Finite Group Databasebeta

← #100400

Subgroups — #100400

Order 512.

542 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 112 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#33149C211yesmaximal, characteristic
2#13933C211yesmaximal, characteristic
3#9908C211yesmaximal, characteristic
4#9679C211yesmaximal, characteristic
5#8707C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C2×#389—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1987C412yescomplemented
12#1947C412yescomplemented
13#1921—22no—
14#1852—21no—
15#1824—21no—
16#1797C2211yescharacteristic
17#1797C2211yescharacteristic
18C4×C22⋊C8C2211yescharacteristic
19C2×#328—22no—
20#1442—22no—
21C4⋊#95—21no—
22C4⋊#95—21no—
23C4⋊#95C2211yescharacteristic
24#1434—21no—
25C4×#95C2211yescharacteristic
26#1294—21no—
27#1294—21no—
28#1279—22no—
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×C22⋊Q8—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4D411yescharacteristic
34C22×C4⋊C4D411yescharacteristic
35C22×C42—21no—
36C22×C42—21no—
37C22×C42C2311yescharacteristic
38C2×#107—22no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8C2×C412yes—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C4×C8—21no—
45#391—21no—
46#389—21no—
47#389—21no—
48#389—42no—
49#389—42no—
50#384C2×C411yescharacteristic
51#380—21no—
52#380—21no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4C2×C412yes—
56C4×C4⋊C4D412yes—
57C2×#95—21no—
58C2×#95—22no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95D411yescharacteristic
64C2×#95D411yescharacteristic
65C2×#95D412yes—
66#340—22no—
67#340—22no—
68#340—22no—
69#336—21no—
70#336—21no—
71#336—21no—
72#328—44no—
73C22×Q8—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—22no—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85C4⋊Q8—42no—
86C22⋊Q8—42no—
87C22⋊Q8—42no—
88C2×C4⋊C4—21no—
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—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—44no—
101C2×C4⋊C4—44no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—82no—
106C2×C4⋊C4C22⋊C412yes—
107C2×C4⋊C4C22⋊C412yes—
108C2×C4⋊C4SD1612yes—
109C2×C4⋊C4Q1612yes—
110C2×C22⋊C4—41no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42—41no—
117C2×C42—41no—
118C2×C42—82no—
119C2×C42C22×C411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42#4111yescharacteristic
123#107—44no—
124C4⋊C8—42no—
125C4⋊C8—42no—
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#95Q1612yes—
148C24#12011yescharacteristic
149C2×Q8—42no—
150C2×Q8—81no—
151C2×Q8—81no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4C4×D411yescharacteristic
195C22×C4C4×D411yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12311yescharacteristic
199C22×C4C2×SD1611yescharacteristic
200C22×C4C2×Q1611yescharacteristic
201C22×C4C2×Q1611yescharacteristic
202C22×C4#13511yescharacteristic
203C22×C4AGL(1,ℤ/8)11yescharacteristic
204C22×C4#13711yescharacteristic
205C22×C4#13711yescharacteristic
206C22×C4#13711yescharacteristic
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—81no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—81no—
215C2×C8—81no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—88no—
231C4⋊C4#10314yes—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C42—22no—
235C42—22no—
236C42—41no—
237C42—41no—
238C42—82no—
239C42—82no—
240C42—82no—
241C42—82no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23#38611yescharacteristic
247C23#45011yescharacteristic
248C23#45111yescharacteristic
249C23#45111yescharacteristic
250C23#45211yescharacteristic
251C23#45311yescharacteristic
252C23#45411yescharacteristic
253Q8—162no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
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—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4#41012yes—
314C2×C4C2×#10311yescharacteristic
315C2×C4#41711yescharacteristic
316C2×C4#42112yes—
317C2×C4C4×Q1611yescharacteristic
318C2×C4#44111yescharacteristic
319C2×C4#46211yescharacteristic
320C2×C4#46411yescharacteristic
321C2×C4#48311yescharacteristic
322C2×C4#48411yescharacteristic
323C8—82no—
324C8—82no—
325C8—161no—
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#163111yescharacteristic
339C22#163311yescharacteristic
340C22#163611yescharacteristic
341C22#163711yescharacteristic
342C22#187211yescharacteristic
343C22#187811yescharacteristic
344C22#188311yescharacteristic
345C22#189711yescharacteristic
346C22#189911yescharacteristic
347C22#190411yescharacteristic
348C22#190611yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—44no—
358C4—82no—
359C4—82no—
360C4—82no—
361C4—162no—
362C4—161no—
363C4—161no—
364C4#151212yes—
365C4#151812yes—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#897511yescharacteristic
371C2#915211yescharacteristic
372C2#917511yescharacteristic
373C2#923611yescharacteristic
374C2#925711yescharacteristic
375C2#932911yescharacteristic
376C2#936611yescharacteristic
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. #1987C4semidirect
12. #1947C4semidirect

Maximal subgroups

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

subgroupindex
#331492
#139332
#99082
#96792
#87072
#70392
#70282