Finite Group Databasebeta

← #141157

Subgroups — #141157

Order 512.

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

normal subgroups only

This group has 670 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#15135C211yesmaximal, characteristic
2C2×#1352C214yesmaximal
3#10801C211yesmaximal, characteristic
4#8559C211yesmaximal, characteristic
5C2×#399C2212yes—
6C2×#389—21no—
7C2×#384C411yescharacteristic, complemented
8C2×#380—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1445C412yescomplemented
15#1444—21no—
16#1439—21no—
17C4⋊#95C411yescharacteristic, complemented
18C22⋊#95—21no—
19#1352—28no—
20#1280—21no—
21C24×C4D411yescharacteristic
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C412yes—
27C22×C22⋊C4—22no—
28C22×C22⋊C4—22no—
29C22×C22⋊C4D411yescharacteristic
30C22×C22⋊C4C2311yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C2×C4⋊C8—41no—
34C2×C22⋊C8—22no—
35#399—24no—
36#399—24no—
37#389—44no—
38#384—41no—
39#384—41no—
40#380—44no—
41C4×C22⋊C4—41no—
42C4×C22⋊C4—41no—
43C2×#95—21no—
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
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4—41no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C22×C8—42no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—82no—
88C2×C4⋊C4—82no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—44no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—44no—
101C2×C22⋊C4—82no—
102C2×C22⋊C4SD1612yes—
103C2×C22⋊C4Q1612yes—
104C2×C22⋊C4C2×D412yes—
105C2×C22⋊C4#4112yes—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—42no—
111C2×C42—42no—
112C4⋊C8—82no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115#95—24no—
116#95—41no—
117#95—41no—
118#95—42no—
119#95—44no—
120#95—44no—
121#95—82no—
122#95—82no—
123C24—21no—
124C24—21no—
125C24—21no—
126C24—21no—
127C24—41no—
128C24—41no—
129C24—41no—
130C24—41no—
131C24C2×C22⋊C411yescharacteristic
132C24#12112yes—
133C24C2×SD1611yescharacteristic
134C24C2×Q1611yescharacteristic
135C24#13712yes—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—24no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—48no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—84no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4#9911yescharacteristic
191C22×C4#9911yescharacteristic
192C22×C4#10312yes—
193C22×C4#10312yes—
194C22×C4C4×D412yes—
195C22×C4#12411yescharacteristic
196C22×C4#12711yescharacteristic
197C2×C8—82no—
198C2×C8—82no—
199C2×C8—82no—
200C4⋊C4—48no—
201C4⋊C4—84no—
202C4⋊C4—82no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C4⋊C4—84no—
206C4⋊C4—82no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C22⋊C4—88no—
214C22⋊C4—82no—
215C22⋊C4—82no—
216C42—42no—
217C42—42no—
218C42—42no—
219C42—81no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—81no—
248C23—81no—
249C23—81no—
250C23#39011yescharacteristic
251C23C2×#9911yescharacteristic
252C23#41011yescharacteristic
253C23C2×#10311yescharacteristic
254C23#41911yescharacteristic
255C23#42111yescharacteristic
256C23#42111yescharacteristic
257C23#46112yes—
258C23#46212yes—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—84no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—162no—
312C2×C4—162no—
313C2×C4—162no—
314C2×C4—162no—
315C2×C4#35212yes—
316C2×C4#35312yes—
317C8—162no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22—82no—
344C22—81no—
345C22—81no—
346C22#151111yescharacteristic
347C22#151511yescharacteristic
348C22#151811yescharacteristic
349C22#152011yescharacteristic
350C22#196311yescharacteristic
351C22#196411yescharacteristic
352C22#198111yescharacteristic
353C22C2×#35211yescharacteristic
354C22#212311yescharacteristic
355C22C2×#35311yescharacteristic
356C22#212811yescharacteristic
357C4—22no—
358C4—22no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—41no—
363C4—41no—
364C4—84no—
365C4—81no—
366C4—164no—
367C4—162no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—21no—
372C2—42no—
373C2—42no—
374C2#971411yescharacteristic
375C2#1180911yescharacteristic
376C2#1181311yescharacteristic
377C2#1189911yescharacteristic
378C2#1190111yescharacteristic
379C2#1322011yescharacteristic
380C2#1322311yescharacteristic
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
7. C2×#384C4semidirect
14. #1445C4semidirect
17. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#151352
C2×#13522
#108012
#85592