Finite Group Databasebeta

← #134453

Subgroups — #134453

Order 512.

674 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C4×#395C211yesmaximal, characteristic
2C2×#1354C212yesmaximal
3#10847C211yesmaximal, characteristic
4#8558C211yesmaximal, characteristic
5#7251C212yesmaximal
6C22×C4×Q8C411yescharacteristic
7C2×#423—21no—
8C2×#395C2211yescharacteristic
9C2×C4×C4⋊C4—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1797—21no—
12C2×#343—21no—
13C2×#343C2212yes—
14C2×#324C2212yes—
15#1446C411yescharacteristic
16#1444—21no—
17#1444C411yescharacteristic
18#1441—21no—
19#1441C2211yescharacteristic
20C4×#95—21no—
21C4×#95C411yescharacteristic
22#1354—24no—
23#1280—21no—
24C23×Q8C2×C411yescharacteristic
25C22×M16—22no—
26C2×C4×Q8—44no—
27C2×C4×Q8—41no—
28C2×C4×Q8—41no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
34C22×C4⋊C4C2311yescharacteristic
35C22×C42—21no—
36C22×C42—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40#423—24no—
41#423—42no—
42C2×C4⋊C8—41no—
43C2×C4⋊C8—41no—
44C2×C4⋊C8—41no—
45C2×C22⋊C8—22no—
46#395—21no—
47#395—21no—
48#395D412yes—
49#395Q812yes—
50C4×C4⋊C4—41no—
51C4×C4⋊C4—41no—
52C4×C4⋊C4—44no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95C2×C411yescharacteristic
63C2×#95D411yescharacteristic
64#343—22no—
65#343—22no—
66#343—42no—
67#336—41no—
68#324—22no—
69#324—22no—
70C22×Q8—21no—
71C22×Q8—42no—
72C22×Q8—42no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×Q8C4⋊C412yes—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C2×M16—24no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—42no—
92C2×M16—42no—
93C22×C8—42no—
94C22×C8—42no—
95C4×Q8—44no—
96C4×Q8—44no—
97C4×Q8—44no—
98C4×Q8—44no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4C2×D411yescharacteristic
118C2×C4⋊C4C2×Q811yescharacteristic
119C2×C4⋊C4#4111yescharacteristic
120C2×C4⋊C4#4111yescharacteristic
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—22no—
124C2×C42—24no—
125C2×C42—24no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42—42no—
137C2×C42SD1612yes—
138C2×C42Q1612yes—
139C4⋊C8—44no—
140C4⋊C8—44no—
141C4⋊C8—82no—
142C22⋊C8—42no—
143C22⋊C8—42no—
144#95—21no—
145#95—22no—
146#95—22no—
147#95—41no—
148#95—41no—
149#95—41no—
150#95—42no—
151#95—42no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—41no—
156#95—41no—
157#95C4⋊C412yes—
158C24C2×C22⋊C411yescharacteristic
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—48no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
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×C4#9911yescharacteristic
225C22×C4#9911yescharacteristic
226C22×C4#10312yes—
227C22×C4#10312yes—
228C22×C4#10412yes—
229C22×C4C2×C4⋊C411yescharacteristic
230C22×C4#11711yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C2×SD1611yescharacteristic
236C22×C4C2×Q1611yescharacteristic
237C22×C4AGL(1,ℤ/8)11yescharacteristic
238C22×C4#13711yescharacteristic
239M16—44no—
240M16—44no—
241M16—82no—
242M16—82no—
243C2×C8—44no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—48no—
259C4⋊C4—48no—
260C4⋊C4—42no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—48no—
266C42—44no—
267C42—42no—
268C42—42no—
269C42—44no—
270C42—42no—
271C42—81no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23#38011yescharacteristic
277C23C2×#9911yescharacteristic
278C23#41011yescharacteristic
279C23C2×#10311yescharacteristic
280C23#41711yescharacteristic
281C23C2×#10411yescharacteristic
282C23#42111yescharacteristic
283Q8—44no—
284Q8—44no—
285Q8—44no—
286Q8—44no—
287C2×C4—21no—
288C2×C4—21no—
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—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
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#32812yes—
361C2×C4#32812yes—
362C2×C4AΣL(1,𝔽16)12yes—
363C2×C4#35612yes—
364C2×C4#46011yescharacteristic
365C2×C4#46211yescharacteristic
366C2×C4#47511yescharacteristic
367C2×C4#47711yescharacteristic
368C8—84no—
369C8—162no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22C2×#32811yescharacteristic
383C22#150911yescharacteristic
384C22#151311yescharacteristic
385C22#151811yescharacteristic
386C22#179911yescharacteristic
387C22#180311yescharacteristic
388C22#180611yescharacteristic
389C22C2×AΣL(1,𝔽16)11yescharacteristic
390C22#212011yescharacteristic
391C22C2×#35611yescharacteristic
392C22#213211yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—22no—
398C4—44no—
399C4—42no—
400C4—41no—
401C4—44no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—82no—
406C4—81no—
407C4—81no—
408C4—82no—
409C4#135414yescomplemented
410C2—21no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2#857211yescharacteristic
415C2C2×#135411yescharacteristic
416C2#1172111yescharacteristic
417C2#1182811yescharacteristic
418C2#1183111yescharacteristic
419C2#1279311yescharacteristic
420C2#1280211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
409. C4#1354semidirect

Maximal subgroups

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

subgroupindex
C4×#3952
C2×#13542
#108472
#85582
#72512