Finite Group Databasebeta

← #136127

Subgroups — #136127

Order 512.

527 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 88 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#14587C211yesmaximal, characteristic
2C2×#1346C211yesmaximal, characteristic
3#11342C211yesmaximal, characteristic
4#9027C211yesmaximal, characteristic
5#8856C211yesmaximal, characteristic
6#7251C211yesmaximal, characteristic
7#7189C211yesmaximal, characteristic
8C2×#428—21no—
9C2×#398C2211yescharacteristic
10#2316C2211yescharacteristic
11C2×#389—21no—
12#1852—21no—
13#1823—21no—
14C2×#343C2211yescharacteristic
15C2×#339—21no—
16C2×#324C2211yescharacteristic
17C2×#324C2211yescharacteristic
18#1444—21no—
19#1441—21no—
20#1441C2211yescharacteristic
21#1436—21no—
22#1436—21no—
23C4⋊#95C2211yescharacteristic
24#1434—21no—
25#1346C414yescomplemented
26#1294—21no—
27#1294—21no—
28C22×M16—21no—
29C2×#117—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4D411yescharacteristic
34C22×C4⋊C4D411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4C2311yescharacteristic
37C22×C42—21no—
38C22×C42D411yescharacteristic
39#428—42no—
40#428—42no—
41C2×#107—41no—
42C2×#107—41no—
43C2×#106—41no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46#398—21no—
47#398—21no—
48#398C2×C412yes—
49#398Q812yes—
50#389—41no—
51#389—41no—
52#389—42no—
53#389—42no—
54#380—21no—
55#380—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95D411yescharacteristic
67C2×#95D411yescharacteristic
68#343—21no—
69#343C2×C412yes—
70#340—41no—
71#340—41no—
72#340—41no—
73#339—42no—
74#324—21no—
75#324—21no—
76#324—21no—
77#324C2×C412yes—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8—41no—
96#117—41no—
97#117—41no—
98#117—41no—
99#117—41no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—81no—
125C2×C4⋊C4—81no—
126C2×C4⋊C4—81no—
127C2×C4⋊C4—81no—
128C2×C4⋊C4C4⋊C412yes—
129C2×C4⋊C4D812yes—
130C2×C4⋊C4C22×C411yescharacteristic
131C2×C4⋊C4C2×Q811yescharacteristic
132C2×C4⋊C4#4111yescharacteristic
133C2×C4⋊C4#4111yescharacteristic
134C2×C22⋊C4—41no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42—81no—
143#107—44no—
144#107—82no—
145#106—44no—
146C22⋊C8—22no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150#95—24no—
151#95—22no—
152#95—22no—
153#95—21no—
154#95—41no—
155#95—42no—
156#95—42no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—81no—
162#95—81no—
163#95—81no—
164#95—81no—
165#95—81no—
166#95C4⋊C412yes—
167C24#12011yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
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—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4C2×C4⋊C411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C22⋊Q811yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4#13511yescharacteristic
232C22×C4AGL(1,ℤ/8)11yescharacteristic
233C22×C4#13711yescharacteristic
234M16—24no—
235M16—42no—
236M16—42no—
237M16—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—81no—
243C2×C8—81no—
244C2×C8—81no—
245C2×C8—81no—
246C2×C8—81no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4—84no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C4⋊C4—84no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C42—42no—
274C42—81no—
275C42—81no—
276C42—81no—
277C42—84no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23#38511yescharacteristic
283C23#44711yescharacteristic
284C23#45211yescharacteristic
285C23#45311yescharacteristic
286C23#45611yescharacteristic
287C23B(4,𝔽2)11yescharacteristic
288C23#45811yescharacteristic
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—24no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4C4×D811yescharacteristic
364C2×C4#44011yescharacteristic
365C2×C4#47411yescharacteristic
366C2×C4#47811yescharacteristic
367C8—44no—
368C8—82no—
369C8—161no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22#159511yescharacteristic
383C22#160311yescharacteristic
384C22#161011yescharacteristic
385C22#161811yescharacteristic
386C22#185911yescharacteristic
387C22#186211yescharacteristic
388C22#186711yescharacteristic
389C22#220011yescharacteristic
390C22#220111yescharacteristic
391C22#220411yescharacteristic
392C22#220511yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—44no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—82no—
403C4—82no—
404C4—82no—
405C4—82no—
406C4—161no—
407C4—161no—
408C4—161no—
409C2—21no—
410C2—21no—
411C2—21no—
412C2—21no—
413C2#886611yescharacteristic
414C2#1209911yescharacteristic
415C2#1210111yescharacteristic
416C2#1224011yescharacteristic
417C2#1224111yescharacteristic
418C2#1296111yescharacteristic
419C2#1297311yescharacteristic
subgroupG/H
center, socleC23B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1346C4semidirect

Maximal subgroups

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

subgroupindex
#145872
C2×#13462
#113422
#90272
#88562
#72512
#71892