Finite Group Databasebeta

← #72222970

Subgroups — #72222970

Order 2592.

726 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C2×C3⋊#6258C211yesmaximal, characteristic
2C3×(C3⋊Dic3)⋊A4C211yesmaximal, characteristic, complemented
3C3⋊((C3⋊SL(2,ℤ/4))⋊C3)C211yesmaximal, characteristic, complemented
4Dic3×C3⋊S4—31nomaximal
5#11723726—31nomaximal
6#6258⋊C4—33nomaximal
7C2×C3×C22⋊H(3,𝔽3)C2211yescharacteristic
8C3⋊#6258C412yescomplemented
9#10619885—41nomaximal
10C2×C3×C3⋊S4—31no—
11C2×C3×C3⋊S4—61no—
12C2×C3×#638—31no—
13C22×C3×C3⋊Dic3D311yescharacteristic, complemented
14C2×#6294—31no—
15C3×Dic3×A4—31no—
16C3⋊(C3⋊SL(2,ℤ/4))D311yescharacteristic, complemented
17C2×#6258—33no—
18C2×#6258—63no—
19C3×#3757—31no—
20#88815—31no—
21#88814—31no—
22#88813—31no—
23(C3⋊Dic3)⋊A4D313yescomplemented
24(C3⋊SL(2,ℤ/4))⋊C3—33no—
25C2×C3⋊#272—41no—
26C3×C22⋊H(3,𝔽3)C2×C411yescharacteristic, complemented
27C3⋊((C3⋊Dic3)⋊C3)—41no—
28C3×(C3⋊Dic3)⋊C3—41no—
29C4×C3⋊S4—91no—
30Dic3×S4—93no—
31#67⋊Dic3—91no—
32Dic3×#67—91no—
33#63952—91no—
34C23×C33C2×D311yescharacteristic
35C22×C3×C3⋊D3—31no—
36C2×C32×A4—21no—
37C2×C32×A4C2×D311yescharacteristic
38C3×C3⋊S4—32no—
39C3×C3⋊S4—62no—
40C3×#638—32no—
41C3×#638—32no—
42C2×C3×C3⋊Dic3—31no—
43C2×C3×C3⋊Dic3—31no—
44C2×C3×C3⋊Dic3—31no—
45C2×C3×C3⋊Dic3—31no—
46C2×C3×C3⋊Dic3—31no—
47#6294—61no—
48#6294—61no—
49Dic3×C3⋊D3—121no—
50C2×C22⋊H(3,𝔽3)—23no—
51C2×C22⋊H(3,𝔽3)C2×D313yes—
52#6258—36no—
53#6258—66no—
54#6199—123no—
55C2×C3×H(3,𝔽3)—41no—
56C3⋊#272—42no—
57C2×C3⋊S4—31no—
58C2×C3⋊S4—61no—
59C2×C3×S4—93no—
60C2×C3×S4—183no—
61C2×#638—31no—
62C22×C3⋊Dic3C3⋊D311yescharacteristic, complemented
63C2×C3×#67—91no—
64C2×C3×#67—91no—
65C2×C3×#67—181no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—91no—
68C3×C4×A4—91no—
69C2×#624—91no—
70C2×D3×Dic3—91no—
71C2×D3×Dic3—91no—
72Dic3×A4—33no—
73C3⋊SL(2,ℤ/4)—33no—
74C3⋊SL(2,ℤ/4)—91no—
75C3⋊SL(2,ℤ/4)—181no—
76#3757—31no—
77C3×#217—91no—
78C3×#217—91no—
79C3×#217—181no—
80#3724—91no—
81Dic3⋊Dic3—91no—
82Dic3⋊Dic3—91no—
83#3722—91no—
84(C2×D3)⋊Dic3—91no—
85(C2×D3)⋊Dic3—91no—
86Dic32—91no—
87Dic32—91no—
88Dic32—91no—
89C22×C33—31no—
90C22×C33—31no—
91C22×C33C4×D311yescharacteristic, complemented
92C2×C3×C3⋊D3—32no—
93C2×C3×C3⋊D3—61no—
94C2×C3×C3⋊D3—61no—
95C2×C3×C3⋊D3—121no—
96C2×C3×C3⋊D3—241no—
97C32×A4—21no—
98C32×A4C4×D311yescharacteristic, complemented
99C3⋊(C3⋊Dic3)—41no—
100C3×C3⋊Dic3—32no—
101C3×C3⋊Dic3—31no—
102C3×C3⋊Dic3—61no—
103C3×C3⋊Dic3—61no—
104C3×C3⋊Dic3S411yescharacteristic, complemented
105C32×Dic3—121no—
106C2×#272—123no—
107C2×#272—243no—
108C22⋊H(3,𝔽3)—23no—
109C22⋊H(3,𝔽3)C4×D313yescomplemented
110(C3⋊Dic3)⋊C3—43no—
111(C3⋊Dic3)⋊C3—123no—
112C4×S4—273no—
113D4×Dic3—271no—
114C4×#67—271no—
115#67⋊C4—271no—
116C3×H(3,𝔽3)—41noSylow
117C23×C32—21no—
118C23×C32—31no—
119C23×C32—61no—
120C23×C32D3211yescharacteristic
121C23×C32C2×C3⋊D311yescharacteristic
122C22×C3⋊D3—31no—
123C22×C3×D3—91no—
124C22×C3×D3—91no—
125C22×C3×D3—181no—
126C2×C3×A4—23no—
127C2×C3×A4—31no—
128C2×C3×A4—61no—
129C2×C3×A4—61no—
130C2×C3×A4—121no—
131C2×C3×A4D3213yes—
132C3⋊S4—32no—
133C3⋊S4—62no—
134C3×S4—96no—
135C3×S4—186no—
136#638—32no—
137#638—32no—
138C2×C3⋊Dic3—31no—
139C2×C3⋊Dic3—31no—
140C2×C3⋊Dic3—31no—
141C2×C3⋊Dic3—31no—
142C2×C3⋊Dic3—31no—
143C4×C3⋊D3—361no—
144C3×#67—92no—
145C3×#67—92no—
146C3×#67—92no—
147C3×#67—92no—
148C3×#67—182no—
149C3×#67—182no—
150C2×C3×Dic3—91no—
151C2×C3×Dic3—91no—
152C2×C3×Dic3—91no—
153C2×C3×Dic3—91no—
154C2×C3×Dic3—91no—
155C2×C3×Dic3—91no—
156C2×C3×Dic3—91no—
157C2×C3×Dic3—91no—
158C2×C3×Dic3—91no—
159C2×C3×Dic3—91no—
160C2×C3×Dic3—181no—
161C2×C3×Dic3—181no—
162C2×C3×Dic3—181no—
163C2×C3×Dic3—181no—
164C2×C3×Dic3—181no—
165#624—181no—
166#624—181no—
167D3×Dic3—181no—
168D3×Dic3—181no—
169D3×Dic3—181no—
170D3×Dic3—181no—
171D3×Dic3—363no—
172C2×C33—31no—
173C2×C33—31no—
174C2×C33—41no—
175C2×C33—81no—
176C2×C33C2×S411yescharacteristic
177C3×C3⋊D3—62no—
178C3×C3⋊D3—122no—
179C3×C3⋊D3—242no—
180C2×H(3,𝔽3)—43no—
181C2×H(3,𝔽3)—83no—
182#272—126no—
183#272—246no—
184C2×S4—93no—
185C2×S4—183no—
186C2×C3×D4—271no—
187C22×C3×C4—91no—
188C2×#67—91no—
189C2×#67—91no—
190C2×#67—181no—
191C22×Dic3—31no—
192C22×Dic3—91no—
193C22×Dic3—271no—
194C2×C4×D3—271no—
195C2×C4×D3—271no—
196C4×A4—93no—
197SL(2,ℤ/4)—273no—
198C3×C22⋊C4—271no—
199#217—91no—
200#217—91no—
201#217—181no—
202#217—271no—
203(C2×D3)⋊C4—271no—
204(C2×D3)⋊C4—271no—
205C4⋊Dic3—271no—
206Dic3⋊C4—271no—
207Dic3⋊C4—271no—
208C4×Dic3—271no—
209C4×Dic3—271no—
210C4×Dic3—271no—
211C22×C32—21no—
212C22×C32—31no—
213C22×C32—31no—
214C22×C32—31no—
215C22×C32—31no—
216C22×C32—31no—
217C22×C32—31no—
218C22×C32—31no—
219C22×C32—61no—
220C22×C32—61no—
221C22×C32—61no—
222C22×C32—61no—
223C22×C32—61no—
224C22×C32—61no—
225C22×C32—61no—
226C22×C32—121no—
227C22×C32—121no—
228C22×C32#62411yescharacteristic
229C22×C32C4×C3⋊D311yescharacteristic, complemented
230C2×C3⋊D3—32no—
231C2×C3⋊D3—61no—
232C2×C3⋊D3—61no—
233C2×C3⋊D3—121no—
234C2×C3⋊D3—241no—
235C2×C3×D3—92no—
236C2×C3×D3—92no—
237C2×C3×D3—182no—
238C2×C3×D3—181no—
239C2×C3×D3—181no—
240C2×C3×D3—181no—
241C2×C3×D3—181no—
242C2×C3×D3—361no—
243C2×C3×D3—363no—
244C2×C3×D3—361no—
245C2×C3×D3—723no—
246C3×A4—23no—
247C3×A4—31no—
248C3×A4—61no—
249C3×A4—61no—
250C3×A4—121no—
251C3×A4#62413yescomplemented
252C32×C4—361no—
253C3⋊Dic3—32no—
254C3⋊Dic3—31no—
255C3⋊Dic3—61no—
256C3⋊Dic3—61no—
257C3⋊Dic3—123no—
258C3⋊Dic3—361no—
259C3⋊Dic3—721no—
260C3⋊Dic3C3⋊S411yescharacteristic, complemented
261C3×Dic3—31no—
262C3×Dic3—91no—
263C3×Dic3—91no—
264C3×Dic3—92no—
265C3×Dic3—92no—
266C3×Dic3—91no—
267C3×Dic3—123no—
268C3×Dic3—181no—
269C3×Dic3—182no—
270C3×Dic3—181no—
271C3×Dic3—181no—
272C3×Dic3—181no—
273C3×Dic3—181no—
274C3×Dic3—361no—
275C3×Dic3—361no—
276C4×D4—811noSylow
277C33—41no—
278C33—81no—
279C33C4×S411yescharacteristic, complemented
280H(3,𝔽3)—43no—
281H(3,𝔽3)—83no—
282C23×C3—21no—
283C23×C3—31no—
284C23×C3—61no—
285C23×C3—271no—
286C23×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic
287C23×C3D3×C3⋊D311yescharacteristic
288C22×D3—91no—
289C22×D3—91no—
290C22×D3—181no—
291C2×A4—33no—
292C2×A4—63no—
293S4—96no—
294S4—186no—
295C3×D4—272no—
296C3×D4—272no—
297C2×C3×C4—271no—
298C2×C3×C4—271no—
299C2×C3×C4—271no—
300C2×C3×C4—271no—
301C2×C3×C4—271no—
302#67—92no—
303#67—92no—
304#67—92no—
305#67—92no—
306#67—182no—
307#67—182no—
308C2×Dic3—91no—
309C2×Dic3—91no—
310C2×Dic3—91no—
311C2×Dic3—91no—
312C2×Dic3—91no—
313C2×Dic3—91no—
314C2×Dic3—91no—
315C2×Dic3—91no—
316C2×Dic3—91no—
317C2×Dic3—91no—
318C2×Dic3—181no—
319C2×Dic3—181no—
320C2×Dic3—181no—
321C2×Dic3—181no—
322C2×Dic3—181no—
323C2×Dic3—541no—
324C2×Dic3—541no—
325C4×D3—541no—
326C4×D3—541no—
327C4×D3—541no—
328C4×D3—541no—
329C4×D3—1083no—
330C2×C32—21no—
331C2×C32—31no—
332C2×C32—31no—
333C2×C32—31no—
334C2×C32—31no—
335C2×C32—31no—
336C2×C32—31no—
337C2×C32—31no—
338C2×C32—43no—
339C2×C32—61no—
340C2×C32—61no—
341C2×C32—61no—
342C2×C32—61no—
343C2×C32—61no—
344C2×C32—61no—
345C2×C32—61no—
346C2×C32—83no—
347C2×C32—121no—
348C2×C32—121no—
349C2×C32—121no—
350C2×C32—241no—
351C2×C32—241no—
352C2×C32—481no—
353C2×C32D3×S411yescharacteristic
354C2×C32C2×C3⋊S411yescharacteristic
355C3⋊D3—62no—
356C3⋊D3—122no—
357C3⋊D3—242no—
358C3×D3—182no—
359C3×D3—182no—
360C3×D3—362no—
361C3×D3—366no—
362C3×D3—726no—
363C2×D4—271no—
364C22×C4—91no—
365C22×C4—811no—
366C4⋊C4—811no—
367C22⋊C4—271no—
368C22⋊C4—811no—
369C42—811no—
370C22×C3—21no—
371C22×C3—31no—
372C22×C3—31no—
373C22×C3—31no—
374C22×C3—31no—
375C22×C3—31no—
376C22×C3—31no—
377C22×C3—31no—
378C22×C3—61no—
379C22×C3—61no—
380C22×C3—61no—
381C22×C3—61no—
382C22×C3—61no—
383C22×C3—61no—
384C22×C3—61no—
385C22×C3—121no—
386C22×C3—121no—
387C22×C3—272no—
388C22×C3—541no—
389C22×C3—541no—
390C22×C3#619911yescharacteristic, complemented
391C22×C3#628911yescharacteristic
392C2×D3—92no—
393C2×D3—92no—
394C2×D3—182no—
395C2×D3—181no—
396C2×D3—181no—
397C2×D3—181no—
398C2×D3—181no—
399C2×D3—361no—
400C2×D3—361no—
401C2×D3—363no—
402C2×D3—723no—
403A4—33no—
404A4—63no—
405C3×C4—91no—
406C3×C4—272no—
407C3×C4—271no—
408C3×C4—363no—
409C3×C4—541no—
410C3×C4—541no—
411Dic3—31no—
412Dic3—91no—
413Dic3—91no—
414Dic3—92no—
415Dic3—92no—
416Dic3—91no—
417Dic3—181no—
418Dic3—182no—
419Dic3—181no—
420Dic3—181no—
421Dic3—181no—
422Dic3—181no—
423Dic3—361no—
424Dic3—361no—
425Dic3—1083no—
426C32—21no—
427C32—31no—
428C32—43no—
429C32—61no—
430C32—83no—
431C32—121no—
432C32—241no—
433C32—241no—
434C32—481no—
435C32(C3⋊S4)⋊C411yescharacteristic
436C32C4×C3⋊S411yescharacteristic, complemented
437C23—271no—
438C23((C3⋊D3)⋊C3)⋊D311yescharacteristic
439D4—272no—
440D4—272no—
441C2×C4—271no—
442C2×C4—271no—
443C2×C4—271no—
444C2×C4—271no—
445C2×C4—271no—
446C2×C4—1621no—
447C2×C4—1621no—
448C2×C3—21no—
449C2×C3—31no—
450C2×C3—31no—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—31no—
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—61no—
457C2×C3—61no—
458C2×C3—61no—
459C2×C3—61no—
460C2×C3—61no—
461C2×C3—61no—
462C2×C3—61no—
463C2×C3—121no—
464C2×C3—121no—
465C2×C3—123no—
466C2×C3—243no—
467C2×C3—542no—
468C2×C3((C3⋊S4)⋊C3)⋊C211yescharacteristic
469C2×C3D3×C3⋊S411yescharacteristic
470D3—182no—
471D3—182no—
472D3—362no—
473D3—366no—
474D3—726no—
475C22—31no—
476C22—31no—
477C22—272no—
478C22—541no—
479C22—541no—
480C22#1061988511yescharacteristic, complemented
481C4—91no—
482C4—272no—
483C4—271no—
484C4—541no—
485C4—541no—
486C3—21no—
487C3—31no—
488C3—61no—
489C3—123no—
490C3—243no—
491C3#6258⋊C411yescharacteristic, complemented
492C3#1172420111yescharacteristic
493C2—31no—
494C2—31no—
495C2—542no—
496C2#1304921511yescharacteristic
subgroupG/H
center, hypercenterC2#13049215
commutator, hypocenterC3×C22⋊H(3,𝔽3)C2×C4
FrattiniC2×C3D3×C3⋊S4
FittingC23×C33C2×D3
socleC23×C32D32
prime radicalC2×C3⋊#6258C2
cosocleC2×C3×C22⋊H(3,𝔽3)C22
LCSG ▷ C3×C22⋊H(3,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C3×(C3⋊Dic3)⋊A4C2semidirect
3. C3⋊((C3⋊SL(2,ℤ/4))⋊C3)C2semidirect
8. C3⋊#6258C4semidirect
13. C22×C3×C3⋊Dic3D3semidirect
16. C3⋊(C3⋊SL(2,ℤ/4))D3semidirect
23. (C3⋊Dic3)⋊A4D3semidirect
26. C3×C22⋊H(3,𝔽3)C2×C4semidirect
62. C22×C3⋊Dic3C3⋊D3semidirect
91. C22×C33C4×D3semidirect
98. C32×A4C4×D3semidirect
104. C3×C3⋊Dic3S4semidirect
109. C22⋊H(3,𝔽3)C4×D3semidirect
229. C22×C32C4×C3⋊D3semidirect
251. C3×A4#624semidirect
260. C3⋊Dic3C3⋊S4semidirect
279. C33C4×S4semidirect
390. C22×C3#6199semidirect
436. C32C4×C3⋊S4semidirect
480. C22#10619885semidirect
491. C3#6258⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊#62582
C3×(C3⋊Dic3)⋊A42
C3⋊((C3⋊SL(2,ℤ/4))⋊C3)2
Dic3×C3⋊S43
#117237263
#6258⋊C43
#106198854