Finite Group Databasebeta

← #13162029

Subgroups — #13162029

Order 1728.

858 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C3×#64364C211yesmaximal, characteristic
2C3×#64153C211yesmaximal, characteristic, complemented
3#11723663C212yesmaximal, complemented
4C3×#63663C211yesmaximal, characteristic, complemented
5#11722077C212yesmaximal, complemented
6#10591650—31nomaximal
7#10591649—31nomaximal
8#10591648—31nomaximal
9C2×C33×Q8C2211yescharacteristic
10C3×#3832—22no—
11C3×Q8×C3⋊D3—22no—
12C3×#3828—22no—
13C3×#3828C2212yes—
14C2×C3×#634C2211yescharacteristic
15C3×#3750—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C3×#3748C2212yes—
18#88812—22no—
19#88811—22no—
20#64364D311yescharacteristic
21C3×#1007—31no—
22C3×#1007—31no—
23C3×#1007—61no—
24#64153D311yescharacteristic, complemented
25C3×#944—31no—
26C3×#944—31no—
27C3×#944—61no—
28#63833—32no—
29#63831—32no—
30#63827—32no—
31#63663D311yescharacteristic, complemented
32C3×#836—31no—
33C3×#836—31no—
34C3×#836—61no—
35#63570—32no—
36#63569—32no—
37#63568—32no—
38C33×Q8—22no—
39C2×C33×C4D412yes—
40C2×C33×C4C2311yescharacteristic
41C3×#638—22no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C3×C4⋊(C3⋊D3)—22no—
45C3×C4⋊(C3⋊D3)D412yes—
46C3×C4×C3⋊D3—24no—
47C3×C4×C3⋊D3—22no—
48C3×#634—22no—
49C3×#634—22no—
50C3×#634D412yes—
51C3×C3⋊(C3⋊C8)—22no—
52#5246—91no—
53#5095—91no—
54#4752—91no—
55C2×C32×Q8—21no—
56C2×C32×Q8—21no—
57C2×C32×Q8—21no—
58C2×C32×Q8—41no—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C2×D311yescharacteristic
62#3832—22no—
63Q8×C3⋊D3—22no—
64#3828—22no—
65#3828C2×D312yes—
66C2×#634C2×D311yescharacteristic
67C3×#239—62no—
68C3×#239—62no—
69C3×#239—64no—
70C3×D3×Q8—62no—
71C3×D3×Q8—62no—
72C3×D3×Q8—64no—
73C3×#235—32no—
74C3×#235—32no—
75C3×#235—62no—
76C3×#235—64no—
77C3×#235—62no—
78C3×#235—62no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82#3750—22no—
83C4×C3⋊Dic3C2×D311yescharacteristic
84#3748C2×D312yes—
85C3×Dic3⋊C4—64no—
86C3×Dic3⋊C4—62no—
87C3×Dic3⋊C4—62no—
88C3×C4×Dic3—31no—
89C3×C4×Dic3—31no—
90C3×C4×Dic3—61no—
91C3×#208—32no—
92C3×#208—32no—
93C3×#208—62no—
94#3719—62no—
95#3719—62no—
96#3718—62no—
97#3717—62no—
98#3716—62no—
99#3715—62no—
100C22×C33C2×D411yescharacteristic
101C2×C3×C3⋊D3—22no—
102C33×C4—22no—
103C33×C4C2×D412yes—
104C3×C3⋊Dic3—22no—
105C3×C3⋊Dic3—22no—
106C3×C3⋊Dic3—41no—
107C3×2−1+4—91no—
108#1007—31no—
109#1007—31no—
110#1007—61no—
111C3×C4⋊Q8—91no—
112#951—92no—
113#944—31no—
114#944—31no—
115#944—61no—
116#942—92no—
117#909—92no—
118C3×#101—91no—
119#837—92no—
120#836—31no—
121#836—31no—
122#836—61no—
123#825—92no—
124#805—92no—
125C32×Q8—22no—
126C32×Q8—24no—
127C32×Q8—22no—
128C32×Q8—22no—
129C32×Q8—42no—
130C32×Q8—42no—
131C32×Q8—44no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—42no—
139C2×C32×C4—41no—
140C2×C32×C4D1212yes—
141C2×C32×C4#6712yes—
142C2×C32×C4#6712yes—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146#638—22no—
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3—21no—
149C4⋊(C3⋊D3)—22no—
150C4⋊(C3⋊D3)#6712yes—
151C4×C3⋊D3—24no—
152C4×C3⋊D3—22no—
153#634—22no—
154#634—22no—
155#634#6712yes—
156C3×#67—64no—
157C3×#67—62no—
158C3×#67—62no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C3×D12—32no—
166C3×D12—32no—
167C3×D12—62no—
168C3×D12—64no—
169C3×D12—62no—
170C3×D12—62no—
171C3×C4×D3—68no—
172C3×C4×D3—64no—
173C3×C4×D3—64no—
174C3×C4×D3—64no—
175C3×C4×D3—62no—
176C3×C4×D3—62no—
177C3×Dic6—32no—
178C3×Dic6—32no—
179C3×Dic6—62no—
180C3×Dic6—64no—
181C3×Dic6—62no—
182C3×Dic6—62no—
183C3×Dic6—62no—
184C3×Dic6—62no—
185C3×Dic6—122no—
186C3⋊(C3⋊C8)—22no—
187C3×C3⋊C8—62no—
188C3×C3⋊C8—62no—
189C3×C3⋊C8—122no—
190#456—271noSylow
191C2×C33—21no—
192C2×C33#12011yescharacteristic
193C3×C3⋊D3—42no—
194C3×#41—92no—
195C3×#41—182no—
196C3×#41—182no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—41no—
201C2×C3×Q8—91no—
202C2×C3×Q8—182no—
203C2×C3×Q8D3211yescharacteristic
204C2×C3×Q8D3211yescharacteristic
205C2×C3×Q8D3211yescharacteristic
206#239—62no—
207#239—62no—
208#239—64no—
209D3×Q8—62no—
210D3×Q8—62no—
211D3×Q8—64no—
212#235—32no—
213#235—32no—
214#235—62no—
215#235—64no—
216#235—62no—
217#235—62no—
218C2×Dic6—31no—
219C2×Dic6—31no—
220C2×Dic6—61no—
221C3×M16—92no—
222C3×C4⋊C4—182no—
223C3×C42—91no—
224#216—182no—
225#216—182no—
226#215—182no—
227#214—182no—
228Dic3⋊C4—64no—
229Dic3⋊C4—62no—
230Dic3⋊C4—62no—
231C4×Dic3—31no—
232C4×Dic3—31no—
233C4×Dic3—61no—
234#208—32no—
235#208—32no—
236#208—62no—
237Dic12—182no—
238#204—182no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—41no—
243C22×C32C2×D1211yescharacteristic
244C22×C32C2×#6711yescharacteristic
245C22×C32C2×#6711yescharacteristic
246C2×C3⋊D3—22no—
247C2×C3×D3—64no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C32×C4—22no—
251C32×C4—24no—
252C32×C4—22no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—42no—
258C32×C4—42no—
259C32×C4—42no—
260C32×C4—44no—
261C32×C4D3×D412yes—
262C32×C4D3×D412yes—
263C32×C4C2×#6712yes—
264C3⋊Dic3—22no—
265C3⋊Dic3—22no—
266C3⋊Dic3—41no—
267C3×Dic3—64no—
268C3×Dic3—64no—
269C3×Dic3—62no—
270C3×Dic3—62no—
271C3×Dic3—62no—
272C3×Dic3—62no—
273C3×Dic3—122no—
274C3×Dic3—121no—
275C3×Dic3—121no—
2762−1+4—91no—
277#137—272no—
278C4⋊Q8—91no—
279#104—272no—
280#101—91no—
281C33#45611yescharacteristic, complemented, Sylow
282C3×Q8—24no—
283C3×Q8—22no—
284C3×Q8—22no—
285C3×Q8—22no—
286C3×Q8—42no—
287C3×Q8—42no—
288C3×Q8—44no—
289C3×Q8—92no—
290C3×Q8—182no—
291C3×Q8—182no—
292C3×D4—92no—
293C3×D4—182no—
294C3×D4—182no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—184no—
304C2×C3×C4—181no—
305C2×C3×C4—182no—
306C2×C3×C4—181no—
307C2×C3×C4#62512yes—
308C2×C3×C4#62612yes—
309C2×C3×C4#62612yes—
310C2×C3×C4C2×D3211yescharacteristic
311C2×C3×C4C2×D3211yescharacteristic
312C2×C3×C4C2×D3211yescharacteristic
313#67—64no—
314#67—62no—
315#67—62no—
316C2×Dic3—62no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—62no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322D12—32no—
323D12—32no—
324D12—62no—
325D12—64no—
326D12—62no—
327D12—62no—
328C4×D3—68no—
329C4×D3—64no—
330C4×D3—64no—
331C4×D3—64no—
332C4×D3—62no—
333C4×D3—62no—
334Dic6—32no—
335Dic6—32no—
336Dic6—62no—
337Dic6—64no—
338Dic6—62no—
339Dic6—62no—
340Dic6—62no—
341Dic6—62no—
342Dic6—122no—
343C3×C8—182no—
344C3⋊C8—62no—
345C3⋊C8—62no—
346C3⋊C8—122no—
347C2×C32—21no—
348C2×C32—22no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—41no—
355C2×C32—41no—
356C2×C32—41no—
357C2×C32—42no—
358C2×C32#88211yescharacteristic
359C2×C32#93711yescharacteristic
360C2×C32#95311yescharacteristic
361C3⋊D3—42no—
362C3×D3—124no—
363C3×D3—122no—
364C3×D3—122no—
365#41—92no—
366#41—182no—
367#41—182no—
368C2×Q8—91no—
369C2×Q8—182no—
370C2×Q8#111311yescharacteristic
371Q16—542no—
372SD16—542no—
373M16—92no—
374C4⋊C4—182no—
375C42—91no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—41no—
380C22×C3—182no—
381C22×C3C2×#62511yescharacteristic
382C22×C3C2×#62611yescharacteristic
383C22×C3C2×#62611yescharacteristic
384C2×D3—64no—
385C2×D3—62no—
386C2×D3—62no—
387C3×C4—24no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—42no—
395C3×C4—42no—
396C3×C4—44no—
397C3×C4—42no—
398C3×C4—182no—
399C3×C4—182no—
400C3×C4—361no—
401C3×C4D3×#6712yes—
402C3×C4D3×#6712yes—
403C3×C4#381112yes—
404Dic3—64no—
405Dic3—64no—
406Dic3—62no—
407Dic3—62no—
408Dic3—62no—
409Dic3—62no—
410Dic3—122no—
411Dic3—121no—
412Dic3—121no—
413C32—21no—
414C32—21no—
415C32—21no—
416C32—41no—
417C32#475211yescharacteristic, complemented
418C32#509511yescharacteristic, complemented
419C32#524611yescharacteristic, complemented
420Q8—22no—
421Q8—92no—
422Q8—182no—
423Q8—182no—
424D4—92no—
425D4—182no—
426D4—182no—
427C2×C4—184no—
428C2×C4—181no—
429C2×C4—182no—
430C2×C4—181no—
431C2×C4#629512yes—
432C2×C4C2×#111311yescharacteristic
433C8—182no—
434C2×C3—22no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3—42no—
444C2×C3—41no—
445C2×C3—362no—
446C2×C3#6397611yescharacteristic
447C2×C3#6397711yescharacteristic
448C2×C3#6398111yescharacteristic
449D3—124no—
450D3—122no—
451D3—122no—
452C22—182no—
453C22C2×#629511yescharacteristic
454C4—22no—
455C4—182no—
456C4—182no—
457C4—361no—
458C4#8904912yes—
459C3—21no—
460C3—21no—
461C3—21no—
462C3—41no—
463C3#1059164811yescharacteristic, complemented
464C3#1059164911yescharacteristic, complemented
465C3#1059165011yescharacteristic, complemented
466C2—21no—
467C2—362no—
468C2#1172372711yescharacteristic
subgroupG/H
centerC2#11723727
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC2×C33#120
prime radicalC3×#64364C2
hypercenterC2×Q8#1113
hypocenterC33#456
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#64153C2semidirect
3. #11723663C2semidirect
4. C3×#63663C2semidirect
5. #11722077C2semidirect
24. #64153D3semidirect
31. #63663D3semidirect
281. C33#456semidirect
417. C32#4752semidirect
418. C32#5095semidirect
419. C32#5246semidirect
463. C3#10591648semidirect
464. C3#10591649semidirect
465. C3#10591650semidirect

Maximal subgroups

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

subgroupindex
C3×#643642
C3×#641532
#117236632
C3×#636632
#117220772
#105916503
#105916493
#105916483