Finite Group Databasebeta

← #13139202

Subgroups — #13139202

Order 1728.

563 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 85 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×#88648C211yesmaximal, characteristic
2M16×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#11721053C211yesmaximal, characteristic, complemented
4#11720561C211yesmaximal, characteristic, complemented
5#63650⋊C3C211yesmaximal, characteristic, complemented
6#11720327C211yesmaximal, characteristic, complemented
7#11720320C211yesmaximal, characteristic, complemented
8#10591010—31nomaximal
9#10591007—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#6198C412yes—
14C2×((C3⋊D3)⋊C3)⋊C4C412yes—
15#88648—21no—
16#88648—21no—
17#88648—21no—
18#88648—21no—
19M16×H(3,𝔽3)C2211yescharacteristic
20#88538C2211yescharacteristic
21#3748⋊C3C2211yescharacteristic
22#3748⋊C3C2211yescharacteristic
23#3743⋊C3—21no—
24C8×(C3⋊D3)⋊C3—21no—
25#88430—21no—
26((C3⋊D3)⋊C3)⋊C8—21no—
27C2×#3799—31no—
28C2×#3796—31no—
29M16×C3⋊D3D311yescharacteristic, complemented
30C3×D3×M16—31no—
31#63818—31no—
32D3×#208—31no—
33#63650D311yescharacteristic, complemented
34C3×#824—31no—
35C3×#824—31no—
36C3×#824—61no—
37C3×#823—31no—
38#63560—31no—
39#63560—31no—
40#63559—31no—
41#63558—31no—
42C22×(C3⋊D3)⋊C3C2×C412yes—
43C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×#1084C2×C412yes—
46#6229—21no—
47#6229—21no—
48C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4×(C3⋊D3)⋊C3D411yescharacteristic
52C4×(C3⋊D3)⋊C3D411yescharacteristic
53C4×(C3⋊D3)⋊C3D411yescharacteristic
54C4×(C3⋊D3)⋊C3D411yescharacteristic
55#6198—22no—
56#6198—22no—
57((C3⋊D3)⋊C3)⋊C4—22no—
58((C3⋊D3)⋊C3)⋊C4—22no—
59C8×H(3,𝔽3)—21no—
60#6180—21no—
61(C3⋊(C3⋊C8))⋊C3—21no—
62(C3⋊(C3⋊C8))⋊C3—21no—
63#5132—91no—
64#4751—91no—
65#4745—91no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C2×C3×D12—31no—
69C2×C3×C4×D3—31no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—61no—
73C2×#626—32no—
74C2×#626—32no—
75C2×#624—32no—
76C2×D3×Dic3—32no—
77#3799—61no—
78#3799—61no—
79#3799—61no—
80#3799—61no—
81#3796—61no—
82#3796—61no—
83#3796—61no—
84#3796—61no—
85C32×M16—21no—
86C32×M16C2×D311yescharacteristic
87C32×M16C2×D311yescharacteristic
88#3748C2×D311yescharacteristic
89#3748C2×D311yescharacteristic
90#3743—21no—
91C8×C3⋊D3—21no—
92C3×#208—31no—
93C3×#208—31no—
94C3×#208—31no—
95C3×#208—31no—
96C3×#208—61no—
97C3×ΓL(1,𝔽25)—61no—
98C3×D3×C8—61no—
99#3712—61no—
100#3711—61no—
101#3710—61no—
102D3×C3⋊C8—61no—
103C22×H(3,𝔽3)C22×C411yescharacteristic
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3—41no—
106C2×(C3⋊D3)⋊C3—42no—
107C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
108C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
109C4×H(3,𝔽3)C2×D411yescharacteristic
110C4×H(3,𝔽3)C2×D411yescharacteristic
111#1084—22no—
112(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
113(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
114C2×#239—91no—
115C2×#237—91no—
116C2×#235—91no—
117C2×C3×M16—91no—
118C2×#208—91no—
119D3×M16—31no—
120D3×M16—91no—
121D3×M16—91no—
122D3×M16—91no—
123C3×#101—91no—
124C3×#100—91no—
125#836—91no—
126#833—91no—
127#824—31no—
128#824—31no—
129#824—61no—
130#824—91no—
131#824—91no—
132#823—31no—
133#823—91no—
134#823—91no—
135#823—91no—
136C22×C3⋊D3C4×D312yes—
137C22×C3⋊D3C4×D311yescharacteristic
138C22×C3×D3—31no—
139C22×C3×D3—32no—
140C2×C32×C4—21no—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C3⋊Dic3C4×D311yescharacteristic
144C4⋊(C3⋊D3)—21no—
145C4⋊(C3⋊D3)—21no—
146C4×C3⋊D3D1211yescharacteristic
147C4×C3⋊D3D1211yescharacteristic
148C4×C3⋊D3#6711yescharacteristic
149C4×C3⋊D3#6711yescharacteristic
150C2×C3×Dic3—31no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—62no—
154C3×D12—61no—
155C3×D12—61no—
156C3×C4×D3—31no—
157C3×C4×D3—31no—
158C3×C4×D3—31no—
159C3×C4×D3—31no—
160C3×Dic6—61no—
161C3×Dic6—61no—
162C3×Dic6—61no—
163C3×Dic6—61no—
164C3×Dic6—121no—
165C3×Dic6—121no—
166#626—62no—
167#626—62no—
168#626—62no—
169#626—62no—
170#624—62no—
171#624—62no—
172D3×Dic3—62no—
173D3×Dic3—62no—
174C32×C8—21no—
175C32×C8—21no—
176C32×C8—41no—
177C3⋊(C3⋊C8)—21no—
178C3⋊(C3⋊C8)—21no—
179C3×C3⋊C8—61no—
180C3×C3⋊C8—61no—
181C3×C3⋊C8—61no—
182C3×C3⋊C8—61no—
183C3×C3⋊C8—121no—
184#413—271noSylow
185C2×H(3,𝔽3)—21no—
186C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
187(C3⋊D3)⋊C3—21no—
188(C3⋊D3)⋊C3—21no—
189(C3⋊D3)⋊C3—42no—
190C2×C3×Q8—91no—
191C2×C3×D4—91no—
192C22×C3×C4—91no—
193C2×#67—92no—
194C2×#67—92no—
195C22×Dic3—92no—
196#239—181no—
197#239—181no—
198#239—181no—
199#239—181no—
200#237—181no—
201#237—181no—
202#237—181no—
203#237—181no—
204#235—181no—
205#235—181no—
206#235—181no—
207#235—181no—
208C2×D12—31no—
209C2×D12—91no—
210C2×D12—92no—
211C2×C4×D3—31no—
212C2×C4×D3—91no—
213C2×C4×D3—92no—
214C2×C4×D3—92no—
215C2×Dic6—31no—
216C2×Dic6—31no—
217C2×Dic6—61no—
218C3×M16—31no—
219C3×M16—31no—
220C3×M16—61no—
221C3×M16—91no—
222C3×M16—91no—
223C3×M16—91no—
224C3×M16—181no—
225C3×M16D3211yescharacteristic
226C2×C3×C8—181no—
227#208—31no—
228#208—31no—
229#208—31no—
230#208—31no—
231#208—61no—
232#208—91no—
233#208—91no—
234#208—181no—
235C2×C3⋊C8—181no—
236ΓL(1,𝔽25)—61no—
237ΓL(1,𝔽25)—181no—
238ΓL(1,𝔽25)—181no—
239ΓL(1,𝔽25)—181no—
240D3×C8—61no—
241D3×C8—181no—
242D3×C8—181no—
243D3×C8—181no—
244C22×C32—21no—
245C22×C32C2×C4×D311yescharacteristic
246C22×C32C2×C4×D311yescharacteristic
247C2×C3⋊D3—22no—
248C2×C3⋊D3—41no—
249C2×C3⋊D3—42no—
250C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
251C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
252C2×C3×D3—31no—
253C2×C3×D3—31no—
254C2×C3×D3—62no—
255C2×C3×D3—121no—
256C2×C3×D3—122no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4C2×D1211yescharacteristic
260C32×C4D3×D411yescharacteristic
261C32×C4D3×D411yescharacteristic
262C32×C4C2×#6711yescharacteristic
263C3⋊Dic3(C2×D3)⋊C411yescharacteristic
264C3⋊Dic3(C2×D3)⋊C411yescharacteristic
265C3×Dic3—31no—
266C3×Dic3—31no—
267C3×Dic3—62no—
268C3×Dic3—62no—
269C3×Dic3—122no—
270C2×#41—271no—
271C2×M16—91no—
272C2×M16—271no—
273#101—91no—
274#101—271no—
275#100—91no—
276#100—271no—
277H(3,𝔽3)#41311yescharacteristic, complemented, Sylow
278C23×C3—91no—
279C23×C3—92no—
280C22×D3—31no—
281C22×D3—32no—
282C22×D3—91no—
283C22×D3—92no—
284C3×Q8—181no—
285C3×Q8—181no—
286C3×D4—181no—
287C3×D4—181no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—61no—
291C2×C3×C4—91no—
292C2×C3×C4—91no—
293C2×C3×C4—91no—
294C2×C3×C4—91no—
295C2×C3×C4—91no—
296C2×C3×C4—92no—
297C2×C3×C4C2×D3211yescharacteristic
298#67—182no—
299#67—182no—
300#67—182no—
301#67—182no—
302C2×Dic3—31no—
303C2×Dic3—32no—
304C2×Dic3—32no—
305C2×Dic3—62no—
306C2×Dic3—91no—
307C2×Dic3—182no—
308C2×Dic3—182no—
309D12—61no—
310D12—61no—
311D12—181no—
312D12—181no—
313D12—182no—
314D12—182no—
315C4×D3—31no—
316C4×D3—31no—
317C4×D3—31no—
318C4×D3—31no—
319C4×D3—91no—
320C4×D3—91no—
321C4×D3—91no—
322C4×D3—91no—
323C4×D3—182no—
324C4×D3—182no—
325C4×D3—182no—
326C4×D3—182no—
327Dic6—61no—
328Dic6—61no—
329Dic6—61no—
330Dic6—61no—
331Dic6—121no—
332Dic6—121no—
333C3×C8—21no—
334C3×C8—61no—
335C3×C8—61no—
336C3×C8—121no—
337C3×C8—181no—
338C3×C8—181no—
339C3×C8—181no—
340C3⋊C8—61no—
341C3⋊C8—61no—
342C3⋊C8—61no—
343C3⋊C8—61no—
344C3⋊C8—121no—
345C3⋊C8—181no—
346C3⋊C8—181no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—41no—
351C2×C32D3×C22⋊C411yescharacteristic
352C2×C32C2×(C2×D3)⋊C411yescharacteristic
353C3⋊D3—21no—
354C3⋊D3—21no—
355C3⋊D3—42no—
356C3×D3—61no—
357C3×D3—61no—
358C3×D3—122no—
359#41—541no—
360#41—541no—
361#41—541no—
362#41—541no—
363C2×Q8—91no—
364C2×D4—91no—
365C2×D4—272no—
366C22×C4—91no—
367C22×C4—272no—
368M16—91no—
369M16—91no—
370M16—91no—
371M16—181no—
372M16—541no—
373M16((C3⋊D3)⋊C3)⋊C211yescharacteristic
374C2×C8—181no—
375C2×C8—541no—
376C22×C3—31no—
377C22×C3—31no—
378C22×C3—61no—
379C22×C3—91no—
380C22×C3—91no—
381C22×C3—182no—
382C22×C3—361no—
383C22×C3—362no—
384C22×C3C4×D3211yescharacteristic
385C2×D3—31no—
386C2×D3—31no—
387C2×D3—62no—
388C2×D3—91no—
389C2×D3—91no—
390C2×D3—121no—
391C2×D3—122no—
392C2×D3—182no—
393C2×D3—361no—
394C2×D3—362no—
395C3×C4—31no—
396C3×C4—31no—
397C3×C4—31no—
398C3×C4—31no—
399C3×C4—61no—
400C3×C4—61no—
401C3×C4—91no—
402C3×C4—91no—
403C3×C4—182no—
404C3×C4D3×D1211yescharacteristic
405C3×C4D3×#6711yescharacteristic
406Dic3—31no—
407Dic3—31no—
408Dic3—62no—
409Dic3—62no—
410Dic3—91no—
411Dic3—91no—
412Dic3—122no—
413C32—21no—
414C32#474511yescharacteristic, complemented
415C32#513211yescharacteristic, complemented
416C23—91no—
417C23—92no—
418Q8—181no—
419Q8—181no—
420D4—181no—
421D4—181no—
422D4—542no—
423D4—542no—
424C2×C4—91no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—91no—
428C2×C4—91no—
429C2×C4—92no—
430C2×C4—542no—
431C2×C4—542no—
432C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
433C8—21no—
434C8—181no—
435C8—181no—
436C8—181no—
437C2×C3—21no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—61no—
441C2×C3—61no—
442C2×C3—61no—
443C2×C3—121no—
444C2×C3—181no—
445C2×C3—181no—
446C2×C3—362no—
447C2×C3D3×(C2×D3)⋊C411yescharacteristic
448D3—61no—
449D3—61no—
450D3—122no—
451D3—181no—
452D3—181no—
453D3—362no—
454C22—91no—
455C22—91no—
456C22—182no—
457C22—361no—
458C22—362no—
459C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
460C4—91no—
461C4—91no—
462C4—182no—
463C4#8865411yescharacteristic
464C4#8867611yescharacteristic
465C3—31no—
466C3—31no—
467C3—61no—
468C3#1059100611yescharacteristic
469C2—21no—
470C2—181no—
471C2—181no—
472C2—362no—
473C2#1172116011yescharacteristic
subgroupG/H
centerC2#11721160
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingM16×H(3,𝔽3)C22
socleC2×C3D3×(C2×D3)⋊C4
prime radicalC2×#88648C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterM16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#413
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. M16×(C3⋊D3)⋊C3C2semidirect
3. #11721053C2semidirect
4. #11720561C2semidirect
5. #63650⋊C3C2semidirect
6. #11720327C2semidirect
7. #11720320C2semidirect
29. M16×C3⋊D3D3semidirect
33. #63650D3semidirect
277. H(3,𝔽3)#413semidirect
414. C32#4745semidirect
415. C32#5132semidirect

Maximal subgroups

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

subgroupindex
C2×#886482
M16×(C3⋊D3)⋊C32
#117210532
#117205612
#63650⋊C32
#117203272
#117203202
#105910103
#105910073