Finite Group Databasebeta

← #142027

Subgroups — #142027

Order 512.

656 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 118 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#15195C211yesmaximal, characteristic
2#10846C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7250C214yesmaximal
5C2×#384C411yescharacteristic, complemented
6C2×#380—21no—
7C2×#380—21no—
8C2×C4×C22⋊C4C2211yescharacteristic
9C4×C22⋊C8—21no—
10C2×#342—21no—
11C2×#342C2212yes—
12C2×#323C2212yes—
13#1442—21no—
14#1442C412yescomplemented
15#1439—21no—
16#1439C2212yes—
17C4⋊#95—21no—
18C4⋊#95C411yescharacteristic, complemented
19#1280—21no—
20C24×C4D411yescharacteristic
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—41no—
25C22×C4⋊C4C2×C412yes—
26C22×C22⋊C4—22no—
27C22×C22⋊C4—22no—
28C22×C22⋊C4D411yescharacteristic
29C22×C22⋊C4C2311yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42D411yescharacteristic
32C2×C22⋊C8—22no—
33C2×C4×C8—41no—
34#384—41no—
35#384—41no—
36#380—24no—
37#380—42no—
38#380—42no—
39#380—42no—
40C4×C22⋊C4—41no—
41C4×C22⋊C4—41no—
42C2×#95—21no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—41no—
47C2×#95C2×C412yes—
48C2×#95C2×C411yescharacteristic
49C2×#95D411yescharacteristic
50#342—24no—
51#342—42no—
52#336—41no—
53#323—22no—
54#323Q814yes—
55C25C2×D411yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4D812yes—
69C23×C4SD1612yes—
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C22×C8—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—44no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—82no—
84C2×C4⋊C4—82no—
85C2×C4⋊C4—82no—
86C2×C4⋊C4—82no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—82no—
100C2×C22⋊C4—82no—
101C2×C22⋊C4SD1612yes—
102C2×C22⋊C4Q1612yes—
103C2×C22⋊C4C2×Q812yes—
104C2×C22⋊C4#4112yes—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—42no—
110C2×C42—42no—
111C22⋊C8—24no—
112C22⋊C8—42no—
113C4×C8—82no—
114#95—41no—
115#95—41no—
116#95—42no—
117#95—42no—
118#95—44no—
119#95—44no—
120#95—82no—
121#95—82no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—21no—
126C24—41no—
127C24—41no—
128C24—41no—
129C24—41no—
130C24C2×C22⋊C411yescharacteristic
131C24#12411yescharacteristic
132C24C4⋊Q811yescharacteristic
133C24C2×D811yescharacteristic
134C24C2×SD1611yescharacteristic
135C24C2×SD1611yescharacteristic
136C24C2×Q1611yescharacteristic
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—84no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4—82no—
191C22×C4#9911yescharacteristic
192C22×C4#9911yescharacteristic
193C22×C4#10212yes—
194C22×C4#10212yes—
195C22×C4#10312yes—
196C22×C4#10312yes—
197C22×C4#10412yes—
198C22×C4C4×Q812yes—
199C22×C4C22⋊Q812yes—
200C2×C8—44no—
201C2×C8—82no—
202C2×C8—82no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C4⋊C4—84no—
206C4⋊C4—88no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—44no—
214C22⋊C4—44no—
215C22⋊C4—82no—
216C22⋊C4—84no—
217C22⋊C4—84no—
218C22⋊C4—82no—
219C42—42no—
220C42—42no—
221C42—42no—
222C42—81no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—81no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23#39111yescharacteristic
262C23C2×#9911yescharacteristic
263C23#41011yescharacteristic
264C23C2×#10211yescharacteristic
265C23C2×#10311yescharacteristic
266C23C2×#10411yescharacteristic
267C23#42111yescharacteristic
268C23#48611yescharacteristic
269C23#48711yescharacteristic
270C23#49811yescharacteristic
271C23#50011yescharacteristic
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—84no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—162no—
323C2×C4—162no—
324C2×C4—162no—
325C2×C4—162no—
326C2×C4#32712yes—
327C2×C4#32712yes—
328C2×C4#35212yes—
329C2×C4#35412yes—
330C8—84no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22C2×#32711yescharacteristic
370C22#150311yescharacteristic
371C22#151711yescharacteristic
372C22#152011yescharacteristic
373C22#198611yescharacteristic
374C22#199011yescharacteristic
375C22#199611yescharacteristic
376C22C2×#35211yescharacteristic
377C22#212211yescharacteristic
378C22C2×#35411yescharacteristic
379C22#212811yescharacteristic
380C4—22no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—162no—
387C4—164no—
388C4—162no—
389C4#134414yes—
390C2—21no—
391C2—21no—
392C2—21no—
393C2—21no—
394C2—41no—
395C2—41no—
396C2—41no—
397C2—41no—
398C2#988511yescharacteristic
399C2C2×#134411yescharacteristic
400C2#1169311yescharacteristic
401C2#1189911yescharacteristic
402C2#1190011yescharacteristic
403C2#1329211yescharacteristic
404C2#1329911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#384C4semidirect
14. #1442C4semidirect
18. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#151952
#108462
C4×#3232
#72502