Finite Group Databasebeta

← #100421

Subgroups — #100421

Order 512.

516 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 104 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#33109C211yesmaximal, characteristic
2#13936C211yesmaximal, characteristic
3#9909C211yesmaximal, characteristic
4#9885C211yesmaximal, characteristic
5#8707C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C2×#484—21no—
9#2857C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1981C412yescomplemented
12#1947C412yescomplemented
13#1928—22no—
14#1872—21no—
15#1797C2211yescharacteristic
16#1797C2211yescharacteristic
17C4×C22⋊C8C2211yescharacteristic
18#1443—22no—
19C4⋊#95—21no—
20C4⋊#95—21no—
21C4⋊#95C2211yescharacteristic
22#1434—21no—
23#1434—21no—
24C4×#95C2211yescharacteristic
25#1294—22no—
26#1294—21no—
27#1294—21no—
28#1279—22no—
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×C22⋊Q8—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
34C22×C42—21no—
35C22×C42—21no—
36C22×C42C2311yescharacteristic
37#484—42no—
38#484—42no—
39C2×#107—22no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8C2×C412yes—
42C2×#103—22no—
43C2×#103—22no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C4×C8—21no—
48#391—21no—
49#389—21no—
50#389—21no—
51#384C2×C411yescharacteristic
52#380—21no—
53#380—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4—21no—
56C4×C4⋊C4C2×C412yes—
57C4×C4⋊C4D412yes—
58C2×#95—21no—
59C2×#95—22no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95D411yescharacteristic
66C2×#95D411yescharacteristic
67C2×#95D412yes—
68#340—22no—
69#336—21no—
70#336—21no—
71#336—21no—
72C22×Q8—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—22no—
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—41no—
84C4⋊Q8—42no—
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87C22⋊Q8—41no—
88C22⋊Q8—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C22⋊C412yes—
107C2×C4⋊C4C22⋊C412yes—
108C2×C22⋊C4—41no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—42no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42—41no—
115C2×C42—41no—
116C2×C42—82no—
117C2×C42C22×C411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121#107—44no—
122C4⋊C8—42no—
123C4⋊C8—42no—
124#103—44no—
125#103—44no—
126C22⋊C8—22no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C4×C8—42no—
133#95—22no—
134#95—22no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—42no—
139#95—42no—
140#95—44no—
141#95—41no—
142#95—41no—
143#95—41no—
144#95—41no—
145#95—82no—
146#95—82no—
147#95—82no—
148#95Q1612yes—
149C24#12011yescharacteristic
150C2×Q8—42no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—42no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4C2×C22⋊C411yescharacteristic
196C22×C4C4×D411yescharacteristic
197C22×C4C4×D411yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12311yescharacteristic
201C22×C4C2×Q1611yescharacteristic
202C22×C4#13511yescharacteristic
203C22×C4#13511yescharacteristic
204C22×C4#13511yescharacteristic
205C22×C4AGL(1,ℤ/8)11yescharacteristic
206C22×C4AGL(1,ℤ/8)11yescharacteristic
207C22×C4#13711yescharacteristic
208C22×C4#13711yescharacteristic
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—81no—
214C2×C8—81no—
215C2×C8—81no—
216C2×C8—81no—
217C2×C8—81no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C42—22no—
238C42—22no—
239C42—41no—
240C42—41no—
241C42—82no—
242C42—82no—
243C42—82no—
244C42—82no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23#38611yescharacteristic
250C23#44911yescharacteristic
251C23#44911yescharacteristic
252C23#45111yescharacteristic
253C23#45211yescharacteristic
254C23#45511yescharacteristic
255C23#45611yescharacteristic
256Q8—82no—
257Q8—82no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4#41012yes—
318C2×C4#41611yescharacteristic
319C2×C4#41811yescharacteristic
320C2×C4#42112yes—
321C2×C4C4×Q1611yescharacteristic
322C2×C4#44111yescharacteristic
323C2×C4#46211yescharacteristic
324C2×C4#46411yescharacteristic
325C2×C4#48211yescharacteristic
326C2×C4#48211yescharacteristic
327C8—82no—
328C8—82no—
329C8—161no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22#164011yescharacteristic
343C22#164411yescharacteristic
344C22#164811yescharacteristic
345C22#165211yescharacteristic
346C22#187511yescharacteristic
347C22#187811yescharacteristic
348C22#188211yescharacteristic
349C22#189711yescharacteristic
350C22#189911yescharacteristic
351C22#190411yescharacteristic
352C22#190611yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—82no—
362C4—82no—
363C4—82no—
364C4—82no—
365C4—162no—
366C4—161no—
367C4—161no—
368C4#151412yes—
369C4#152112yes—
370C2—21no—
371C2—21no—
372C2—21no—
373C2—21no—
374C2#898611yescharacteristic
375C2#915711yescharacteristic
376C2#918111yescharacteristic
377C2#923611yescharacteristic
378C2#925711yescharacteristic
379C2#932711yescharacteristic
380C2#936311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #1981C4semidirect
12. #1947C4semidirect

Maximal subgroups

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

subgroupindex
#331092
#139362
#99092
#98852
#87072
#70392
#70282