic
91C2×#97D411yescharacteristic
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389D412yes—
100#389D412yes—
101#385—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C2×#95—22no—
105#343—24no—
106#343—24no—
107#343—42no—
108#343C2×C418yes—
109#341—22no—
110#341C2×C414yes—
111#341Q818yes—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#336—21no—
119#336—21no—
120#336—21no—
121#336—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4#4112yes—
125C23×C4C2411yescharacteristic
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—21no—
130C2×M16—21no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16C22×C412yes—
141C2×M16C22×C414yes—
142C2×M16C2×Q814yes—
143C2×M16C2×Q814yes—
144C2×M16#4112yes—
145C2×M16#4114yes—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8C22×C412yes—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×Q814yes—
161C22×C8#4112yes—
162#117—21no—
163#117—21no—
164#117—22no—
165#117—22no—
166#117C22×C412yes—
167#117#4112yes—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—84no—
184C2×C4⋊C4C22×C414yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D412yes—
197#107—24no—
198#107—24no—
199#107—28no—
200#107—44no—
201#106—28no—
202#106—44no—
203#106—42no—
204#106—42no—
205C4⋊C8—24no—
206C4⋊C8—24no—
207#97—24no—
208#97—41no—
209#97—41no—
210#97—41no—
211#97—41no—
212#97—44no—
213#97—42no—
214C4×C8—24no—
215C4×C8—44no—
216C4×C8—42no—
217#95—22no—
218#95—22no—
219#95—42no—
220#95—42no—
221C24C2×#4111yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4#12112yes—
240C22×C4#12112yes—
241C22×C4C23×C411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C22×Q812yes—
245C22×C4C2×#4111yescharacteristic
246C22×C4C2×#4111yescharacteristic
247C22×C4C2×#4112yes—
248C22×C4C2×#4111yescharacteristic
249C22×C4H(5,𝔽2)12yes—
250C22×C42−1+412yes—
251M16—24no—
252M16—24no—
253M16—24no—
254M16—24no—
255M16—24no—
256M16—24no—
257M16—24no—
258M16—42no—
259M16—42no—
260M16—42no—
261M16—42no—
262M16C4×Q818yes—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8C4×D412yes—
288C2×C8C4×D412yes—
289C2×C8C4⋊Q812yes—
290C2×C8C4⋊Q812yes—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C4⋊C4C4×D412yes—
308C4⋊C4C4×D412yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—42no—
312C42—22no—
313C42—22no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—22no—
319C42—22no—
320C42—41no—
321C42—41no—
322C23—22no—
323C23—22no—
324C23C4×#4111yescharacteristic
325C23C2×#12111yescharacteristic
326C23#52911yescharacteristic
327C23#55312yes—
328C23#55712yes—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—84no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—162no—
353C2×C4—162no—
354C2×C4C2×C4×D411yescharacteristic
355C2×C4C2×C4×D411yescharacteristic
356C2×C4C2×C4×Q812yes—
357C2×C4#52012yes—
358C2×C4#52511yescharacteristic
359C2×C4C2×C4⋊Q811yescharacteristic
360C2×C4#53211yescharacteristic
361C2×C4#53211yescharacteristic
362C2×C4#53312yes—
363C2×C4#53411yescharacteristic
364C2×C4#53611yescharacteristic
365C2×C4#53912yes—
366C2×C4#54612yes—
367C2×C4D4×Q812yes—
368C2×C4#55212yes—
369C2×C4#55212yes—
370C2×C4#55412yes—
371C2×C4#55412yes—
372C8—22no—
373C8—22no—
374C8—22no—
375C8—22no—
376C8—28no—
377C8—42no—
378C8—42no—
379C8—44no—
380C22—24no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22C4×#12111yescharacteristic
388C22#237012yes—
389C22#254311yescharacteristic
390C22#254611yescharacteristic
391C22#258512yes—
392C22#306812yes—
393C22#306912yes—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—84no—
399C4—82no—
400C4—162no—
401C4C4×C4⋊Q811yescharacteristic
402C4#232211yescharacteristic
403C4#236012yes—
404C4#255612yes—
405C4#258311yescharacteristic
406C4#261111yescharacteristic
407C2—22no—
408C2—22no—
409C2#1422611yescharacteristic
410C2#2155511yescharacteristic
411C2#2155611yescharacteristic
412C2#2859612yes—
413C2#2870312yes—
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×#19432
#204362
#113652
#113302
#112382
#109132
#108962
#107052