Finite Group Databasebeta

← #128795

Subgroups — #128795

Order 512.

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

normal subgroups only

This group has 664 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
1C4×#397C211yesmaximal, characteristic
2C2×#1344C212yesmaximal
3#10801C211yesmaximal, characteristic
4#8152C211yesmaximal, characteristic
5#7250C212yesmaximal
6C2×#397C2211yescharacteristic
7C2×C4×C4⋊C4C411yescharacteristic, complemented
8C2×C4×C22⋊C4—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1749—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1445C411yescharacteristic, complemented
15#1443—21no—
16#1442C411yescharacteristic, complemented
17#1440—21no—
18#1439C2211yescharacteristic
19C22⋊#95—21no—
20C4×#95—21no—
21C4×#95C411yescharacteristic, complemented
22#1344C418yescomplemented
23#1280—21no—
24C24×C4D411yescharacteristic
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C22⋊C4—21no—
28C22×C22⋊C4—22no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4C2311yescharacteristic
32C22×C42—21no—
33C22×C42—21no—
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C22⋊C8—22no—
37C2×#97—41no—
38#397—22no—
39#397C2×C414yes—
40C4×C4⋊C4—41no—
41C4×C4⋊C4—41no—
42C4×C22⋊C4—41no—
43C4×C22⋊C4—41no—
44C4×C22⋊C4—44no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—41no—
54C2×#95—41no—
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95D411yescharacteristic
59#342—22no—
60#342—42no—
61#342C2×C414yes—
62#336—41no—
63#323—22no—
64#323C2×C414yes—
65C25C2×D411yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—41no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C22×C8—42no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4C4212yes—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4C22×C412yes—
110C2×C22⋊C4#4112yes—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42—42no—
119C2×C42—42no—
120C2×C42—42no—
121C2×C42—42no—
122C2×C42—41no—
123C2×C42—41no—
124C22⋊C8—24no—
125C22⋊C8—42no—
126#97—82no—
127#95—21no—
128#95—24no—
129#95—41no—
130#95—41no—
131#95—41no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—82no—
140#95—82no—
141#95C4212yes—
142C24—21no—
143C24—21no—
144C24—21no—
145C24—21no—
146C24—41no—
147C24—41no—
148C24—41no—
149C24—41no—
150C24C2×C22⋊C411yescharacteristic
151C24C4×D412yes—
152C24AGL(1,ℤ/8)11yescharacteristic
153C24AGL(1,ℤ/8)12yes—
154C24#13711yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—48no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
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—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—44no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—84no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4#9911yescharacteristic
226C22×C4#9911yescharacteristic
227C22×C4C2×C4211yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4C4×D412yes—
230C2×C8—44no—
231C2×C8—82no—
232C2×C8—82no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—48no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C42—42no—
249C42—42no—
250C42—42no—
251C42—81no—
252C42—84no—
253C42—82no—
254C42—82no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23C4×C22⋊C411yescharacteristic
286C23C2×#9911yescharacteristic
287C23#41011yescharacteristic
288C23#41711yescharacteristic
289C23#41811yescharacteristic
290C23#42111yescharacteristic
291C23#42111yescharacteristic
292C23#44012yes—
293C23#44212yes—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
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—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—84no—
335C2×C4—84no—
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—84no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—162no—
361C2×C4—162no—
362C2×C4—162no—
363C2×C4—162no—
364C2×C4#35212yes—
365C2×C4#35412yes—
366C8—84no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22—42no—
391C22—82no—
392C22—82no—
393C22—81no—
394C22—81no—
395C22#151011yescharacteristic
396C22#151211yescharacteristic
397C22#151311yescharacteristic
398C22#151411yescharacteristic
399C22C4×#9911yescharacteristic
400C22#175911yescharacteristic
401C22#176211yescharacteristic
402C22C2×#35211yescharacteristic
403C22#212011yescharacteristic
404C22C2×#35411yescharacteristic
405C22#212611yescharacteristic
406C4—22no—
407C4—22no—
408C4—41no—
409C4—41no—
410C4—42no—
411C4—82no—
412C4—82no—
413C4—81no—
414C4—82no—
415C4—82no—
416C4—81no—
417C4—81no—
418C4—164no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—42no—
424C2—42no—
425C2#821611yescharacteristic
426C2#1179711yescharacteristic
427C2#1179811yescharacteristic
428C2#1182211yescharacteristic
429C2#1182311yescharacteristic
430C2C4×#35211yescharacteristic
431C2C4×#35411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C22×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
7. C2×C4×C4⋊C4C4semidirect
14. #1445C4semidirect
16. #1442C4semidirect
21. C4×#95C4semidirect
22. #1344C4semidirect

Maximal subgroups

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

subgroupindex
C4×#3972
C2×#13442
#108012
#81522
#72502