Finite Group Databasebeta

← #13161771

Subgroups — #13161771

Order 1728.

792 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 111 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#11724618C211yesmaximal, characteristic
2C3×M16×C3⋊D3C211yesmaximal, characteristic, complemented
3#11723673C211yesmaximal, characteristic, complemented
4C3×#63652C212yesmaximal, complemented
5#11722077C212yesmaximal, complemented
6#10591009—31nomaximal
7#10591005—32nomaximal
8C3×#3828C2212yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10#89049—21no—
11#89046—21no—
12C4⋊#1113C411yescharacteristic, complemented
13C4×#1113—21no—
14#89042—21no—
15#89041C412yescomplemented
16#89040C411yescharacteristic, complemented
17C33×M16C2211yescharacteristic, complemented
18C3×C4×C3⋊Dic3C2212yescomplemented
19C3×#3748C2211yescharacteristic, complemented
20C3×#3743—21no—
21C3×C8×C3⋊D3—21no—
22#88815—21no—
23#88814—21no—
24#64306—31no—
25D3×#235—32no—
26M16×C3⋊D3D311yescharacteristic, complemented
27C3×D3×M16—32no—
28C3×D3×M16—61no—
29#63874—31no—
30#63855—32no—
31#63652D312yescomplemented
32C3×#825—32no—
33C3×#825—32no—
34C3×#825—62no—
35#63570—32no—
36#63569—32no—
37#63568—32no—
38C22×C3×C3⋊D3D411yescharacteristic
39C2×#1113—21no—
40C2×C33×C4C2311yescharacteristic
41C3×#638—22no—
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4⋊(C3⋊D3)C2×C412yes—
45C3×C4×C3⋊D3—22no—
46C3×C4×C3⋊D3C2×C411yescharacteristic
47C3×C4×C3⋊D3C2×C411yescharacteristic
48C3×C4×C3⋊D3D411yescharacteristic
49C3×C4×C3⋊D3D411yescharacteristic
50C3×#634C2×C412yes—
51#6296—21no—
52#6295—22no—
53#6295—21no—
54#6294—21no—
55#6294—22no—
56C33×C8—21no—
57C3×C3⋊(C3⋊C8)—21no—
58#5139—91no—
59#4821—92no—
60#3828C2×D312yes—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C3×#235—32no—
63C3×#235—32no—
64C3×#235—62no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—61no—
67#3811—61no—
68D3×#67—62no—
69#3805—61no—
70#3804—62no—
71#3802—31no—
72D3×D12—32no—
73C4×D32—62no—
74C4×D32—61no—
75#3799—32no—
76#3798—62no—
77#3798—61no—
78#3797—31no—
79#3796—32no—
80#3795—32no—
81D3×Dic6—32no—
82C32×M16—21no—
83C32×M16—22no—
84C32×M16—41no—
85C32×M16C2×D312yescomplemented
86C32×M16C2×D311yescharacteristic, complemented
87C4×C3⋊Dic3C2×D312yescomplemented
88#3748C2×D311yescharacteristic, complemented
89#3743—21no—
90C8×C3⋊D3—21no—
91C3×C4×Dic3—32no—
92C3×C4×Dic3—32no—
93C3×C4×Dic3—62no—
94C3×#208—32no—
95C3×#208—61no—
96C3×ΓL(1,𝔽25)—62no—
97C3×ΓL(1,𝔽25)—62no—
98C3×D3×C8—62no—
99C3×D3×C8—62no—
100#3724—61no—
101Dic3⋊Dic3—62no—
102#3722—61no—
103(C2×D3)⋊Dic3—62no—
104C22×C33C2×D411yescharacteristic
105C2×C3×C3⋊D3—22no—
106C2×C3×C3⋊D3—41no—
107C2×C3×C3⋊D3C22⋊C411yescharacteristic
108C2×C3×C3⋊D3C22⋊C411yescharacteristic
109#1113—42no—
110C33×C4C22×C411yescharacteristic
111C33×C4C2×D411yescharacteristic
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3—42no—
114C3×C3⋊Dic3C22⋊C411yescharacteristic
115C3×C3⋊Dic3C22⋊C411yescharacteristic
116D3×#41—92no—
117C2×#235—91no—
118C2×C3×M16—91no—
119#926—91no—
120D3×M16—32no—
121D3×M16—61no—
122#872—92no—
123C3×#104—92no—
124#837—92no—
125#825—32no—
126#825—32no—
127#825—62no—
128#825—92no—
129#805—92no—
130C22×C3⋊D3D1211yescharacteristic
131C22×C3×D3—32no—
132C22×C3×D3—61no—
133C2×D32—62no—
134C2×D32—61no—
135C2×C32×C4—21no—
136C2×C32×C4—22no—
137C2×C32×C4—41no—
138C2×C32×C4C22×D312yes—
139C2×C32×C4C22×D311yescharacteristic
140#638—22no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3D1211yescharacteristic
143C4⋊(C3⋊D3)C4×D312yes—
144C4×C3⋊D3—22no—
145C4×C3⋊D3C4×D311yescharacteristic
146C4×C3⋊D3C4×D311yescharacteristic
147C4×C3⋊D3#6711yescharacteristic
148C4×C3⋊D3#6711yescharacteristic
149#634C4×D312yes—
150C3×#67—62no—
151C3×#67—62no—
152C3×#67—122no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—122no—
158C3×D12—32no—
159C3×D12—32no—
160C3×D12—62no—
161C3×C4×D3—32no—
162C3×C4×D3—32no—
163C3×C4×D3—32no—
164C3×C4×D3—32no—
165C3×C4×D3—61no—
166C3×C4×D3—61no—
167C3×C4×D3—61no—
168C3×C4×D3—61no—
169C3×C4×D3—62no—
170C3×C4×D3—62no—
171C3×C4×D3—122no—
172C3×Dic6—32no—
173C3×Dic6—32no—
174C3×Dic6—62no—
175#627—62no—
176#627—61no—
177#626—62no—
178#626—62no—
179#626—62no—
180#625—62no—
181#625—61no—
182#624—62no—
183#624—61no—
184D3×Dic3—62no—
185D3×Dic3—62no—
186D3×Dic3—62no—
187C32×C8—21no—
188C32×C8—22no—
189C32×C8—22no—
190C32×C8—42no—
191C32×C8—42no—
192C3⋊(C3⋊C8)—21no—
193C3×C3⋊C8—62no—
194C3×C3⋊C8—62no—
195#421—271noSylow
196C2×C33—21no—
197C2×C33C2×C22⋊C411yescharacteristic
198C3×C3⋊D3—21no—
199C3×C3⋊D3—21no—
200C3×C3⋊D3—42no—
201C3×#41—92no—
202C22×C3×C4—91no—
203C2×#67—181no—
204#239—92no—
205D3×Q8—92no—
206#237—92no—
207#237—182no—
208D3×D4—92no—
209D3×D4—182no—
210#235—32no—
211#235—32no—
212#235—62no—
213#235—92no—
214#235—182no—
215#235—181no—
216#235—181no—
217C2×D12—91no—
218C2×C4×D3—32no—
219C2×C4×D3—61no—
220C2×C4×D3—182no—
221C2×C4×D3—181no—
222C2×Dic6—91no—
223C3×M16—21no—
224C3×M16—22no—
225C3×M16—41no—
226C3×M16—91no—
227C3×M16—181no—
228C3×M16D3211yescharacteristic, complemented
229C3×M16D3212yescomplemented
230C2×C3×C8—181no—
231C3×C42—92no—
232#217—181no—
233(C2×D3)⋊C4—182no—
234C4⋊Dic3—181no—
235Dic3⋊C4—182no—
236C4×Dic3—32no—
237C4×Dic3—32no—
238C4×Dic3—62no—
239#208—32no—
240#208—61no—
241ΓL(1,𝔽25)—62no—
242ΓL(1,𝔽25)—62no—
243D3×C8—62no—
244D3×C8—62no—
245C22×C32—22no—
246C22×C32—21no—
247C22×C32—41no—
248C22×C32C2×D1211yescharacteristic
249C22×C32D3×D412yes—
250C2×C3⋊D3—22no—
251C2×C3⋊D3—41no—
252C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
253C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
254C2×C3×D3—32no—
255C2×C3×D3—32no—
256C2×C3×D3—61no—
257C2×C3×D3—61no—
258C2×C3×D3—62no—
259C2×C3×D3—62no—
260C2×C3×D3—122no—
261C2×C3×D3—122no—
262C2×C3×D3—122no—
263D32—124no—
264D32—122no—
265C32×C4—22no—
266C32×C4—21no—
267C32×C4—22no—
268C32×C4—21no—
269C32×C4—41no—
270C32×C4—41no—
271C32×C4C2×C4×D312yes—
272C32×C4C2×C4×D311yescharacteristic
273C32×C4D3×D412yes—
274C32×C4C2×#6711yescharacteristic
275C3⋊Dic3—22no—
276C3⋊Dic3—42no—
277C3⋊Dic3(C2×D3)⋊C411yescharacteristic
278C3⋊Dic3(C2×D3)⋊C411yescharacteristic
279C3×Dic3—32no—
280C3×Dic3—32no—
281C3×Dic3—61no—
282C3×Dic3—61no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3—122no—
286C3×Dic3—122no—
287C3×Dic3—122no—
288C3×Dic3—124no—
289C2×#41—271no—
290C2×M16—91no—
291#117—271no—
292#104—92no—
293#104—272no—
294C33#42111yescharacteristic, complemented, Sylow
295C23×C3—91no—
296C22×D3—32no—
297C22×D3—61no—
298C22×D3—182no—
299C22×D3—181no—
300C3×Q8—92no—
301C3×D4—92no—
302C3×D4—182no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—41no—
306C2×C3×C4—91no—
307C2×C3×C4—91no—
308C2×C3×C4—91no—
309C2×C3×C4—91no—
310C2×C3×C4—91no—
311C2×C3×C4—182no—
312C2×C3×C4—182no—
313C2×C3×C4C2×D3212yes—
314C2×C3×C4C2×D3211yescharacteristic
315#67—62no—
316#67—62no—
317#67—122no—
318#67—182no—
319#67—182no—
320#67—182no—
321C2×Dic3—32no—
322C2×Dic3—61no—
323C2×Dic3—62no—
324C2×Dic3—62no—
325C2×Dic3—122no—
326C2×Dic3—182no—
327C2×Dic3—181no—
328D12—32no—
329D12—32no—
330D12—62no—
331D12—181no—
332D12—182no—
333C4×D3—32no—
334C4×D3—32no—
335C4×D3—32no—
336C4×D3—32no—
337C4×D3—61no—
338C4×D3—61no—
339C4×D3—61no—
340C4×D3—61no—
341C4×D3—62no—
342C4×D3—62no—
343C4×D3—122no—
344C4×D3—182no—
345C4×D3—182no—
346C4×D3—182no—
347Dic6—32no—
348Dic6—32no—
349Dic6—62no—
350Dic6—182no—
351Dic6—181no—
352C3×C8—22no—
353C3×C8—22no—
354C3×C8—21no—
355C3×C8—42no—
356C3×C8—42no—
357C3×C8—181no—
358C3⋊C8—62no—
359C3⋊C8—62no—
360C2×C32—21no—
361C2×C32—22no—
362C2×C32—22no—
363C2×C32—21no—
364C2×C32—41no—
365C2×C32—42no—
366C2×C32—41no—
367C2×C32—81no—
368C2×C32D3×C22⋊C412yes—
369C2×C32C2×(C2×D3)⋊C411yescharacteristic
370C3⋊D3—21no—
371C3⋊D3—21no—
372C3⋊D3—42no—
373C3×D3—62no—
374C3×D3—62no—
375C3×D3—62no—
376C3×D3—121no—
377C3×D3—122no—
378C3×D3—122no—
379C3×D3—242no—
380#41—92no—
381#41—272no—
382#41—541no—
383#41—541no—
384C2×Q8—271no—
385C2×D4—271no—
386C2×D4—541no—
387C22×C4—91no—
388C22×C4—541no—
389M16—91no—
390M16—181no—
391M16#111311yescharacteristic, complemented
392C2×C8—181no—
393C4⋊C4—541no—
394C22⋊C4—541no—
395C42—92no—
396C22×C3—21no—
397C22×C3—22no—
398C22×C3—41no—
399C22×C3—91no—
400C22×C3—91no—
401C22×C3—182no—
402C22×C3—361no—
403C22×C3D3×D1212yes—
404C22×C3#380211yescharacteristic
405C2×D3—32no—
406C2×D3—32no—
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—62no—
410C2×D3—62no—
411C2×D3—122no—
412C2×D3—122no—
413C2×D3—122no—
414C2×D3—364no—
415C2×D3—362no—
416C3×C4—22no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—22no—
420C3×C4—41no—
421C3×C4—41no—
422C3×C4—91no—
423C3×C4—91no—
424C3×C4—182no—
425C3×C4—362no—
426C3×C4C4×D3212yes—
427C3×C4C4×D3211yescharacteristic
428C3×C4D3×#6712yes—
429C3×C4#381111yescharacteristic
430Dic3—32no—
431Dic3—32no—
432Dic3—61no—
433Dic3—61no—
434Dic3—62no—
435Dic3—62no—
436Dic3—122no—
437Dic3—122no—
438Dic3—122no—
439Dic3—124no—
440C32—22no—
441C32—21no—
442C32—41no—
443C32#482112yescomplemented
444C32#513911yescharacteristic, complemented
445C23—91no—
446C23—541no—
447Q8—92no—
448Q8—541no—
449D4—92no—
450D4—182no—
451D4—541no—
452D4—542no—
453C2×C4—91no—
454C2×C4—91no—
455C2×C4—91no—
456C2×C4—91no—
457C2×C4—91no—
458C2×C4—182no—
459C2×C4—182no—
460C2×C4—542no—
461C2×C4—541no—
462C2×C4C2×#111311yescharacteristic
463C8—21no—
464C8—181no—
465C2×C3—22no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—22no—
469C2×C3—42no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—81no—
473C2×C3—181no—
474C2×C3—181no—
475C2×C3—362no—
476C2×C3D3×(C2×D3)⋊C412yes—
477C2×C3#6392111yescharacteristic
478D3—62no—
479D3—62no—
480D3—62no—
481D3—121no—
482D3—122no—
483D3—122no—
484D3—242no—
485C22—91no—
486C22—91no—
487C22—182no—
488C22—361no—
489C22—1082no—
490C22C4⋊#111311yescharacteristic
491C4—91no—
492C4—91no—
493C4—182no—
494C4—362no—
495C4C4×#111311yescharacteristic
496C4#8904911yescharacteristic
497C3—21no—
498C3—22no—
499C3—41no—
500C3#1059100512yescomplemented
501C3#1059100911yescharacteristic, complemented
502C2—21no—
503C2—181no—
504C2—181no—
505C2—362no—
506C2#1172370011yescharacteristic
subgroupG/H
centerC4#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC33×M16C22
socleC2×C33C2×C22⋊C4
prime radical#11724618C2
cosocleC2×C33×C4C23
hypercenterM16#1113
hypocenterC33#421
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C3×M16×C3⋊D3C2semidirect
3. #11723673C2semidirect
4. C3×#63652C2semidirect
5. #11722077C2semidirect
12. C4⋊#1113C4semidirect
15. #89041C4semidirect
16. #89040C4semidirect
17. C33×M16C22semidirect
18. C3×C4×C3⋊Dic3C22semidirect
19. C3×#3748C22semidirect
26. M16×C3⋊D3D3semidirect
31. #63652D3semidirect
85. C32×M16C2×D3semidirect
86. C32×M16C2×D3semidirect
87. C4×C3⋊Dic3C2×D3semidirect
88. #3748C2×D3semidirect
228. C3×M16D32semidirect
229. C3×M16D32semidirect
294. C33#421semidirect
391. M16#1113semidirect
443. C32#4821semidirect
444. C32#5139semidirect
500. C3#10591005semidirect
501. C3#10591009semidirect

Maximal subgroups

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

subgroupindex
#117246182
C3×M16×C3⋊D32
#117236732
C3×#636522
#117220772
#105910093
#105910053