Finite Group Databasebeta

← #13147780

Subgroups — #13147780

Order 1728.

1090 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88648C212yesmaximal, complemented
2#11721715C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
5#11721141C212yesmaximal, complemented
6#11721131C214yesmaximal, complemented
7#11721129C212yesmaximal, complemented
8#11721107C212yesmaximal, complemented
9#10593896—31nomaximal
10#10593869—31nomaximal
11C2×#6229C2212yes—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2212yes—
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×#6198C2214yes—
17C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
18#88648—28no—
19C42×H(3,𝔽3)C2211yescharacteristic
20C4⋊#1084C2212yes—
21C4⋊#1084C2212yes—
22#3752⋊C3C2212yes—
23#3752⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26#88450C2214yes—
27#88448C2214yes—
28C2×#3799—32no—
29C2×#3796—32no—
30C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
31C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
32C3×C4×D12—31no—
33C3×C4×D12—31no—
34C3×#869—31no—
35C3×#869—31no—
36C3×#869—61no—
37#63916—32no—
38#63909—32no—
39#63900—32no—
40#63890—34no—
41D3×C4⋊Dic3—32no—
42#63871—32no—
43#63856—32no—
44#63854—34no—
45C22×(C3⋊D3)⋊C3C2312yes—
46C22×(C3⋊D3)⋊C3C2312yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×#1084C2314yes—
51#6229—24no—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C4⋊((C3⋊D3)⋊C3)—22no—
55C4⋊((C3⋊D3)⋊C3)D414yescomplemented
56C4×(C3⋊D3)⋊C3—24no—
57C4×(C3⋊D3)⋊C3—24no—
58#6198—28no—
59((C3⋊D3)⋊C3)⋊C4—28no—
60#5726—91no—
61#5555—91no—
62#5551—91no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C4×C3⋊D3C2×D312yes—
66C2×C4×C3⋊D3C2×D312yes—
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—32no—
71C2×C3×Dic6—32no—
72C2×C3×Dic6—32no—
73C2×C3×Dic6—62no—
74C2×#626—34no—
75C2×#626—34no—
76C2×#624—34no—
77C2×D3×Dic3—34no—
78#3799—68no—
79#3796—68no—
80C32×C42—21no—
81C32×C42C2×D311yescharacteristic
82C32×C42C2×D311yescharacteristic
83#3752C2×D312yes—
84#3752C2×D312yes—
85C4⋊(C3⋊Dic3)C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)C2×D311yescharacteristic
87C3×(C2×D3)⋊C4—32no—
88C3×(C2×D3)⋊C4—32no—
89C3×C4⋊Dic3—32no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—32no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—32no—
95C3×C4⋊Dic3—62no—
96C3×C4⋊Dic3—62no—
97#3724—34no—
98Dic3⋊Dic3—34no—
99#3722—34no—
100(C2×D3)⋊Dic3—34no—
101C22×H(3,𝔽3)C2411yescharacteristic
102C2×(C3⋊D3)⋊C3—22no—
103C2×(C3⋊D3)⋊C3—22no—
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3—22no—
106C2×(C3⋊D3)⋊C3—22no—
107C2×(C3⋊D3)⋊C3C2×D414yes—
108C4×H(3,𝔽3)—21no—
109C4×H(3,𝔽3)C2×D412yes—
110C4×H(3,𝔽3)#4112yescomplemented
111#1084—24no—
112(C3⋊Dic3)⋊C3—22no—
113(C3⋊Dic3)⋊C3—22no—
114C2×#239—92no—
115C2×#237—92no—
116C2×#235—92no—
117C3×C4⋊Q8—91no—
118C3×C4×D4—91no—
119C3×C4×D4—91no—
120#946—92no—
121#938—92no—
122#930—92no—
123C2×C4⋊Dic3—92no—
124#924—92no—
125#897—92no—
126#895—92no—
127#894—94no—
128D3×C4⋊C4—92no—
129D3×C4⋊C4—92no—
130#886—94no—
131#883—94no—
132C4×D12—31no—
133C4×D12—31no—
134C4×D12—91no—
135C4×D12—91no—
136#869—31no—
137#869—31no—
138#869—61no—
139C22×C3⋊D3C22×D312yes—
140C22×C3⋊D3C22×D312yes—
141C22×C3×D3—32no—
142C22×C3×D3—32no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C3⋊Dic3C22×D312yes—
153C2×C3⋊Dic3C22×D312yes—
154C4⋊(C3⋊D3)—22no—
155C4⋊(C3⋊D3)D1214yescomplemented
156C4×C3⋊D3—24no—
157C4×C3⋊D3—24no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—34no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—64no—
163C3×D12—34no—
164C3×D12—62no—
165C3×C4×D3—64no—
166C3×C4×D3—64no—
167C3×Dic6—64no—
168C3×Dic6—64no—
169C3×Dic6—124no—
170#626—68no—
171#626—68no—
172#624—68no—
173D3×Dic3—68no—
174#547—271noSylow
175C2×H(3,𝔽3)C22×D411yescharacteristic
176C2×H(3,𝔽3)C2×#4111yescharacteristic
177C2×H(3,𝔽3)2−1+411yescharacteristic
178(C3⋊D3)⋊C3—24no—
179(C3⋊D3)⋊C3—42no—
180C2×C3×Q8—92no—
181C2×C3×D4—91no—
182C2×C3×D4—91no—
183C22×C3×C4—92no—
184C22×C3×C4—92no—
185C2×#67—94no—
186C2×#67—94no—
187C22×Dic3—94no—
188#239—188no—
189#237—188no—
190#235—188no—
191C2×D12—31no—
192C2×D12—31no—
193C2×D12—91no—
194C2×D12—91no—
195C2×D12—94no—
196C2×C4×D3—32no—
197C2×C4×D3—32no—
198C2×C4×D3—92no—
199C2×C4×D3—92no—
200C2×C4×D3—94no—
201C2×C4×D3—94no—
202C2×Dic6—32no—
203C2×Dic6—32no—
204C2×Dic6—62no—
205C3×C4⋊C4—91no—
206C3×C4⋊C4—91no—
207C3×C4⋊C4—92no—
208C3×C4⋊C4—92no—
209C3×C22⋊C4—92no—
210C3×C22⋊C4—92no—
211C3×C42—31no—
212C3×C42—31no—
213C3×C42—61no—
214C3×C42D3211yescharacteristic
215#217—94no—
216(C2×D3)⋊C4—32no—
217(C2×D3)⋊C4—32no—
218(C2×D3)⋊C4—92no—
219(C2×D3)⋊C4—92no—
220(C2×D3)⋊C4—94no—
221(C2×D3)⋊C4—94no—
222C4⋊Dic3—32no—
223C4⋊Dic3—32no—
224C4⋊Dic3—32no—
225C4⋊Dic3—32no—
226C4⋊Dic3—31no—
227C4⋊Dic3—31no—
228C4⋊Dic3—62no—
229C4⋊Dic3—62no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—94no—
233Dic3⋊C4—94no—
234Dic3⋊C4—94no—
235C22×C32—21no—
236C22×C32C23×D311yescharacteristic
237C22×C32C23×D311yescharacteristic
238C2×C3⋊D3—22no—
239C2×C3⋊D3—22no—
240C2×C3⋊D3—22no—
241C2×C3⋊D3—22no—
242C2×C3⋊D3—22no—
243C2×C3⋊D3C2×D1214yes—
244C2×C3×D3—34no—
245C2×C3×D3—62no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C32×C4—22no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4C2×D1212yes—
256C32×C4#23512yescomplemented
257C32×C4D3×D412yes—
258C32×C4#23712yescomplemented
259C3⋊Dic3—22no—
260C3⋊Dic3—22no—
261C3×Dic3—62no—
262C3×Dic3—64no—
263C3×Dic3—62no—
264C3×Dic3—64no—
265C3×Dic3—124no—
266C2×#41—272no—
267C4⋊Q8—91no—
268#123—274no—
269C22⋊Q8—272no—
270#121—272no—
271C4×D4—91no—
272C4×D4—91no—
273C2×C4⋊C4—272no—
274H(3,𝔽3)#54711yescharacteristic, complemented, Sylow
275C23×C3—92no—
276C23×C3—92no—
277C22×D3—32no—
278C22×D3—32no—
279C22×D3—92no—
280C22×D3—92no—
281C3×Q8—184no—
282C3×D4—94no—
283C3×D4—182no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—92no—
294C2×C3×C4—92no—
295C2×C3×C4—94no—
296C2×C3×C4—184no—
297C2×C3×C4—184no—
298C2×C3×C4C2×D3211yescharacteristic
299C2×C3×C4C2×D3211yescharacteristic
300C2×C3×C4C2×D3211yescharacteristic
301#67—188no—
302#67—188no—
303C2×Dic3—34no—
304C2×Dic3—34no—
305C2×Dic3—32no—
306C2×Dic3—32no—
307C2×Dic3—64no—
308C2×Dic3—92no—
309C2×Dic3—92no—
310C2×Dic3—188no—
311D12—34no—
312D12—62no—
313D12—94no—
314D12—182no—
315D12—188no—
316C4×D3—64no—
317C4×D3—64no—
318C4×D3—184no—
319C4×D3—184no—
320C4×D3—188no—
321C4×D3—188no—
322Dic6—64no—
323Dic6—64no—
324Dic6—124no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32C22×D1211yescharacteristic
329C2×C32C2×#23511yescharacteristic
330C2×C32C2×D3×D411yescharacteristic
331C2×C32C2×#23711yescharacteristic
332C2×C32#101011yescharacteristic
333C2×C32#101011yescharacteristic
334C3⋊D3—24no—
335C3⋊D3—42no—
336C3×D3—64no—
337C3×D3—122no—
338#41—548no—
339C2×Q8—92no—
340C2×D4—91no—
341C2×D4—91no—
342C2×D4—274no—
343C22×C4—92no—
344C22×C4—92no—
345C22×C4—274no—
346C4⋊C4—91no—
347C4⋊C4—92no—
348C4⋊C4—91no—
349C4⋊C4—92no—
350C4⋊C4—274no—
351C22⋊C4—92no—
352C22⋊C4—92no—
353C22⋊C4—274no—
354C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
355C22×C3—31no—
356C22×C3—31no—
357C22×C3—61no—
358C22×C3—94no—
359C22×C3—182no—
360C22×C3—182no—
361C22×C3—182no—
362C22×C3—182no—
363C22×C3—182no—
364C22×C3C22×D3211yescharacteristic
365C2×D3—34no—
366C2×D3—62no—
367C2×D3—62no—
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—94no—
372C2×D3—182no—
373C2×D3—182no—
374C2×D3—182no—
375C2×D3—182no—
376C2×D3—182no—
377C3×C4—21no—
378C3×C4—32no—
379C3×C4—32no—
380C3×C4—32no—
381C3×C4—32no—
382C3×C4—61no—
383C3×C4—61no—
384C3×C4—62no—
385C3×C4—62no—
386C3×C4—62no—
387C3×C4—182no—
388C3×C4—182no—
389C3×C4—184no—
390C3×C4#379512yes—
391C3×C4D3×D1212yes—
392Dic3—64no—
393Dic3—64no—
394Dic3—62no—
395Dic3—62no—
396Dic3—124no—
397Dic3—182no—
398Dic3—182no—
399C32—21no—
400C32#555111yescharacteristic, complemented
401C32#555511yescharacteristic, complemented
402C23—92no—
403C23—92no—
404Q8—184no—
405D4—94no—
406D4—182no—
407D4—548no—
408C2×C4—92no—
409C2×C4—92no—
410C2×C4—94no—
411C2×C4—184no—
412C2×C4—184no—
413C2×C4—548no—
414C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
415C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
416C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—31no—
422C2×C3—31no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—61no—
426C2×C3—184no—
427C2×C3—362no—
428C2×C3C2×#379511yescharacteristic
429C2×C3#6429811yescharacteristic
430C2×C3C2×D3×D1211yescharacteristic
431D3—64no—
432D3—122no—
433D3—184no—
434D3—362no—
435C22—94no—
436C22—182no—
437C22—182no—
438C22—182no—
439C22—182no—
440C22—182no—
441C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
442C4—21no—
443C4—182no—
444C4—182no—
445C4—184no—
446C4#8864812yescomplemented
447C4#8865412yes—
448C3—31no—
449C3—31no—
450C3—61no—
451C3#1059389511yescharacteristic
452C2—184no—
453C2—362no—
454C2C2×#8864811yescharacteristic
455C2#1172248311yescharacteristic
456C2C2×#8865411yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#547
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88648C2semidirect
2. #11721715C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
5. #11721141C2semidirect
6. #11721131C2semidirect
7. #11721129C2semidirect
8. #11721107C2semidirect
30. C4×C4⋊(C3⋊D3)D3semidirect
31. C4×C4⋊(C3⋊D3)D3semidirect
55. C4⋊((C3⋊D3)⋊C3)D4semidirect
110. C4×H(3,𝔽3)#41semidirect
155. C4⋊(C3⋊D3)D12semidirect
256. C32×C4#235semidirect
258. C32×C4#237semidirect
274. H(3,𝔽3)#547semidirect
400. C32#5551semidirect
401. C32#5555semidirect
446. C4#88648semidirect

Maximal subgroups

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

subgroupindex
C2×#886482
#117217152
C4×C4⋊((C3⋊D3)⋊C3)2
#117211412
#117211312
#117211292
#117211072
#105938963
#105938693