Finite Group Databasebeta

← #10592831⋊C3 #13143215

Subgroups — #10592831⋊C3 #13143215

Order 1728.

593 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2C2×M16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4#63650⋊C3C212yesmaximal, complemented
5#63651⋊C3C212yesmaximal, complemented
6#10592831C311yesmaximal, characteristic, complemented
7C3×#5132—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3C412yes—
10#3828⋊C3—21no—
11#3828⋊C3—21no—
12#3828⋊C3—21no—
13#3828⋊C3—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C412yes—
16C2×#634⋊C3C2211yescharacteristic
17M16×H(3,𝔽3)—21no—
18M16×H(3,𝔽3)C2212yes—
19C2×C8×H(3,𝔽3)—21no—
20#3748⋊C3—21no—
21#3748⋊C3C2212yes—
22C2×(C3⋊(C3⋊C8))⋊C3—21no—
23C2×#3828C2×C311yescharacteristic
24C2×C3×#235—31no—
25C2×C32×M16—21no—
26C2×C32×M16D311yescharacteristic, complemented
27C2×C32×M16C2×C311yescharacteristic, complemented
28C2×#3748C2×C311yescharacteristic, complemented
29C2×C3×#208—31no—
30#63651C2×C312yescomplemented
31#63650C2×C312yescomplemented
32C3×#824—32no—
33C3×#823—32no—
34C23×H(3,𝔽3)C2×C411yescharacteristic
35C22×(C3⋊D3)⋊C3C2×C412yes—
36C2×C4×H(3,𝔽3)C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42#638⋊C3—22no—
43#638⋊C3—22no—
44C2×(C3⋊Dic3)⋊C3C2×C412yes—
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4×(C3⋊D3)⋊C3—22no—
48C4×(C3⋊D3)⋊C3—22no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51C8×H(3,𝔽3)—22no—
52(C3⋊(C3⋊C8))⋊C3—22no—
53C3×#413—91no—
54#5132—31no—
55#5132—91no—
56C22×C32×C4—21no—
57C22×C32×C4C2×D311yescharacteristic
58C22×C32×C4C22×C311yescharacteristic
59C2×#638C3×C412yes—
60#3828—21no—
61#3828—21no—
62#3828—21no—
63#3828—21no—
64C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
65C2×C4×C3⋊D3C3×C412yes—
66C2×#634C22×C311yescharacteristic
67C2×C3×#67—32no—
68C3×#235—61no—
69C3×#235—61no—
70C3×#235—61no—
71C3×#235—61no—
72C2×C3×D12—31no—
73C2×C3×C4×D3—32no—
74C2×C3×Dic6—31no—
75C32×M16—22no—
76C32×M16—22no—
77C32×M16—21no—
78C32×M16—21no—
79C32×M16C2×D312yes—
80C32×M16C22×C312yes—
81C2×C32×C8—22no—
82C2×C32×C8—21no—
83C2×C32×C8—21no—
84#3748—21no—
85#3748C22×C312yes—
86C2×C3⋊(C3⋊C8)—21no—
87C3×#208—32no—
88C3×#208—61no—
89C2×C3×C3⋊C8—61no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)C22⋊C411yescharacteristic
92C22×H(3,𝔽3)C22⋊C411yescharacteristic
93C22×H(3,𝔽3)C22×C411yescharacteristic
94C2×(C3⋊D3)⋊C3—22no—
95C2×(C3⋊D3)⋊C3—42no—
96C4×H(3,𝔽3)C22⋊C411yescharacteristic
97C4×H(3,𝔽3)C22⋊C411yescharacteristic
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)C2×D411yescharacteristic
100(C3⋊Dic3)⋊C3—22no—
101C2×C3×#41—91no—
102C2×#235—31no—
103C2×#235—91no—
104C2×C3×M16—31no—
105C2×C3×M16—31no—
106C2×C3×M16—61no—
107C2×C3×M16—91no—
108C2×C3×M16C3×D311yescharacteristic
109C2×#208—31no—
110C2×#208—91no—
111C3×#101—92no—
112C3×#100—92no—
113#824—32no—
114#824—92no—
115#823—32no—
116#823—92no—
117C23×C32—21no—
118C23×C32C4×D311yescharacteristic
119C23×C32C2×C3×C411yescharacteristic
120C22×C3⋊D3C2×C3×C412yes—
121C22×C3×D3—32no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4C4×D311yescharacteristic
129C2×C32×C4D1211yescharacteristic
130C2×C32×C4D1211yescharacteristic
131C2×C32×C4#6711yescharacteristic
132C2×C32×C4#6711yescharacteristic
133C2×C32×C4C2×C3×C411yescharacteristic
134C2×C32×C4C3×D411yescharacteristic
135C2×C32×C4C3×D411yescharacteristic
136C2×C32×C4C3×D411yescharacteristic
137C2×C32×C4C3×D411yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C23×C311yescharacteristic
140#638—22no—
141#638—22no—
142C2×C3⋊Dic3C2×C3×C412yes—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—21no—
145C4×C3⋊D3—22no—
146C4×C3⋊D3—22no—
147#634—21no—
148#634—21no—
149C3×#67—62no—
150C3×#67—62no—
151C2×C3×Dic3—32no—
152C3×D12—61no—
153C3×D12—61no—
154C3×C4×D3—62no—
155C3×C4×D3—62no—
156C3×Dic6—61no—
157C3×Dic6—61no—
158C32×C8—24no—
159C32×C8—22no—
160C32×C8—22no—
161C3⋊(C3⋊C8)—22no—
162C3×C3⋊C8—62no—
163#413—91noSylow
164C2×H(3,𝔽3)—21no—
165C2×H(3,𝔽3)—21no—
166C2×H(3,𝔽3)—21no—
167C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
168(C3⋊D3)⋊C3—42no—
169C3×#41—181no—
170C3×#41—181no—
171C3×#41—181no—
172C3×#41—181no—
173C2×C3×Q8—91no—
174C2×C3×D4—91no—
175C2×C3×D4—92no—
176C22×C3×C4—31no—
177C22×C3×C4—31no—
178C22×C3×C4—61no—
179C22×C3×C4—92no—
180C22×C3×C4C2×C3×D311yescharacteristic
181C2×#67—32no—
182C2×#67—92no—
183#235—61no—
184#235—61no—
185#235—61no—
186#235—61no—
187#235—181no—
188#235—181no—
189#235—181no—
190#235—181no—
191C2×D12—31no—
192C2×D12—91no—
193C2×C4×D3—32no—
194C2×C4×D3—92no—
195C2×Dic6—31no—
196C2×Dic6—91no—
197C3×M16—21no—
198C3×M16—32no—
199C3×M16—32no—
200C3×M16—61no—
201C3×M16—61no—
202C3×M16—62no—
203C3×M16—62no—
204C3×M16—92no—
205C3×M16—181no—
206C3×M16C2×C3×D312yes—
207C2×C3×C8—21no—
208C2×C3×C8—61no—
209C2×C3×C8—61no—
210C2×C3×C8—62no—
211C2×C3×C8—181no—
212#208—32no—
213#208—61no—
214#208—92no—
215#208—181no—
216C2×C3⋊C8—61no—
217C2×C3⋊C8—181no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—42no—
224C22×C32(C2×D3)⋊C411yescharacteristic
225C22×C32(C2×D3)⋊C411yescharacteristic
226C22×C32C3×C22⋊C411yescharacteristic
227C22×C32C3×C22⋊C411yescharacteristic
228C22×C32C2×C4×D311yescharacteristic
229C22×C32C22×C3×C411yescharacteristic
230C2×C3⋊D3—22no—
231C2×C3⋊D3—42no—
232C2×C3×D3—62no—
233C2×C3×D3—122no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4(C2×D3)⋊C411yescharacteristic
239C32×C4(C2×D3)⋊C411yescharacteristic
240C32×C4C3×C22⋊C411yescharacteristic
241C32×C4C3×C22⋊C411yescharacteristic
242C32×C4C2×D1211yescharacteristic
243C32×C4C2×#6711yescharacteristic
244C32×C4C2×C3×D411yescharacteristic
245C32×C4C2×C3×D411yescharacteristic
246C3⋊Dic3—22no—
247C3×Dic3—62no—
248C2×#41—91no—
249C2×M16—91no—
250C2×M16(C3⋊D3)⋊C311yescharacteristic, complemented
251#101—92no—
252#100—92no—
253H(3,𝔽3)#41311yescharacteristic, complemented, Sylow
254C23×C3—31no—
255C23×C3—31no—
256C23×C3—61no—
257C23×C3—92no—
258C23×C3C3×C4×D311yescharacteristic
259C22×D3—32no—
260C22×D3—92no—
261C3×Q8—181no—
262C3×Q8—181no—
263C3×D4—181no—
264C3×D4—181no—
265C3×D4—182no—
266C3×D4—182no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—92no—
286C2×C3×C4—182no—
287C2×C3×C4—182no—
288C2×C3×C4C3×C4×D311yescharacteristic
289C2×C3×C4C3×D1211yescharacteristic
290C2×C3×C4C3×D1211yescharacteristic
291C2×C3×C4C3×#6711yescharacteristic
292C2×C3×C4C3×#6711yescharacteristic
293C2×C3×C4C22×C3×D311yescharacteristic
294#67—62no—
295#67—62no—
296#67—182no—
297#67—182no—
298C2×Dic3—32no—
299C2×Dic3—92no—
300D12—61no—
301D12—61no—
302D12—181no—
303D12—181no—
304C4×D3—62no—
305C4×D3—62no—
306C4×D3—182no—
307C4×D3—182no—
308Dic6—61no—
309Dic6—61no—
310Dic6—181no—
311Dic6—181no—
312C3×C8—22no—
313C3×C8—62no—
314C3×C8—62no—
315C3×C8—64no—
316C3×C8—182no—
317C3⋊C8—62no—
318C3⋊C8—182no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—41no—
328C2×C32—41no—
329C2×C32C2×(C2×D3)⋊C411yescharacteristic
330C2×C32C2×C3×C22⋊C411yescharacteristic
331C3⋊D3—42no—
332C3×D3—122no—
333#41—181no—
334#41—181no—
335#41—181no—
336#41—181no—
337C2×Q8—91no—
338C2×D4—92no—
339C2×D4—91no—
340C22×C4—92no—
341C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
342M16—21no—
343M16—92no—
344M16—181no—
345M16C2×(C3⋊D3)⋊C312yes—
346C2×C8—21no—
347C2×C8—181no—
348C22×C3—31no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—31no—
353C22×C3—31no—
354C22×C3—41no—
355C22×C3—61no—
356C22×C3—61no—
357C22×C3—61no—
358C22×C3—121no—
359C22×C3—121no—
360C22×C3—122no—
361C22×C3—182no—
362C22×C3—362no—
363C22×C3C3×(C2×D3)⋊C411yescharacteristic
364C22×C3C3×(C2×D3)⋊C411yescharacteristic
365C22×C3C2×C3×C4×D311yescharacteristic
366C2×D3—62no—
367C2×D3—122no—
368C2×D3—182no—
369C2×D3—362no—
370C3×C4—31no—
371C3×C4—31no—
372C3×C4—31no—
373C3×C4—31no—
374C3×C4—31no—
375C3×C4—31no—
376C3×C4—31no—
377C3×C4—31no—
378C3×C4—61no—
379C3×C4—61no—
380C3×C4—61no—
381C3×C4—61no—
382C3×C4—182no—
383C3×C4C3×(C2×D3)⋊C411yescharacteristic
384C3×C4C3×(C2×D3)⋊C411yescharacteristic
385C3×C4C2×C3×D1211yescharacteristic
386C3×C4C2×C3×#6711yescharacteristic
387Dic3—62no—
388Dic3—182no—
389C32—21no—
390C32#513211yescharacteristic, complemented
391C32C3×#41311yescharacteristic, complemented
392C23—92no—
393C23C4×(C3⋊D3)⋊C311yescharacteristic
394Q8—181no—
395Q8—181no—
396D4—182no—
397D4—182no—
398D4—181no—
399D4—181no—
400C2×C4—92no—
401C2×C4—182no—
402C2×C4—182no—
403C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
404C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
405C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
406C2×C4#638⋊C311yescharacteristic
407C2×C4#638⋊C311yescharacteristic
408C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
409C8—22no—
410C8—182no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—31no—
415C2×C3—31no—
416C2×C3—61no—
417C2×C3—61no—
418C2×C3—61no—
419C2×C3—61no—
420C2×C3—61no—
421C2×C3—61no—
422C2×C3—62no—
423C2×C3—61no—
424C2×C3—121no—
425C2×C3—121no—
426C2×C3—362no—
427C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
428D3—122no—
429D3—362no—
430C22—41no—
431C22—182no—
432C22—362no—
433C22#3752⋊C311yescharacteristic
434C22#3752⋊C311yescharacteristic
435C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
436C4—182no—
437C4#3752⋊C311yescharacteristic
438C4#3752⋊C311yescharacteristic
439C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
440C4C2×#638⋊C311yescharacteristic
441C3—31no—
442C3—31no—
443C3—61no—
444C3C3×#513211yescharacteristic
445C2—21no—
446C2—21no—
447C2—21no—
448C2—362no—
449C2C2×#3752⋊C311yescharacteristic
subgroupG/H
centerC4#3752⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×H(3,𝔽3)C2
socleC2×C3C2×C3×(C2×D3)⋊C4
prime radicalC2×#3828⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenterC2×M16(C3⋊D3)⋊C3
hypocenterC32C3×#413
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×M16×H(3,𝔽3)C2semidirect
3. C2×#3748⋊C3C2semidirect
4. #63650⋊C3C2semidirect
5. #63651⋊C3C2semidirect
6. #10592831C3semidirect
26. C2×C32×M16D3semidirect
27. C2×C32×M16C2×C3semidirect
28. C2×#3748C2×C3semidirect
30. #63651C2×C3semidirect
31. #63650C2×C3semidirect
250. C2×M16(C3⋊D3)⋊C3semidirect
253. H(3,𝔽3)#413semidirect
390. C32#5132semidirect
391. C32C3×#413semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C2×M16×H(3,𝔽3)2
C2×#3748⋊C32
#63650⋊C32
#63651⋊C32
#105928313
C3×#51323