Finite Group Databasebeta

← C4×#1351 #128784

Subgroups — C4×#1351 #128784

Order 512.

836 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 132 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×#393C211yesmaximal, characteristic
2C2×#1351C212yesmaximal
3C4×#342C211yesmaximal, characteristic
4C4×#323C211yesmaximal, characteristic
5#7250C212yesmaximal
6C22×C4×Q8C411yescharacteristic, complemented
7C2×#393C2211yescharacteristic
8C2×C4×C22⋊C4—21no—
9C2×C4×C22⋊C4—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11C4×C22⋊C8—21no—
12C2×#342—21no—
13C2×#342C2212yes—
14C2×#323C2212yes—
15#1446C411yescharacteristic, complemented
16#1442—21no—
17#1442C411yescharacteristic, complemented
18#1439—21no—
19#1439C2211yescharacteristic
20C4×#95—21no—
21C4×#95C411yescharacteristic, complemented
22#1351C418yesdirect product factor
23#1280—21no—
24C23×Q8C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C2×C4×Q8—44no—
27C2×C4×Q8—41no—
28C2×C4×Q8—41no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—22no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4D411yescharacteristic
35C22×C22⋊C4C2311yescharacteristic
36C22×C42—21no—
37C22×C42—21no—
38C22×C42—41no—
39C22×C42C2×C411yescharacteristic
40C22×C42D411yescharacteristic
41C2×C22⋊C8—22no—
42C2×C4×C8—41no—
43#393—22no—
44#393C2×C414yes—
45C4×C22⋊C4—24no—
46C4×C22⋊C4—41no—
47C4×C22⋊C4—41no—
48C4×C22⋊C4—44no—
49C4×C22⋊C4—42no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—41no—
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95D411yescharacteristic
62#342—24no—
63#342—22no—
64#342C2×C414yes—
65#336—41no—
66#323—22no—
67#323C2×C414yes—
68C25C2×D411yescharacteristic
69C22×Q8—21no—
70C22×Q8—42no—
71C22×Q8—42no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8C4212yescomplemented
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—22no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4SD1612yes—
90C23×C4Q1612yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—42no—
94C4×Q8—44no—
95C4×Q8—44no—
96C4×Q8—44no—
97C4×Q8—44no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4C22×C412yes—
121C2×C22⋊C4#4112yes—
122C2×C42—24no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—42no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42—44no—
133C2×C42—42no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42—82no—
137C2×C42—82no—
138C22⋊C8—24no—
139C22⋊C8—42no—
140C4×C8—82no—
141#95—21no—
142#95—24no—
143#95—41no—
144#95—41no—
145#95—41no—
146#95—42no—
147#95—42no—
148#95—42no—
149#95—42no—
150#95—42no—
151#95—44no—
152#95—42no—
153#95C4212yescomplemented
154C24—21no—
155C24—21no—
156C24—21no—
157C24—21no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24C2×C22⋊C411yescharacteristic
163C24C4×D412yes—
164C24C2×SD1611yescharacteristic
165C24C2×Q1611yescharacteristic
166C24#13512yes—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—48no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
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—44no—
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—44no—
225C22×C4—44no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—84no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4#9911yescharacteristic
246C22×C4#9911yescharacteristic
247C22×C4#10312yes—
248C22×C4#10312yes—
249C22×C4#10412yes—
250C22×C4#10412yes—
251C22×C4C2×C4211yescharacteristic
252C22×C4#11711yescharacteristic
253C22×C4C4×D412yes—
254C2×C8—44no—
255C2×C8—82no—
256C2×C8—82no—
257C4⋊C4—42no—
258C4⋊C4—48no—
259C4⋊C4—42no—
260C22⋊C4—28no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—44no—
264C22⋊C4—48no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—48no—
268C22⋊C4—44no—
269C42—44no—
270C42—42no—
271C42—44no—
272C42—42no—
273C42—42no—
274C42—81no—
275C42—84no—
276C42—88no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—21no—
288C23—21no—
289C23—42no—
290C23—42no—
291C23—41no—
292C23—41no—
293C23—42no—
294C23—41no—
295C23—41no—
296C23—42no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23C4×C22⋊C411yescharacteristic
308C23C2×#9911yescharacteristic
309C23#41011yescharacteristic
310C23C2×#10311yescharacteristic
311C23#41611yescharacteristic
312C23C2×#10411yescharacteristic
313C23C2×#10411yescharacteristic
314C23C4×SD1612yes—
315C23C4×Q1612yes—
316Q8—44no—
317Q8—44no—
318Q8—44no—
319Q8—44no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—48no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4#32812yes—
391C2×C4#32812yes—
392C2×C4AΣL(1,𝔽16)12yes—
393C2×C4#35412yes—
394C8—84no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—22no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—42no—
419C22—82no—
420C22—82no—
421C22—81no—
422C22—81no—
423C22C2×#32811yescharacteristic
424C22#150011yescharacteristic
425C22#150111yescharacteristic
426C22#150311yescharacteristic
427C22C4×#9911yescharacteristic
428C22C4×#10411yescharacteristic
429C22C4×#10311yescharacteristic
430C22C2×AΣL(1,𝔽16)11yescharacteristic
431C22#212011yescharacteristic
432C22C2×#35411yescharacteristic
433C22#212611yescharacteristic
434C4—22no—
435C4—44no—
436C4—41no—
437C4—41no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—82no—
442C4—82no—
443C4—88no—
444C4—81no—
445C4—82no—
446C4#135114yesdirect product factor
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—42no—
452C2—42no—
453C2C4×#32811yescharacteristic
454C2C2×#135111yescharacteristic
455C2#1171511yescharacteristic
456C2#1172511yescharacteristic
457C2#1172811yescharacteristic
458C2C4×AΣL(1,𝔽16)11yescharacteristic
459C2C4×#35411yescharacteristic
subgroupG/H
centerC22×C4#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C22⋊C4C23
socleC23C2×#99
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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
6. C22×C4×Q8C4semidirect
15. #1446C4semidirect
17. #1442C4semidirect
21. C4×#95C4semidirect
22. #1351C4direct (N × H)
74. C22×Q8C42semidirect
153. #95C42semidirect
446. C4#1351direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#3932
C2×#13512
C4×#3422
C4×#3232
#72502