Finite Group Databasebeta

← #141158

Subgroups — #141158

Order 512.

726 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15125C211yesmaximal, characteristic
2C2×#1351C214yesmaximal
3#10801C211yesmaximal, characteristic
4#8559C211yesmaximal, characteristic
5C22×C4⋊Q8C411yescharacteristic, complemented
6C2×#393C2212yes—
7C2×#380—21no—
8C2×C4×C22⋊C4C2211yescharacteristic
9#1797—21no—
10C2×#342—21no—
11C2×#342C2212yes—
12C2×#323C2212yes—
13#1446C412yescomplemented
14#1442—21no—
15#1439—21no—
16C4⋊#95—21no—
17C4⋊#95C411yescharacteristic, complemented
18C22⋊#95—21no—
19#1351—28no—
20#1280—21no—
21C23×Q8C2×C412yes—
22C24×C4D411yescharacteristic
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—41no—
25C2×C4⋊Q8—44no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C22⋊C4—22no—
30C22×C22⋊C4—22no—
31C22×C22⋊C4D411yescharacteristic
32C22×C22⋊C4C2311yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C4⋊C8—41no—
36C2×C22⋊C8—22no—
37#393—24no—
38#393—24no—
39#380—44no—
40C4×C22⋊C4—41no—
41C4×C22⋊C4—41no—
42C2×#95—21no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—41no—
47C2×#95—41no—
48C2×#95C2×C412yes—
49C2×#95C2×C411yescharacteristic
50C2×#95D411yescharacteristic
51#342—22no—
52#342—42no—
53#342D414yes—
54#336—41no—
55#323—24no—
56C25C2×D411yescharacteristic
57C22×Q8—24no—
58C22×Q8—44no—
59C22×Q8—42no—
60C22×Q8—42no—
61C22×Q8—82no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4—41no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C22×C8—42no—
78C4⋊Q8—48no—
79C4⋊Q8—44no—
80C4⋊Q8—44no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—44no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—44no—
102C2×C22⋊C4—82no—
103C2×C22⋊C4SD1612yes—
104C2×C22⋊C4Q1612yes—
105C2×C22⋊C4C2×D412yes—
106C2×C22⋊C4#4112yes—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—42no—
112C2×C42—42no—
113C4⋊C8—82no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116#95—24no—
117#95—41no—
118#95—41no—
119#95—42no—
120#95—44no—
121#95—42no—
122#95—42no—
123#95—82no—
124#95—82no—
125#95—82no—
126C24—21no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—41no—
131C24—41no—
132C24—41no—
133C24—41no—
134C24C2×C22⋊C411yescharacteristic
135C24#12112yes—
136C24C2×SD1611yescharacteristic
137C24C2×Q1611yescharacteristic
138C24#13712yes—
139C2×Q8—44no—
140C2×Q8—44no—
141C2×Q8—84no—
142C2×Q8—88no—
143C2×Q8—88no—
144C2×Q8—84no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—24no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—48no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—84no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4—82no—
199C22×C4#9911yescharacteristic
200C22×C4#9911yescharacteristic
201C22×C4#10312yes—
202C22×C4#10312yes—
203C22×C4C4×D412yes—
204C22×C4#12411yescharacteristic
205C22×C4#12711yescharacteristic
206C2×C8—82no—
207C2×C8—82no—
208C2×C8—82no—
209C4⋊C4—44no—
210C4⋊C4—48no—
211C4⋊C4—44no—
212C4⋊C4—84no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—44no—
218C22⋊C4—44no—
219C22⋊C4—88no—
220C22⋊C4—82no—
221C22⋊C4—82no—
222C42—42no—
223C42—42no—
224C42—42no—
225C42—81no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23—81no—
256C23#39011yescharacteristic
257C23C2×#9911yescharacteristic
258C23#41011yescharacteristic
259C23C2×#10311yescharacteristic
260C23#41911yescharacteristic
261C23#42111yescharacteristic
262C23#42111yescharacteristic
263C23#46112yes—
264C23#46212yes—
265Q8—88no—
266Q8—88no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—84no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—82no—
319C2×C4—162no—
320C2×C4—162no—
321C2×C4—162no—
322C2×C4—162no—
323C2×C4AΣL(1,𝔽16)12yes—
324C2×C4#35412yes—
325C8—162no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—42no—
349C22—42no—
350C22—82no—
351C22—82no—
352C22—81no—
353C22—81no—
354C22#151111yescharacteristic
355C22#151511yescharacteristic
356C22#151811yescharacteristic
357C22#152011yescharacteristic
358C22#196311yescharacteristic
359C22#196411yescharacteristic
360C22#198111yescharacteristic
361C22C2×AΣL(1,𝔽16)11yescharacteristic
362C22#212211yescharacteristic
363C22C2×#35411yescharacteristic
364C22#212911yescharacteristic
365C4—22no—
366C4—22no—
367C4—42no—
368C4—42no—
369C4—42no—
370C4—41no—
371C4—41no—
372C4—81no—
373C4—84no—
374C4—162no—
375C4—164no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2#971411yescharacteristic
383C2#1180811yescharacteristic
384C2#1181211yescharacteristic
385C2#1189811yescharacteristic
386C2#1190011yescharacteristic
387C2#1322111yescharacteristic
388C2#1322411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C22×C4⋊Q8C4semidirect
13. #1446C4semidirect
17. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#151252
C2×#13512
#108012
#85592