Finite Group Databasebeta

← #100412

Subgroups — #100412

Order 512.

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

normal subgroups only

This group has 556 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#33146C211yesmaximal, characteristic
2#13918C211yesmaximal, characteristic
3#9884C211yesmaximal, characteristic
4#9711C211yesmaximal, characteristic
5#8706C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C2×#389—21no—
10C2×#382—21no—
11C2×C4×C4⋊C4C2211yescharacteristic
12#1988C412yescomplemented
13#1947C412yescomplemented
14#1918—22no—
15#1852—21no—
16#1823—21no—
17#1797C2211yescharacteristic
18#1797C2211yescharacteristic
19C4×C22⋊C8C2211yescharacteristic
20C2×#328—21no—
21C2×#328—21no—
22#1442—22no—
23C4⋊#95—21no—
24C4⋊#95C2211yescharacteristic
25#1434—21no—
26C4×#95C2211yescharacteristic
27#1294—22no—
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#382—42no—
52#382—42no—
53#380—21no—
54#380—21no—
55C4×C4⋊C4—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4C2×C412yes—
58C4×C4⋊C4D412yes—
59C2×#95—21no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95D411yescharacteristic
66C2×#95D411yescharacteristic
67C2×#95D412yes—
68#340—22no—
69#340—22no—
70#340—22no—
71#336—21no—
72#336—21no—
73#336—21no—
74#328—42no—
75#328—42no—
76C22×Q8—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—22no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C22×C8—22no—
86C22×C8—41no—
87C22×C8—41no—
88C4⋊Q8—42no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91C22⋊Q8—41no—
92C22⋊Q8—41no—
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—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—82no—
111C2×C4⋊C4C22⋊C412yes—
112C2×C4⋊C4C22⋊C412yes—
113C2×C4⋊C4SD1612yes—
114C2×C4⋊C4Q1612yes—
115C2×C22⋊C4—41no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—42no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42C22×C411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42#4111yescharacteristic
130C2×C42#4111yescharacteristic
131#107—44no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134C22⋊C8—22no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C4×C8—42no—
141#95—22no—
142#95—22no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—42no—
147#95—42no—
148#95—44no—
149#95—44no—
150#95—44no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95D812yes—
156C24#12011yescharacteristic
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—81no—
160C2×Q8—81no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—41no—
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—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4C2×C22⋊C411yescharacteristic
205C22×C4C4×D411yescharacteristic
206C22×C4C4×D411yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12311yescharacteristic
210C22×C4C2×D811yescharacteristic
211C22×C4C2×SD1611yescharacteristic
212C22×C4C2×Q1611yescharacteristic
213C22×C4#13511yescharacteristic
214C22×C4AGL(1,ℤ/8)11yescharacteristic
215C22×C4AGL(1,ℤ/8)11yescharacteristic
216C22×C4AGL(1,ℤ/8)11yescharacteristic
217C22×C4#13711yescharacteristic
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—81no—
225C2×C8—81no—
226C2×C8—81no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—88no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4#10314yes—
245C22⋊C4—81no—
246C22⋊C4—81no—
247C42—22no—
248C42—22no—
249C42—41no—
250C42—41no—
251C42—42no—
252C42—42no—
253C42—42no—
254C42—42no—
255C42—82no—
256C42—82no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23#38611yescharacteristic
262C23#44711yescharacteristic
263C23#44911yescharacteristic
264C23#45011yescharacteristic
265C23#45111yescharacteristic
266C23#45511yescharacteristic
267C23#45611yescharacteristic
268Q8—161no—
269Q8—161no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4#41012yes—
332C2×C4C2×#10311yescharacteristic
333C2×C4#41711yescharacteristic
334C2×C4#42112yes—
335C2×C4C4×D811yescharacteristic
336C2×C4#44211yescharacteristic
337C2×C4#45911yescharacteristic
338C2×C4#46311yescharacteristic
339C2×C4#48311yescharacteristic
340C2×C4#48411yescharacteristic
341C8—82no—
342C8—82no—
343C8—161no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22#163911yescharacteristic
357C22#164411yescharacteristic
358C22#164911yescharacteristic
359C22#165411yescharacteristic
360C22#187211yescharacteristic
361C22#187911yescharacteristic
362C22#188211yescharacteristic
363C22#189711yescharacteristic
364C22#189911yescharacteristic
365C22#190511yescharacteristic
366C22#190611yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—44no—
374C4—41no—
375C4—41no—
376C4—82no—
377C4—82no—
378C4—82no—
379C4—162no—
380C4—161no—
381C4—161no—
382C4#151212yes—
383C4#151812yes—
384C2—21no—
385C2—21no—
386C2—21no—
387C2—21no—
388C2#898111yescharacteristic
389C2#915211yescharacteristic
390C2#917611yescharacteristic
391C2#923711yescharacteristic
392C2#926011yescharacteristic
393C2#932811yescharacteristic
394C2#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
12. #1988C4semidirect
13. #1947C4semidirect

Maximal subgroups

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

subgroupindex
#331462
#139182
#98842
#97112
#87062
#70392
#70282