Finite Group Databasebeta

← #13139198

Subgroups — #13139198

Order 1728.

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

normal subgroups only

This group has 571 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×#88652C211yesmaximal, characteristic
2M16×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#11721053C211yesmaximal, characteristic, complemented
4#11720560C211yesmaximal, characteristic, complemented
5#63651⋊C3C211yesmaximal, characteristic, complemented
6#11720325C211yesmaximal, characteristic, complemented
7#11720321C211yesmaximal, characteristic, complemented
8#10591010—31nomaximal
9#10591006—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#6200C412yes—
14C2×#6199C412yes—
15#88652—21no—
16#88652—21no—
17#88652—21no—
18#88652—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×#3796—31no—
28C2×#3795—31no—
29M16×C3⋊D3D311yescharacteristic, complemented
30C3×D3×M16—31no—
31#63818—31no—
32D3×#208—31no—
33#63651D311yescharacteristic, complemented
34C3×#824—31no—
35C3×#823—31no—
36C3×#823—31no—
37C3×#823—61no—
38#63561—31no—
39#63560—31no—
40#63558—31no—
41#63558—31no—
42C22×#272C2×C412yes—
43C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C4⋊#272—21no—
46C4⋊#272—21no—
47C2×(C3⋊Dic3)⋊C3C2×C412yes—
48C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
49C4×(C3⋊D3)⋊C3D411yescharacteristic
50C4×(C3⋊D3)⋊C3D411yescharacteristic
51C4×(C3⋊D3)⋊C3D411yescharacteristic
52C4×(C3⋊D3)⋊C3D411yescharacteristic
53#634⋊C3—21no—
54#634⋊C3—21no—
55#6200—22no—
56#6200—22no—
57#6199—22no—
58#6199—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⋊D3C2×D311yescharacteristic
67C2×#634C2×D311yescharacteristic
68C2×C3×D12—31no—
69C2×C3×D12—31no—
70C2×C3×D12—61no—
71C2×C3×C4×D3—31no—
72C2×C3×Dic6—31no—
73C2×#626—32no—
74C2×#625—32no—
75C2×D3×Dic3—32no—
76C2×D3×Dic3—32no—
77#3796—61no—
78#3796—61no—
79#3796—61no—
80#3796—61no—
81#3795—61no—
82#3795—61no—
83#3795—61no—
84#3795—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×#272—22no—
105C2×#272—42no—
106C2×(C3⋊D3)⋊C3—41no—
107C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
108C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
109C4×H(3,𝔽3)C2×D411yescharacteristic
110C4×H(3,𝔽3)C2×D411yescharacteristic
111(C3⋊Dic3)⋊C3—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—91no—
129#824—91no—
130#824—91no—
131#823—31no—
132#823—31no—
133#823—61no—
134#823—91no—
135#823—91no—
136C22×C3⋊D3C4×D311yescharacteristic
137C22×C3×D3—31no—
138C22×C3×D3—32no—
139C22×C3×D3—32no—
140C22×C3×D3—62no—
141C2×C32×C4—21no—
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C3⋊Dic3C4×D312yes—
145C2×C3⋊Dic3C4×D311yescharacteristic
146C4×C3⋊D3D1211yescharacteristic
147C4×C3⋊D3D1211yescharacteristic
148C4×C3⋊D3#6711yescharacteristic
149C4×C3⋊D3#6711yescharacteristic
150#634—21no—
151#634—21no—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—32no—
154C3×D12—61no—
155C3×D12—61no—
156C3×D12—61no—
157C3×D12—61no—
158C3×D12—121no—
159C3×D12—121no—
160C3×C4×D3—31no—
161C3×C4×D3—31no—
162C3×C4×D3—31no—
163C3×C4×D3—31no—
164C3×Dic6—61no—
165C3×Dic6—61no—
166#626—62no—
167#626—62no—
168#625—62no—
169#625—62no—
170D3×Dic3—62no—
171D3×Dic3—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#272—42no—
188(C3⋊D3)⋊C3—21no—
189(C3⋊D3)⋊C3—21no—
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—31no—
210C2×D12—61no—
211C2×D12—92no—
212C2×C4×D3—31no—
213C2×C4×D3—91no—
214C2×C4×D3—92no—
215C2×C4×D3—92no—
216C2×Dic6—31no—
217C2×Dic6—91no—
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—41no—
248C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
249C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
250C2×C3×D3—31no—
251C2×C3×D3—31no—
252C2×C3×D3—62no—
253C2×C3×D3—62no—
254C2×C3×D3—122no—
255C2×C3×D3—121no—
256C2×C3×D3—122no—
257C2×C3×D3—122no—
258C2×C3×D3—242no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4C2×D1211yescharacteristic
262C32×C4D3×D411yescharacteristic
263C32×C4D3×D411yescharacteristic
264C32×C4C2×#6711yescharacteristic
265C3⋊Dic3—22no—
266C3⋊Dic3(C2×D3)⋊C411yescharacteristic
267C3⋊Dic3(C2×D3)⋊C411yescharacteristic
268C3×Dic3—31no—
269C3×Dic3—31no—
270C3×Dic3—62no—
271C2×#41—271no—
272C2×M16—91no—
273C2×M16—271no—
274#101—91no—
275#101—271no—
276#100—91no—
277#100—271no—
278H(3,𝔽3)#41311yescharacteristic, complemented, Sylow
279C23×C3—91no—
280C23×C3—92no—
281C22×D3—31no—
282C22×D3—32no—
283C22×D3—32no—
284C22×D3—62no—
285C22×D3—91no—
286C3×Q8—181no—
287C3×Q8—181no—
288C3×D4—181no—
289C3×D4—181no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—61no—
293C2×C3×C4—91no—
294C2×C3×C4—91no—
295C2×C3×C4—91no—
296C2×C3×C4—91no—
297C2×C3×C4—91no—
298C2×C3×C4—92no—
299C2×C3×C4C2×D3211yescharacteristic
300#67—182no—
301#67—182no—
302#67—182no—
303#67—182no—
304C2×Dic3—31no—
305C2×Dic3—32no—
306C2×Dic3—91no—
307C2×Dic3—92no—
308C2×Dic3—182no—
309C2×Dic3—182no—
310D12—61no—
311D12—61no—
312D12—61no—
313D12—61no—
314D12—121no—
315D12—121no—
316D12—182no—
317D12—182no—
318C4×D3—31no—
319C4×D3—31no—
320C4×D3—31no—
321C4×D3—31no—
322C4×D3—91no—
323C4×D3—91no—
324C4×D3—91no—
325C4×D3—91no—
326C4×D3—182no—
327C4×D3—182no—
328C4×D3—182no—
329C4×D3—182no—
330Dic6—61no—
331Dic6—61no—
332Dic6—181no—
333Dic6—181no—
334C3×C8—21no—
335C3×C8—61no—
336C3×C8—61no—
337C3×C8—121no—
338C3×C8—181no—
339C3×C8—181no—
340C3×C8—181no—
341C3⋊C8—61no—
342C3⋊C8—61no—
343C3⋊C8—61no—
344C3⋊C8—61no—
345C3⋊C8—121no—
346C3⋊C8—181no—
347C3⋊C8—181no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—41no—
352C2×C32D3×C22⋊C411yescharacteristic
353C2×C32C2×(C2×D3)⋊C411yescharacteristic
354C3⋊D3—21no—
355C3⋊D3—21no—
356C3×D3—61no—
357C3×D3—61no—
358C3×D3—122no—
359C3×D3—122no—
360C3×D3—242no—
361#41—541no—
362#41—541no—
363#41—541no—
364#41—541no—
365C2×Q8—91no—
366C2×D4—91no—
367C2×D4—272no—
368C22×C4—91no—
369C22×C4—272no—
370M16—91no—
371M16—91no—
372M16—91no—
373M16—181no—
374M16—541no—
375M16((C3⋊D3)⋊C3)⋊C211yescharacteristic
376C2×C8—181no—
377C2×C8—541no—
378C22×C3—31no—
379C22×C3—31no—
380C22×C3—61no—
381C22×C3—91no—
382C22×C3—91no—
383C22×C3—182no—
384C22×C3—362no—
385C22×C3—361no—
386C22×C3C4×D3211yescharacteristic
387C2×D3—31no—
388C2×D3—31no—
389C2×D3—62no—
390C2×D3—62no—
391C2×D3—91no—
392C2×D3—91no—
393C2×D3—122no—
394C2×D3—121no—
395C2×D3—122no—
396C2×D3—122no—
397C2×D3—242no—
398C2×D3—361no—
399C3×C4—31no—
400C3×C4—31no—
401C3×C4—31no—
402C3×C4—31no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—91no—
406C3×C4—91no—
407C3×C4—182no—
408C3×C4D3×D1211yescharacteristic
409C3×C4D3×#6711yescharacteristic
410Dic3—31no—
411Dic3—31no—
412Dic3—62no—
413Dic3—91no—
414Dic3—91no—
415Dic3—182no—
416C32—21no—
417C32#475111yescharacteristic, complemented
418C32#513211yescharacteristic, complemented
419C23—91no—
420C23—92no—
421Q8—181no—
422Q8—181no—
423D4—181no—
424D4—181no—
425D4—542no—
426D4—542no—
427C2×C4—91no—
428C2×C4—91no—
429C2×C4—91no—
430C2×C4—91no—
431C2×C4—91no—
432C2×C4—92no—
433C2×C4—542no—
434C2×C4—542no—
435C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
436C8—21no—
437C8—181no—
438C8—181no—
439C8—181no—
440C2×C3—21no—
441C2×C3—31no—
442C2×C3—31no—
443C2×C3—61no—
444C2×C3—61no—
445C2×C3—61no—
446C2×C3—121no—
447C2×C3—181no—
448C2×C3—181no—
449C2×C3—362no—
450C2×C3D3×(C2×D3)⋊C411yescharacteristic
451D3—61no—
452D3—61no—
453D3—122no—
454D3—122no—
455D3—181no—
456D3—181no—
457D3—242no—
458C22—91no—
459C22—91no—
460C22—182no—
461C22—361no—
462C22—362no—
463C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
464C4—91no—
465C4—91no—
466C4—182no—
467C4#8865411yescharacteristic
468C4#8867611yescharacteristic
469C3—31no—
470C3—31no—
471C3—61no—
472C3#1059100711yescharacteristic
473C2—21no—
474C2—181no—
475C2—181no—
476C2—362no—
477C2#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×#88652C2
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. #11720560C2semidirect
5. #63651⋊C3C2semidirect
6. #11720325C2semidirect
7. #11720321C2semidirect
29. M16×C3⋊D3D3semidirect
33. #63651D3semidirect
278. H(3,𝔽3)#413semidirect
417. C32#4751semidirect
418. C32#5132semidirect

Maximal subgroups

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

subgroupindex
C2×#886522
M16×(C3⋊D3)⋊C32
#117210532
#117205602
#63651⋊C32
#117203252
#117203212
#105910103
#105910063