Finite Group Databasebeta

← C23×#5948 #72267895

Subgroups — C23×#5948 #72267895

Order 1536.

152803 conjugacy classes of subgroups; only the 638 normal subgroups are shown.

normal subgroups only

This group has 152803 conjugacy classes of subgroups — too many to draw. Only its 638 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
1C22×#5948C2114yesmaximal, direct product factor
2C23×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C25×#41C311yesmaximal, characteristic, complemented, Sylow
4C2×#5948C22128yesdirect product factor
5C22×C22⋊SL(2,𝔽3)C2217yescomplemented
6C23×#41⋊C3—44nomaximal
7C23×C4×A4—41nomaximal
8C24×#41—356no—
9C24×#41—34no—
10C24×#41—34no—
11C24×#41—314no—
12C24×#41C2×C3114yescomplemented
13C25×Q8C2×C311yescharacteristic
14C25×D4—31no—
15C26×C4—31no—
16C24×A4—41no—
17#5948C2318yesdirect product factor
18C2×C22⋊SL(2,𝔽3)C2317yescomplemented
19C22×#41⋊C3—456no—
20C23×SL(2,𝔽3)—44no—
21C22×C4×A4—414no—
22C27—31no—
23C23×#41—3336no—
24C23×#41—356no—
25C23×#41—312no—
26C23×#41—3168no—
27C23×#41—3672no—
28C23×#41—3168no—
29C23×#41—31008no—
30C23×#41—3168no—
31C23×#41—3168no—
32C23×#41—3224no—
33C23×#41—356no—
34C23×#41—356no—
35C23×#41—34no—
36C23×#41—356no—
37C23×#41—356no—
38C23×#41—34no—
39C23×#41—384no—
40C23×#41A414yescomplemented
41C23×#41C22×C3128yescomplemented
42C24×Q8—328no—
43C24×Q8—34no—
44C24×Q8—37no—
45C24×Q8C22×C317yes—
46C24×D4—314no—
47C24×D4—312no—
48C24×D4—384no—
49C24×D4—314no—
50C25×C4—384no—
51C25×C4—36no—
52C25×C4—36no—
53C25×C4—31no—
54C25×C4—314no—
55C25×C4—314no—
56C25×C4A411yescharacteristic
57C23×A4—48no—
58C23×A4—47no—
59C23×C3×C4—161no—
60C22⋊SL(2,𝔽3)C2411yescharacteristic, complemented
61C2×#41⋊C3—4112no—
62C22×SL(2,𝔽3)—428no—
63C2×C4×A4—428no—
64C26—342no—
65C26—36no—
66C26—314no—
67C26—624no—
68C26—68no—
69C26C2×A411yescharacteristic
70C22×#41—3224no—
71C22×#41—3112no—
72C22×#41—3168no—
73C22×#41—3168no—
74C22×#41—31008no—
75C22×#41—31344no—
76C22×#41—31008no—
77C22×#41—32016no—
78C22×#41—32016no—
79C22×#41—35376no—
80C22×#41—32016no—
81C22×#41—31344no—
82C22×#41—3672no—
83C22×#41—32016no—
84C22×#41—3672no—
85C22×#41—3336no—
86C22×#41—31008no—
87C22×#41—31008no—
88C22×#41—32016no—
89C22×#41—3448no—
90C22×#41—3336no—
91C22×#41—3168no—
92C22×#41—3112no—
93C22×#41—3168no—
94C22×#41—356no—
95C22×#41—3336no—
96C22×#41—3112no—
97C22×#41—356no—
98C22×#41—356no—
99C22×#41—356no—
100C22×#41C2×A4156yescomplemented
101C22×#41C23×C318yescomplemented
102C23×Q8—384no—
103C23×Q8—3168no—
104C23×Q8—3252no—
105C23×Q8—384no—
106C23×Q8—384no—
107C23×Q8—356no—
108C23×Q8—314no—
109C23×Q8—34no—
110C23×Q8—328no—
111C23×Q8—328no—
112C23×Q8—321no—
113C23×Q8C2×A414yes—
114C23×Q8C23×C317yes—
115C23×D4—34no—
116C23×D4—328no—
117C23×D4—3168no—
118C23×D4—3168no—
119C23×D4—3504no—
120C23×D4—3504no—
121C23×D4—3672no—
122C23×D4—3168no—
123C23×D4—3168no—
124C23×D4—384no—
125C23×D4—312no—
126C24×C4—342no—
127C24×C4—36no—
128C24×C4—3168no—
129C24×C4—3672no—
130C24×C4—3504no—
131C24×C4—384no—
132C24×C4—3504no—
133C24×C4—384no—
134C24×C4—3168no—
135C24×C4—384no—
136C24×C4—384no—
137C24×C4—312no—
138C24×C4—37no—
139C24×C4—37no—
140C24×C4—384no—
141C24×C4—328no—
142C24×C4—31no—
143C24×C4—34no—
144C24×C4—31no—
145C24×C4—314no—
146C24×C4C2×A4114yes—
147C24×C3—161no—
148C22×A4—428no—
149C22×A4—47no—
150C22×C3×C4—1614no—
151#41⋊C3—432no—
152C2×SL(2,𝔽3)—428no—
153C4×A4—48no—
154C25—37no—
155C25—38no—
156C25—31no—
157C25—3252no—
158C25—3168no—
159C25—384no—
160C25—384no—
161C25—328no—
162C25—34no—
163C25—6336no—
164C25—6168no—
165C25—6336no—
166C25—648no—
167C25—648no—
168C25—656no—
169C25#41⋊C318yescomplemented
170C25C22×A417yes—
171C2×#41—332no—
172C2×#41—3336no—
173C2×#41—31008no—
174C2×#41—3672no—
175C2×#41—3672no—
176C2×#41—34032no—
177C2×#41—34032no—
178C2×#41—35376no—
179C2×#41—31344no—
180C2×#41—35376no—
181C2×#41—34032no—
182C2×#41—31344no—
183C2×#41—32016no—
184C2×#41—3672no—
185C2×#41—32016no—
186C2×#41—3672no—
187C2×#41—3672no—
188C2×#41—34032no—
189C2×#41—3224no—
190C2×#41—31008no—
191C2×#41—3672no—
192C2×#41—3224no—
193C2×#41—31008no—
194C2×#41—3112no—
195C2×#41—3224no—
196C2×#41—332no—
197C2×#41—3112no—
198C2×#41—3336no—
199C2×#41C22×A41112yescomplemented
200C22×Q8—328no—
201C22×Q8—3168no—
202C22×Q8—3252no—
203C22×Q8—3168no—
204C22×Q8—3672no—
205C22×Q8—3504no—
206C22×Q8—3168no—
207C22×Q8—3252no—
208C22×Q8—3252no—
209C22×Q8—3504no—
210C22×Q8—356no—
211C22×Q8—314no—
212C22×Q8—384no—
213C22×Q8—328no—
214C22×Q8—328no—
215C22×Q8—384no—
216C22×Q8—328no—
217C22×Q8—37no—
218C22×Q8C22×A4128yes—
219C22×Q8C24×C311yescharacteristic
220C22×D4—356no—
221C22×D4—38no—
222C22×D4—3336no—
223C22×D4—31008no—
224C22×D4—3336no—
225C22×D4—31008no—
226C22×D4—31344no—
227C22×D4—31344no—
228C22×D4—32016no—
229C22×D4—31008no—
230C22×D4—31008no—
231C22×D4—356no—
232C22×D4—356no—
233C22×D4—3168no—
234C22×D4—3168no—
235C23×C4—314no—
236C23×C4—356no—
237C23×C4—3126no—
238C23×C4—3126no—
239C23×C4—3168no—
240C23×C4—342no—
241C23×C4—384no—
242C23×C4—342no—
243C23×C4—31008no—
244C23×C4—31008no—
245C23×C4—31344no—
246C23×C4—31344no—
247C23×C4—31008no—
248C23×C4—3504no—
249C23×C4—3336no—
250C23×C4—3168no—
251C23×C4—31008no—
252C23×C4—31008no—
253C23×C4—3504no—
254C23×C4—3168no—
255C23×C4—3168no—
256C23×C4—3168no—
257C23×C4—321no—
258C23×C4—37no—
259C23×C4—31no—
260C23×C4—34no—
261C23×C4—356no—
262C23×C4—38no—
263C23×C4—314no—
264C23×C4—314no—
265C23×C4—356no—
266C23×C4—356no—
267C23×C4—314no—
268C23×C4—384no—
269C23×C4C22×A4128yes—
270C23×C4C22⋊A411yescharacteristic
271C23×C3—168no—
272C23×C3—167no—
273C2×A4—414no—
274C2×A4—41no—
275C2×C3×C4—1628no—
276SL(2,𝔽3)—44no—
277C24—321no—
278C24—356no—
279C24—328no—
280C24—314no—
281C24—356no—
282C24—38no—
283C24—38no—
284C24—37no—
285C24—37no—
286C24—3168no—
287C24—3336no—
288C24—3504no—
289C24—3168no—
290C24—38no—
291C24—356no—
292C24—61008no—
293C24—6168no—
294C24—61008no—
295C24—61344no—
296C24—6672no—
297C24—6336no—
298C24—6336no—
299C24—656no—
300C24—632no—
301C24C2×#41⋊C3128yescomplemented
302C24C23×A417yes—
303C24C2×C22⋊A411yescharacteristic
304#41—396no—
305#41—3672no—
306#41—3672no—
307#41—34032no—
308#41—31344no—
309#41—31344no—
310#41—31344no—
311#41—3448no—
312#41—3672no—
313#41—332no—
314#41—332no—
315#41—3224no—
316#41C23×A4132yescomplemented
317C2×Q8—384no—
318C2×Q8—384no—
319C2×Q8—3504no—
320C2×Q8—3672no—
321C2×Q8—3504no—
322C2×Q8—3168no—
323C2×Q8—3168no—
324C2×Q8—356no—
325C2×Q8—3252no—
326C2×Q8—328no—
327C2×Q8—34no—
328C2×Q8—328no—
329C2×Q8—384no—
330C2×Q8C23×A4128yes—
331C2×D4—3112no—
332C2×D4—396no—
333C2×D4—3672no—
334C2×D4—34032no—
335C2×D4—3672no—
336C2×D4—3672no—
337C2×D4—3336no—
338C2×D4—31008no—
339C2×D4—3336no—
340C22×C4—384no—
341C22×C4—3112no—
342C22×C4—3126no—
343C22×C4—342no—
344C22×C4—3168no—
345C22×C4—3168no—
346C22×C4—3252no—
347C22×C4—3126no—
348C22×C4—3252no—
349C22×C4—342no—
350C22×C4—32016no—
351C22×C4—3672no—
352C22×C4—3336no—
353C22×C4—348no—
354C22×C4—32016no—
355C22×C4—3672no—
356C22×C4—3336no—
357C22×C4—348no—
358C22×C4—31008no—
359C22×C4—3336no—
360C22×C4—37no—
361C22×C4—31no—
362C22×C4—314no—
363C22×C4—328no—
364C22×C4—328no—
365C22×C4—384no—
366C22×C4—328no—
367C22×C4—3336no—
368C22×C4—3112no—
369C22×C4—328no—
370C22×C4—356no—
371C22×C4C23×A418yes—
372C22×C4C2×C22⋊A4114yes—
373C22×C3—167no—
374C22×C3—1628no—
375A4—41no—
376C3×C4—168no—
377C23—37no—
378C23—384no—
379C23—314no—
380C23—314no—
381C23—384no—
382C23—3112no—
383C23—321no—
384C23—37no—
385C23—356no—
386C23—328no—
387C23—328no—
388C23—3336no—
389C23—348no—
390C23—3112no—
391C23—6336no—
392C23—624no—
393C23—6336no—
394C23—61344no—
395C23—62016no—
396C23—6336no—
397C23—6336no—
398C23—68no—
399C23—6224no—
400C23C22×#41⋊C3114yescomplemented
401C23#594818yesdirect product factor
402C23C24×A411yescharacteristic
403C23C22×C22⋊A417yes—
404Q8—328no—
405Q8—3168no—
406Q8—356no—
407Q8—384no—
408Q8—34no—
409Q8C24×A414yes—
410D4—332no—
411D4—3224no—
412D4—3672no—
413D4—396no—
414C2×C4—356no—
415C2×C4—3252no—
416C2×C4—342no—
417C2×C4—384no—
418C2×C4—36no—
419C2×C4—3672no—
420C2×C4—396no—
421C2×C4—328no—
422C2×C4—384no—
423C2×C4—328no—
424C2×C4—356no—
425C2×C4—38no—
426C2×C4—3224no—
427C2×C4—332no—
428C2×C4—38no—
429C2×C4C22×C22⋊A4128yes—
430C2×C3—1614no—
431C2×C3—161no—
432C22—314no—
433C22—31no—
434C22—314no—
435C22—356no—
436C22—384no—
437C22—314no—
438C22—314no—
439C22—37no—
440C22—31no—
441C22—332no—
442C22—6672no—
443C22—648no—
444C22—648no—
445C22—6224no—
446C22C23×#41⋊C311yescharacteristic, complemented
447C22C2×#5948128yesdirect product factor
448C22C23×C22⋊A417yes—
449C4—38no—
450C4—328no—
451C4—34no—
452C4C23×C22⋊A418yes—
453C3—161noSylow
454C2—314no—
455C2—31no—
456C2—31no—
457C2—632no—
458C2C22×#5948114yesdirect product factor
459C2C24×C22⋊A411yescharacteristic
subgroupG/H
center, hypercenterC23×C4C22⋊A4
commutator, cosocle, hypocenterC22×Q8C24×C3
FrattiniC2C24×C22⋊A4
FittingC25×#41C3
socleC26C2×A4
LCSG ▷ C22×Q8
UCSC1 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×#5948C2direct (N × H)
2. C23×C22⋊SL(2,𝔽3)C2semidirect
3. C25×#41C3semidirect
4. C2×#5948C22direct (N × H)
5. C22×C22⋊SL(2,𝔽3)C22semidirect
12. C24×#41C2×C3semidirect
17. #5948C23direct (N × H)
18. C2×C22⋊SL(2,𝔽3)C23semidirect
40. C23×#41A4semidirect
41. C23×#41C22×C3semidirect
60. C22⋊SL(2,𝔽3)C24semidirect
100. C22×#41C2×A4semidirect
101. C22×#41C23×C3semidirect
169. C25#41⋊C3semidirect
199. C2×#41C22×A4semidirect
301. C24C2×#41⋊C3semidirect
316. #41C23×A4semidirect
400. C23C22×#41⋊C3semidirect
401. C23#5948direct (N × H)
446. C22C23×#41⋊C3semidirect
447. C22C2×#5948direct (N × H)
458. C2C22×#5948direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#59482
C23×C22⋊SL(2,𝔽3)2
C25×#413
C23×#41⋊C34
C23×C4×A44