Finite Group Databasebeta

← #378798

Subgroups — #378798

Order 512.

820 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2089C211yesmaximal, characteristic
2#21033C212yesmaximal
3C2×#1776C211yesmaximal, characteristic
4#20436C211yesmaximal, characteristic
5#11365C211yesmaximal, characteristic
6#11271C211yesmaximal, characteristic
7#11238C211yesmaximal, characteristic
8#10896C212yesmaximal
9#10827C212yesmaximal
10#10823C211yesmaximal, characteristic
11#10802C212yesmaximal
12C2×#428C2212yes—
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C2×#424C2211yescharacteristic
16C2×#405C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C22×#97C2211yescharacteristic
19#2296C2211yescharacteristic
20C2×C4×C4⋊C4C2212yes—
21#2089C418yescomplemented
22#1921—22no—
23#1920—22no—
24#1919—22no—
25#1918—22no—
26#1776—21no—
27#1776—21no—
28#1776—21no—
29#1776—21no—
30#1771—21no—
31#1771—21no—
32#1770—21no—
33#1770—21no—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#341C2212yes—
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41#1736C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#340C2212yes—
45#1729C2211yescharacteristic
46#1728C2212yes—
47#1728C2211yescharacteristic
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C42—22no—
59C22×C42C2311yescharacteristic
60#428—24no—
61#428C2×C418yes—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107—22no—
65C2×#107—22no—
66C2×#107C2×C414yes—
67C2×#106—21no—
68C2×#106—21no—
69C2×#106—22no—
70C2×#106—24no—
71#424—22no—
72#424C2×C414yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75#405—22no—
76#405—22no—
77#405—24no—
78C2×#97—21no—
79C2×#97—24no—
80C2×#97—21no—
81C2×#97D411yescharacteristic
82C2×#97D411yescharacteristic
83C2×#97D411yescharacteristic
84C2×#97D411yescharacteristic
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#389D412yes—
88#389D412yes—
89#385—22no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C2×#95—22no—
97#343—24no—
98#343—24no—
99#343—24no—
100#343C2×C418yes—
101#341—22no—
102#341—24no—
103#341C2×C414yes—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#336—21no—
113#336—21no—
114#336—22no—
115#336—22no—
116#336—21no—
117#336—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4#4112yes—
121C23×C4C2411yescharacteristic
122C2×M16—21no—
123C2×M16—21no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
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×M16C22×C414yes—
140C2×M16C22×C412yes—
141C2×M16#4112yes—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8C22×C412yes—
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8#4112yes—
157C22×C8#4114yes—
158#117—21no—
159#117—21no—
160#117—22no—
161#117—22no—
162#117C22×C412yes—
163#117#4112yes—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
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⋊C4C22×C414yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4#4114yes—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—84no—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193#107—24no—
194#107—24no—
195#107—28no—
196#106—24no—
197#106—24no—
198#106—28no—
199C4⋊C8—24no—
200C4⋊C8—24no—
201#97—24no—
202#97—22no—
203#97—22no—
204#97—22no—
205#97—22no—
206#97—42no—
207C4×C8—24no—
208C4×C8—42no—
209#95—22no—
210#95—22no—
211#95—42no—
212#95—42no—
213C24C2×#4111yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4#12112yes—
231C22×C4#12112yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C2×#4111yescharacteristic
236C22×C4C2×#4111yescharacteristic
237C22×C4C2×#4112yes—
238C22×C4C2×#4112yes—
239C22×C4C2×#4111yescharacteristic
240C22×C42−1+412yes—
241C22×C42−1+412yes—
242M16—24no—
243M16—24no—
244M16—24no—
245M16—24no—
246M16—42no—
247M16—42no—
248M16—42no—
249M16—42no—
250M16—42no—
251M16—42no—
252M16—42no—
253M16—42no—
254C2×C8—24no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—24no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—22no—
267C2×C8—24no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8C4×D412yes—
279C2×C8C4×D412yes—
280C2×C8#12312yes—
281C2×C8#12312yes—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4C4×D412yes—
296C4⋊C4C4×D412yes—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—42no—
300C42—22no—
301C42—21no—
302C42—21no—
303C42—22no—
304C42—21no—
305C42—21no—
306C42—24no—
307C42—41no—
308C42—41no—
309C42—84no—
310C42—84no—
311C23—22no—
312C23—22no—
313C23C4×#4111yescharacteristic
314C23C2×#12111yescharacteristic
315C23#52911yescharacteristic
316C23#55212yes—
317C23#55212yes—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—48no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4C2×C4×D411yescharacteristic
343C2×C4C2×C4×D411yescharacteristic
344C2×C4C4×#4112yes—
345C2×C4#51912yes—
346C2×C4C2×#12311yescharacteristic
347C2×C4#52511yescharacteristic
348C2×C4#52511yescharacteristic
349C2×C4#52912yes—
350C2×C4#53211yescharacteristic
351C2×C4#53611yescharacteristic
352C2×C4#53611yescharacteristic
353C2×C4#54112yes—
354C2×C4#54712yes—
355C2×C4#54812yes—
356C2×C4#55212yes—
357C2×C4#55212yes—
358C2×C4#55212yes—
359C2×C4#55612yes—
360C8—22no—
361C8—22no—
362C8—22no—
363C8—22no—
364C8—42no—
365C8—44no—
366C8—42no—
367C8—82no—
368C22—24no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22C4×#12111yescharacteristic
376C22#237112yes—
377C22#248611yescharacteristic
378C22#252412yes—
379C22#253611yescharacteristic
380C22#306812yes—
381C22#306912yes—
382C4—42no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4—84no—
388C4—84no—
389C4C4×#12311yescharacteristic
390C4#232711yescharacteristic
391C4#236212yes—
392C4#248411yescharacteristic
393C4#251912yes—
394C4#259911yescharacteristic
395C2—22no—
396C2—22no—
397C2#1422311yescharacteristic
398C2#2155511yescharacteristic
399C2#2155611yescharacteristic
400C2#2767612yes—
401C2#2822112yes—
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
21. #2089C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20892
#210332
C2×#17762
#204362
#113652
#112712
#112382
#108962
#108272
#108232
#108022