Finite Group Databasebeta

← #13140038

Subgroups — #13140038

Order 1728.

510 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C22×#6180C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic
3#64133⋊C3C211yesmaximal, characteristic, complemented
4C2×#88431C211yesmaximal, characteristic, complemented
5#11720318C211yesmaximal, characteristic, complemented
6#11720314C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10591177—31nomaximal
9#10591175—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C2×#6180—22no—
15C2×#6180C2211yescharacteristic
16C2×#6180C2211yescharacteristic
17#3757⋊C3—21no—
18#3752⋊C3—21no—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21#3750⋊C3—21no—
22#88431—22no—
23#64139D311yescharacteristic
24#64133D311yescharacteristic, complemented
25C3×#930—31no—
26C3×#924—31no—
27C22×C3×C3⋊C8—31no—
28C22×C3×C3⋊C8—31no—
29C22×C3×C3⋊C8—61no—
30C2×#3717—31no—
31C2×#3716—31no—
32#63573—31no—
33#63573—31no—
34#63567—31no—
35#63566—31no—
36#63564—31no—
37#63563—31no—
38C23×H(3,𝔽3)D411yescharacteristic
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)D411yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#638⋊C3—42no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52#6180—21no—
53#6180D412yescomplemented
54#5114—91no—
55#5048—91no—
56#5041—91no—
57C22×C32×C4—21no—
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C2×D311yescharacteristic
60C2×#638—21no—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×#634C2×D311yescharacteristic
63C2×C3×#67—61no—
64C2×C3×D12—31no—
65C2×C3×Dic6—31no—
66#3757—21no—
67#3752—21no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70#3750—21no—
71C3×#217—61no—
72C3×(C2×D3)⋊C4—61no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×Dic3⋊C4—61no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—62no—
81C2×C3×C3⋊C8—62no—
82C2×C3×C3⋊C8—61no—
83C2×C3×C3⋊C8—61no—
84C2×C3×C3⋊C8—64no—
85#3717—62no—
86#3716—62no—
87C22×H(3,𝔽3)—21no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)SD1612yescomplemented
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C4×H(3,𝔽3)—21no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97(C3⋊Dic3)⋊C3—41no—
98(C3⋊Dic3)⋊C3—41no—
99(C3⋊Dic3)⋊C3—41no—
100C22×C3×C8—91no—
101C3×C22⋊Q8—91no—
102C3×#121—91no—
103C2×#215—91no—
104C2×#214—91no—
105#930—31no—
106#930—91no—
107#924—31no—
108#924—91no—
109C22×C3⋊C8—31no—
110C22×C3⋊C8—31no—
111C22×C3⋊C8—61no—
112C2×#204—91no—
113#835—91no—
114#832—91no—
115#821—91no—
116#817—91no—
117#816—91no—
118#810—91no—
119#809—91no—
120#808—91no—
121#808—91no—
122C23×C32—21no—
123C23×C32#6711yescharacteristic
124C23×C32#6711yescharacteristic
125C22×C3⋊D3—21no—
126C22×C3×D3—61no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4#6711yescharacteristic
136C2×C32×C4#6711yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139#638—42no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—21no—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—21no—
145#634—21no—
146#634—21no—
147C3×#67—122no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C3×C3⋊C8—32no—
156C3×C3⋊C8—32no—
157C3×C3⋊C8—62no—
158C3×C3⋊C8—62no—
159C3×C3⋊C8—61no—
160C3×C3⋊C8—61no—
161#465—271noSylow
162C2×H(3,𝔽3)—22no—
163C2×H(3,𝔽3)#12111yescharacteristic
164C2×H(3,𝔽3)C2×SD1611yescharacteristic
165C2×H(3,𝔽3)#13511yescharacteristic
166(C3⋊D3)⋊C3—81no—
167C2×C3×Q8—91no—
168C2×C3×D4—91no—
169C2×C3×D4—181no—
170C22×C3×C4—31no—
171C22×C3×C4—31no—
172C22×C3×C4—61no—
173C22×C3×C4D3211yescharacteristic
174C2×#67—61no—
175C2×#67—181no—
176C2×D12—31no—
177C2×D12—91no—
178C2×Dic6—31no—
179C2×Dic6—91no—
180C2×C3×C8—91no—
181C2×C3×C8—91no—
182C2×C3×C8—182no—
183C3×C4⋊C4—91no—
184C3×C4⋊C4—91no—
185C3×C4⋊C4—181no—
186C3×C22⋊C4—181no—
187C3×C22⋊C4—181no—
188#217—61no—
189#217—181no—
190#215—182no—
191#214—182no—
192(C2×D3)⋊C4—61no—
193(C2×D3)⋊C4—181no—
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196C4⋊Dic3—91no—
197C4⋊Dic3—91no—
198Dic3⋊C4—61no—
199Dic3⋊C4—181no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—61no—
205C2×C3⋊C8—61no—
206C2×C3⋊C8—62no—
207C2×C3⋊C8—62no—
208C2×C3⋊C8—64no—
209#204—182no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—22no—
217C22×C32—21no—
218C22×C32#21412yescomplemented
219C22×C32#21512yescomplemented
220C22×C32C2×#6711yescharacteristic
221C22×C32C2×#6711yescharacteristic
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—41no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C32×C4—22no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4D3×D411yescharacteristic
234C32×C4D3×D411yescharacteristic
235C32×C4#23711yescharacteristic
236C32×C4#23711yescharacteristic
237C3⋊Dic3—41no—
238C3⋊Dic3—41no—
239C3⋊Dic3—41no—
240C3×Dic3—121no—
241C3×Dic3—121no—
242C3×Dic3—121no—
243C2×SD16—271no—
244C22×C8—91no—
245C22⋊Q8—91no—
246#121—91no—
247#106—271no—
248#103—271no—
249#102—271no—
250H(3,𝔽3)#46511yescharacteristic, complemented, Sylow
251C23×C3—31no—
252C23×C3—31no—
253C23×C3—61no—
254C23×C3—181no—
255C23×C3#62511yescharacteristic
256C22×D3—61no—
257C22×D3—181no—
258C3×Q8—181no—
259C3×Q8—181no—
260C3×D4—181no—
261C3×D4—181no—
262C3×D4—362no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—62no—
274C2×C3×C4—62no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4#62511yescharacteristic
281C2×C3×C4C2×D3211yescharacteristic
282#67—122no—
283#67—362no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—181no—
288C2×Dic3—181no—
289C2×Dic3—181no—
290D12—61no—
291D12—61no—
292D12—181no—
293D12—181no—
294Dic6—61no—
295Dic6—61no—
296Dic6—181no—
297Dic6—181no—
298C3×C8—92no—
299C3×C8—181no—
300C3⋊C8—32no—
301C3⋊C8—32no—
302C3⋊C8—62no—
303C3⋊C8—61no—
304C3⋊C8—61no—
305C3⋊C8—62no—
306C2×C32—24no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32C2×#21411yescharacteristic
313C2×C32#93911yescharacteristic
314C2×C32#93911yescharacteristic
315C2×C32C2×#21511yescharacteristic
316C2×C32#95011yescharacteristic
317C2×C32#95011yescharacteristic
318C3⋊D3—81no—
319C3×D3—241no—
320C2×Q8—91no—
321C2×D4—91no—
322C2×D4—181no—
323C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
324SD16—542no—
325C2×C8—91no—
326C2×C8—91no—
327C2×C8—182no—
328C4⋊C4—91no—
329C4⋊C4—91no—
330C4⋊C4—181no—
331C22⋊C4—181no—
332C22⋊C4—181no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—32no—
338C22×C3—32no—
339C22×C3—61no—
340C22×C3—61no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3—62no—
344C22×C3—62no—
345C22×C3—61no—
346C22×C3—62no—
347C22×C3—361no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3#371512yes—
351C22×C3C2×#62511yescharacteristic
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—361no—
356C2×D3—361no—
357C2×D3—361no—
358C3×C4—21no—
359C3×C4—31no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—61no—
364C3×C4—61no—
365C3×C4—62no—
366C3×C4—61no—
367C3×C4—61no—
368C3×C4—361no—
369C3×C4—361no—
370C3×C4—361no—
371C3×C4#380511yescharacteristic
372C3×C4#381111yescharacteristic
373Dic3—121no—
374Dic3—121no—
375Dic3—121no—
376Dic3—361no—
377Dic3—361no—
378Dic3—361no—
379C32—21no—
380C32#504111yescharacteristic, complemented
381C32#504811yescharacteristic, complemented
382C23—181no—
383C23#619811yescharacteristic
384Q8—181no—
385Q8—181no—
386D4—181no—
387D4—181no—
388D4—362no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—181no—
392C2×C4—181no—
393C2×C4—181no—
394C2×C4#619811yescharacteristic
395C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
396C8—92no—
397C8—181no—
398C2×C3—22no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—62no—
406C2×C3—62no—
407C2×C3—64no—
408C2×C3—61no—
409C2×C3—61no—
410C2×C3—61no—
411C2×C3—721no—
412C2×C3#6382211yescharacteristic
413C2×C3C2×#371511yescharacteristic
414C2×C3#6398011yescharacteristic
415D3—241no—
416D3—721no—
417C22—21no—
418C22—21no—
419C22—361no—
420C22—361no—
421C22—361no—
422C22#8843112yescomplemented
423C22C2×#619811yescharacteristic
424C4—21no—
425C4—361no—
426C4—361no—
427C4—361no—
428C4#8867211yescharacteristic
429C4#8867711yescharacteristic
430C3—31no—
431C3—31no—
432C3—61no—
433C3#1059117911yescharacteristic
434C2—22no—
435C2—721no—
436C2C2×#8843111yescharacteristic
437C2#1172106011yescharacteristic
438C2#1172131811yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radical#64139⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#465
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×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. C22×#6180C2semidirect
3. #64133⋊C3C2semidirect
4. C2×#88431C2semidirect
5. #11720318C2semidirect
6. #11720314C2semidirect
7. #11720313C2semidirect
24. #64133D3semidirect
53. #6180D4semidirect
89. C22×H(3,𝔽3)SD16semidirect
218. C22×C32#214semidirect
219. C22×C32#215semidirect
250. H(3,𝔽3)#465semidirect
380. C32#5041semidirect
381. C32#5048semidirect
422. C22#88431semidirect

Maximal subgroups

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

subgroupindex
C22×#61802
#64139⋊C32
#64133⋊C32
C2×#884312
#117203182
#117203142
#117203132
#105911773
#105911753