Finite Group Databasebeta

← #13150752

Subgroups — #13150752

Order 1728.

855 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 855 conjugacy classes of subgroups — too many to draw. Only its 139 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×C4⋊#1084C211yesmaximal, characteristic, complemented
2#64138⋊C3C212yesmaximal, complemented
3#11721318C211yesmaximal, characteristic
4#11721315C212yesmaximal, complemented
5#11721309C211yesmaximal, characteristic, complemented
6#11721133C212yesmaximal, complemented
7#11721131C212yesmaximal, complemented
8#11721111C212yesmaximal, complemented
9#11721085C212yesmaximal, complemented
10#10594987—32nomaximal
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C2×#638⋊C3C2212yes—
15C2×#6198C2211yescharacteristic
16C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
17C2×#6196C2211yescharacteristic
18C4⋊#1084C2212yes—
19C4⋊#1084C2212yes—
20#3757⋊C3C2212yes—
21#3752⋊C3C2212yes—
22#3752⋊C3C2212yes—
23#3750⋊C3C2212yes—
24#3750⋊C3C2212yes—
25#88450C2212yes—
26#88450C2212yes—
27#88449C2211yescharacteristic
28#88449C2211yescharacteristic
29#88448C2212yes—
30#88448C2212yes—
31#88447C2212yes—
32#64138D312yescomplemented
33C3×#929—32no—
34C2×C3×C4⋊Dic3—32no—
35C2×C3×C4⋊Dic3—61no—
36#63978—32no—
37Dic3×#67—32no—
38#63964—32no—
39#63952—32no—
40#63899—32no—
41#63895—32no—
42#63890—32no—
43#63858—32no—
44#63854—32no—
45#63848—32no—
46#63847—32no—
47#63846—32no—
48C23×H(3,𝔽3)C2311yescharacteristic
49C22×(C3⋊D3)⋊C3C2312yes—
50C2×C4×H(3,𝔽3)—22no—
51C2×C4×H(3,𝔽3)C2312yes—
52C2×#1084—41no—
53C2×#1084—41no—
54C2×#1084C2311yescharacteristic
55C2×#1084C2312yes—
56C2×#1084C2311yescharacteristic
57#638⋊C3—24no—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×(C3⋊Dic3)⋊C3C2312yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61#6198—41no—
62((C3⋊D3)⋊C3)⋊C4—42no—
63#6196—41no—
64#5665—92no—
65#5515—91no—
66C22×C32×C4—21no—
67C22×C32×C4C2×D312yes—
68C2×#638C2×D312yes—
69C2×C3×#67—32no—
70C22×C3×Dic3—32no—
71C22×C3×Dic3—32no—
72C22×C3×Dic3—61no—
73C22×C3×Dic3—61no—
74C2×#627—32no—
75C2×#626—32no—
76C2×#624—32no—
77C2×D3×Dic3—32no—
78#3757C2×D312yes—
79#3752C2×D312yes—
80#3752C2×D312yes—
81#3750C2×D312yes—
82#3750C2×D312yes—
83C3×#217—32no—
84C3×(C2×D3)⋊C4—32no—
85C3×(C2×D3)⋊C4—32no—
86C3×C4⋊Dic3—32no—
87C3×C4⋊Dic3—32no—
88C3×C4⋊Dic3—32no—
89C3×C4⋊Dic3—32no—
90C3×C4⋊Dic3—62no—
91C3×C4⋊Dic3—62no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94#3724—32no—
95#3724—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—32no—
100#3722—32no—
101#3722—32no—
102(C2×D3)⋊Dic3—32no—
103(C2×D3)⋊Dic3—32no—
104Dic32—32no—
105Dic32—32no—
106C22×H(3,𝔽3)—21no—
107C22×H(3,𝔽3)—21no—
108C22×H(3,𝔽3)#4112yes—
109C22×H(3,𝔽3)C2411yescharacteristic
110C2×(C3⋊D3)⋊C3—22no—
111C2×(C3⋊D3)⋊C3—22no—
112C2×(C3⋊D3)⋊C3—22no—
113C4×H(3,𝔽3)—22no—
114#1084—21no—
115#1084—22no—
116#1084—21no—
117(C3⋊Dic3)⋊C3—22no—
118(C3⋊Dic3)⋊C3—22no—
119(C3⋊Dic3)⋊C3—22no—
120C3×#123—92no—
121C2×C3×C4⋊C4—91no—
122#939—92no—
123D4×Dic3—92no—
124#930—91no—
125#929—32no—
126#929—92no—
127C4×#67—92no—
128C2×C4⋊Dic3—32no—
129C2×C4⋊Dic3—61no—
130#924—91no—
131#898—92no—
132#898—92no—
133#897—92no—
134#896—92no—
135#894—92no—
136#890—92no—
137#889—92no—
138#889—92no—
139#886—92no—
140#883—92no—
141#67⋊C4—92no—
142#879—92no—
143#878—92no—
144#878—92no—
145C23×C32—21no—
146C23×C32C22×D312yes—
147C22×C3⋊D3C22×D312yes—
148C22×C3×D3—32no—
149C2×C32×C4—24no—
150C2×C32×C4—24no—
151C2×C32×C4—22no—
152C2×C32×C4C22×D312yes—
153C2×C32×C4C22×D312yes—
154#638—24no—
155C2×C3⋊Dic3C22×D312yes—
156C2×C3⋊Dic3C22×D312yes—
157C2×C3⋊Dic3C22×D312yes—
158C3×#67—64no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—122no—
170C2×C3×Dic3—122no—
171C2×C3×Dic3—122no—
172C2×C3×Dic3—122no—
173#627—122no—
174#626—122no—
175#624—122no—
176D3×Dic3—122no—
177#539—271noSylow
178C2×H(3,𝔽3)—22no—
179C2×H(3,𝔽3)C2×#4111yescharacteristic
180C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
181C2×H(3,𝔽3)2−1+411yescharacteristic
182(C3⋊D3)⋊C3—42no—
183C2×C3×D4—92no—
184C22×C3×C4—32no—
185C22×C3×C4—61no—
186C22×C3×C4—91no—
187C22×C3×C4—91no—
188C22×C3×C4D3211yescharacteristic
189C2×#67—32no—
190C2×#67—92no—
191C2×#67—92no—
192C22×Dic3—32no—
193C22×Dic3—32no—
194C22×Dic3—61no—
195C22×Dic3—61no—
196C22×Dic3—92no—
197C2×D12—91no—
198C2×C4×D3—92no—
199C2×C4×D3—92no—
200C2×Dic6—91no—
201C2×Dic6—92no—
202C3×C4⋊C4—92no—
203C3×C4⋊C4—92no—
204C3×C4⋊C4—92no—
205C3×C4⋊C4—92no—
206C3×C22⋊C4—92no—
207C3×C22⋊C4—92no—
208C3×C22⋊C4—92no—
209#217—32no—
210#217—92no—
211#217—92no—
212#217—92no—
213(C2×D3)⋊C4—32no—
214(C2×D3)⋊C4—32no—
215(C2×D3)⋊C4—92no—
216(C2×D3)⋊C4—92no—
217(C2×D3)⋊C4—92no—
218(C2×D3)⋊C4—92no—
219(C2×D3)⋊C4—92no—
220(C2×D3)⋊C4—92no—
221C4⋊Dic3—32no—
222C4⋊Dic3—32no—
223C4⋊Dic3—32no—
224C4⋊Dic3—32no—
225C4⋊Dic3—62no—
226C4⋊Dic3—62no—
227C4⋊Dic3—91no—
228C4⋊Dic3—91no—
229C4⋊Dic3—92no—
230C4⋊Dic3—92no—
231Dic3⋊C4—32no—
232Dic3⋊C4—32no—
233Dic3⋊C4—92no—
234Dic3⋊C4—92no—
235Dic3⋊C4—92no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241C4×Dic3—92no—
242C4×Dic3—92no—
243C4×Dic3—92no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—21no—
249C22×C32—22no—
250C22×C32#23714yes—
251C22×C32C23×D312yes—
252C2×C3⋊D3—22no—
253C2×C3⋊D3—22no—
254C2×C3⋊D3—22no—
255C2×C3×D3—62no—
256C2×C3×D3—62no—
257C2×C3×D3—62no—
258C32×C4—24no—
259C32×C4—22no—
260C32×C4—22no—
261C3⋊Dic3—22no—
262C3⋊Dic3—22no—
263C3⋊Dic3—22no—
264C3×Dic3—62no—
265C3×Dic3—62no—
266C3×Dic3—62no—
267C3×Dic3—62no—
268C3×Dic3—62no—
269C3×Dic3—62no—
270C3×Dic3—62no—
271C3×Dic3—121no—
272C3×Dic3—122no—
273C3×Dic3—121no—
274#126—272no—
275#125—272no—
276#123—92no—
277#123—272no—
278C22⋊Q8—271no—
279C22⋊Q8—272no—
280#121—271no—
281C4×D4—272no—
282C2×C4⋊C4—91no—
283H(3,𝔽3)#53911yescharacteristic, complemented, Sylow
284C23×C3—32no—
285C23×C3—61no—
286C23×C3—92no—
287C23×C3C2×D3211yescharacteristic
288C22×D3—32no—
289C22×D3—92no—
290C3×D4—184no—
291C2×C3×C4—22no—
292C2×C3×C4—32no—
293C2×C3×C4—32no—
294C2×C3×C4—64no—
295C2×C3×C4—64no—
296C2×C3×C4—62no—
297C2×C3×C4—92no—
298C2×C3×C4—92no—
299C2×C3×C4—92no—
300C2×C3×C4—92no—
301C2×C3×C4—91no—
302C2×C3×C4—91no—
303C2×C3×C4—361no—
304C2×C3×C4—361no—
305C2×C3×C4C2×D3212yes—
306#67—64no—
307#67—184no—
308#67—362no—
309C2×Dic3—32no—
310C2×Dic3—32no—
311C2×Dic3—32no—
312C2×Dic3—32no—
313C2×Dic3—32no—
314C2×Dic3—32no—
315C2×Dic3—32no—
316C2×Dic3—62no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—92no—
320C2×Dic3—92no—
321C2×Dic3—92no—
322C2×Dic3—122no—
323C2×Dic3—122no—
324C2×Dic3—122no—
325C2×Dic3—122no—
326C2×Dic3—362no—
327D12—361no—
328C4×D3—362no—
329C4×D3—362no—
330Dic6—361no—
331Dic6—362no—
332C2×C32—24no—
333C2×C32—24no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32C2×#23712yes—
338C2×C32#100912yes—
339C2×C32#101012yes—
340C3⋊D3—42no—
341C3×D3—122no—
342C2×Q8—271no—
343C2×D4—92no—
344C2×D4—271no—
345C22×C4—91no—
346C22×C4—91no—
347C22×C4—272no—
348C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
349C4⋊C4—92no—
350C4⋊C4—92no—
351C4⋊C4—92no—
352C4⋊C4—92no—
353C4⋊C4—272no—
354C4⋊C4—271no—
355C4⋊C4—272no—
356C4⋊C4—271no—
357C22⋊C4—92no—
358C22⋊C4—92no—
359C22⋊C4—92no—
360C22⋊C4—272no—
361C22⋊C4—272no—
362C42—272no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—32no—
366C22×C3—34no—
367C22×C3—62no—
368C22×C3—62no—
369C22×C3—62no—
370C22×C3—62no—
371C22×C3—61no—
372C22×C3—62no—
373C22×C3—182no—
374C22×C3—182no—
375C22×C3—182no—
376C22×C3#380512yes—
377C22×C3C22×D3211yescharacteristic
378C2×D3—62no—
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—182no—
382C2×D3—182no—
383C2×D3—182no—
384C3×C4—22no—
385C3×C4—62no—
386C3×C4—62no—
387C3×C4—64no—
388C3×C4—182no—
389C3×C4—182no—
390C3×C4—182no—
391C3×C4—181no—
392C3×C4—182no—
393C3×C4—181no—
394Dic3—62no—
395Dic3—62no—
396Dic3—62no—
397Dic3—62no—
398Dic3—62no—
399Dic3—62no—
400Dic3—62no—
401Dic3—122no—
402Dic3—121no—
403Dic3—121no—
404Dic3—182no—
405Dic3—182no—
406Dic3—182no—
407C32—21no—
408C32#566512yescomplemented
409C23—92no—
410C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
411Q8—1081no—
412D4—184no—
413D4—1081no—
414C2×C4—22no—
415C2×C4—92no—
416C2×C4—92no—
417C2×C4—92no—
418C2×C4—92no—
419C2×C4—91no—
420C2×C4—91no—
421C2×C4—361no—
422C2×C4—361no—
423C2×C4—1082no—
424C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
425C2×C3—22no—
426C2×C3—32no—
427C2×C3—32no—
428C2×C3—32no—
429C2×C3—64no—
430C2×C3—64no—
431C2×C3—61no—
432C2×C3—61no—
433C2×C3—61no—
434C2×C3—362no—
435C2×C3#6429811yescharacteristic
436C2×C3#6430811yescharacteristic
437C2×C3C2×#380511yescharacteristic
438D3—122no—
439D3—362no—
440C22—21no—
441C22—21no—
442C22—182no—
443C22—182no—
444C22—182no—
445C22#8867212yes—
446C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
447C4—22no—
448C4—182no—
449C4—182no—
450C4—182no—
451C4—181no—
452C4—181no—
453C4—182no—
454C3—32no—
455C3—61no—
456C3#1059499411yescharacteristic
457C2—22no—
458C2—362no—
459C2#1172248311yescharacteristic
460C2#1172249311yescharacteristic
461C2C2×#8867211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical#11721318C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#539
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#1084C2semidirect
2. #64138⋊C3C2semidirect
4. #11721315C2semidirect
5. #11721309C2semidirect
6. #11721133C2semidirect
7. #11721131C2semidirect
8. #11721111C2semidirect
9. #11721085C2semidirect
32. #64138D3semidirect
283. H(3,𝔽3)#539semidirect
408. C32#5665semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
#64138⋊C32
#117213182
#117213152
#117213092
#117211332
#117211312
#117211112
#117210852
#105949873