Finite Group Databasebeta

← #142191

Subgroups — #142191

Order 512.

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

normal subgroups only

This group has 812 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#15178C211yesmaximal, characteristic
2#10801C211yesmaximal, characteristic
3#8559C211yesmaximal, characteristic
4#7250C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic, complemented
6C22×C22⋊Q8—21no—
7C2×#391—21no—
8C2×C4×C22⋊C4C2211yescharacteristic
9#1797—21no—
10C2×#342—21no—
11C2×#342C2212yes—
12C2×#323C2212yes—
13#1444—21no—
14#1444C412yescomplemented
15#1439—21no—
16#1439C2212yes—
17C4⋊#95C411yescharacteristic, complemented
18C22⋊#95—21no—
19#1280—21no—
20C23×Q8—21no—
21C24×C4D411yescharacteristic
22C2×C4⋊Q8—41no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—44no—
25C2×C22⋊Q8—44no—
26C2×C22⋊Q8—44no—
27C2×C22⋊Q8—44no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C2×C412yes—
32C22×C22⋊C4—22no—
33C22×C22⋊C4—22no—
34C22×C22⋊C4D411yescharacteristic
35C22×C22⋊C4C2311yescharacteristic
36C22×C42C2×C411yescharacteristic
37C22×C42D411yescharacteristic
38C2×C4⋊C8—41no—
39C2×C22⋊C8—22no—
40#391—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#342Q814yescomplemented
54#336—41no—
55#323—24no—
56C25C2×D411yescharacteristic
57C22×Q8—24no—
58C22×Q8—42no—
59C22×Q8—42no—
60C22×Q8—42no—
61C22×Q8—42no—
62C22×Q8—42no—
63C22×Q8—41no—
64C22×Q8—41no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—41no—
71C23×C4—42no—
72C23×C4—42no—
73C23×C4—42no—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C22×C8—42no—
81C4⋊Q8—44no—
82C4⋊Q8—48no—
83C4⋊Q8—44no—
84C22⋊Q8—432no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—44no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—44no—
101C2×C22⋊C4—44no—
102C2×C22⋊C4—44no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4SD1612yes—
109C2×C22⋊C4Q1612yes—
110C2×C22⋊C4C2×Q812yes—
111C2×C22⋊C4#4112yes—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—42no—
117C2×C42—42no—
118C4⋊C8—82no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121#95—41no—
122#95—41no—
123#95—42no—
124#95—42no—
125#95—44no—
126#95—44no—
127#95—82no—
128#95—82no—
129C24—21no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—41no—
134C24—41no—
135C24—41no—
136C24—41no—
137C24C2×C22⋊C411yescharacteristic
138C24C22⋊Q812yes—
139C24C2×SD1611yescharacteristic
140C24C2×Q1611yescharacteristic
141C24AGL(1,ℤ/8)12yes—
142C2×Q8—48no—
143C2×Q8—44no—
144C2×Q8—48no—
145C2×Q8—44no—
146C2×Q8—84no—
147C2×Q8—84no—
148C2×Q8—82no—
149C2×Q8—82no—
150C2×Q8—82no—
151C2×Q8—82no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—48no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—84no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4#9911yescharacteristic
207C22×C4#9911yescharacteristic
208C22×C4#10312yes—
209C22×C4#10312yes—
210C22×C4C4×Q812yes—
211C22×C4#12411yescharacteristic
212C22×C4C4⋊Q811yescharacteristic
213C2×C8—82no—
214C2×C8—82no—
215C2×C8—82no—
216C4⋊C4—48no—
217C4⋊C4—44no—
218C4⋊C4—416no—
219C4⋊C4—44no—
220C4⋊C4—48no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—416no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C42—42no—
231C42—42no—
232C42—42no—
233C42—81no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—21no—
241C23—21no—
242C23—22no—
243C23—22no—
244C23—21no—
245C23—21no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—42no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23#39111yescharacteristic
265C23C2×#9911yescharacteristic
266C23#41011yescharacteristic
267C23C2×#10311yescharacteristic
268C23#41811yescharacteristic
269C23#42111yescharacteristic
270C23#42111yescharacteristic
271C23#47612yes—
272C23#47712yes—
273Q8—88no—
274Q8—88no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—162no—
328C2×C4—162no—
329C2×C4—162no—
330C2×C4—162no—
331C2×C4AΣL(1,𝔽16)12yes—
332C2×C4#35312yes—
333C8—162no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—22no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—81no—
361C22—81no—
362C22#151011yescharacteristic
363C22#151411yescharacteristic
364C22#151811yescharacteristic
365C22#152011yescharacteristic
366C22#198711yescharacteristic
367C22#199511yescharacteristic
368C22#199611yescharacteristic
369C22C2×AΣL(1,𝔽16)11yescharacteristic
370C22#212311yescharacteristic
371C22C2×#35311yescharacteristic
372C22#212911yescharacteristic
373C4—22no—
374C4—22no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—41no—
379C4—41no—
380C4—84no—
381C4—81no—
382C4—84no—
383C4—164no—
384C2—21no—
385C2—21no—
386C2—21no—
387C2—21no—
388C2—42no—
389C2—42no—
390C2#991111yescharacteristic
391C2#1179611yescharacteristic
392C2#1180111yescharacteristic
393C2#1189811yescharacteristic
394C2#1190111yescharacteristic
395C2#1329611yescharacteristic
396C2#1329811yescharacteristic
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
14. #1444C4semidirect
17. C4⋊#95C4semidirect
53. #342Q8semidirect

Maximal subgroups

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

subgroupindex
#151782
#108012
#85592
#72502