Finite Group Databasebeta

← #13140218

Subgroups — #13140218

Order 1728.

1069 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1069 conjugacy classes of subgroups — too many to draw. Only its 123 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×C4⋊#272C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic, complemented
3C2×#88451C212yesmaximal, complemented
4C2×#88451C212yesmaximal, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#10591302—31nomaximal
7#10591300—31nomaximal
8C23×#272C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272—24no—
11C2×C4⋊#272—22no—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15#3750⋊C3—22no—
16#88451—24no—
17#88451—24no—
18C22×C3×D12—31no—
19C22×C3×D12—31no—
20C22×C3×D12—61no—
21C2×#3750D311yescharacteristic, complemented
22C2×C3×Dic3⋊C4—31no—
23C2×(C2×D3)⋊Dic3—32no—
24C2×(C2×D3)⋊Dic3—32no—
25C2×(C2×D3)⋊Dic3—32no—
26C2×(C2×D3)⋊Dic3—32no—
27#63658D311yescharacteristic, complemented
28C3×C3⋊#95—31no—
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×#272—22no—
31C22×#272—22no—
32C22×#272—22no—
33C22×#272—22no—
34C22×#272—22no—
35C22×#272—22no—
36C22×#272D414yes—
37C2×C4×H(3,𝔽3)—22no—
38C2×C4×H(3,𝔽3)D412yes—
39C4⋊#272—28no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3—22no—
44C2×(C3⋊Dic3)⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—22no—
47C2×(C3⋊Dic3)⋊C3D412yescomplemented
48#5222—91no—
49#4991—91no—
50#4672—91no—
51C23×C3×D3—32no—
52C23×C3×D3—32no—
53C23×C3×D3—62no—
54C22×C32×C4—21no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C22×C3⋊Dic3C2×D311yescharacteristic
58C22×C3⋊Dic3C2×D312yes—
59C22×C3⋊Dic3C2×D311yescharacteristic
60C22×C3×Dic3—31no—
61C22×C3×Dic3—31no—
62C22×C3×Dic3—32no—
63C2×C3×D12—64no—
64C2×C3×D12—64no—
65C2×C3×D12—68no—
66C2×C3×D12—64no—
67C2×C3×D12—62no—
68C2×C3×D12—62no—
69#3750—22no—
70C3×Dic3⋊C4—62no—
71(C2×D3)⋊Dic3—64no—
72(C2×D3)⋊Dic3—64no—
73(C2×D3)⋊Dic3—64no—
74(C2×D3)⋊Dic3—64no—
75C22×H(3,𝔽3)C2×D412yes—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C22×H(3,𝔽3)#4111yescharacteristic
79C22×H(3,𝔽3)#4112yes—
80C2×#272—24no—
81C2×#272—24no—
82C2×#272—24no—
83C2×#272—42no—
84C2×#272—42no—
85C2×#272—42no—
86C2×#272—42no—
87C4×H(3,𝔽3)—22no—
88(C3⋊Dic3)⋊C3—22no—
89(C3⋊Dic3)⋊C3—41no—
90(C3⋊Dic3)⋊C3—42no—
91C22×C3×D4—91no—
92C22×D12—31no—
93C22×D12—31no—
94C22×D12—61no—
95C2×C3×C4⋊C4—91no—
96C2×#217—92no—
97C2×#217—92no—
98C2×(C2×D3)⋊C4—92no—
99C2×(C2×D3)⋊C4—92no—
100C2×(C2×D3)⋊C4—92no—
101C2×(C2×D3)⋊C4—92no—
102C2×Dic3⋊C4—31no—
103C2×Dic3⋊C4—91no—
104C3×#95—91no—
105C3⋊#95—31no—
106C3⋊#95—91no—
107C23×C32—21no—
108C23×C32C22×D311yescharacteristic
109C23×C32C22×D311yescharacteristic
110C22×C3×D3—34no—
111C22×C3×D3—34no—
112C22×C3×D3—64no—
113C22×C3×D3—64no—
114C22×C3×D3—64no—
115C22×C3×D3—62no—
116C22×C3×D3—62no—
117C22×C3×D3—62no—
118C22×C3×D3—62no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C22×C3×D3—62no—
123C22×C3×D3—62no—
124C22×C3×D3—62no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C22×C3×D3—122no—
128C22×C3×D3—122no—
129C22×C3×D3—122no—
130C22×C3×D3—122no—
131C2×C32×C4—24no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4D1212yes—
136C2×C32×C4#6712yes—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3D1212yescomplemented
145C2×C3×Dic3—32no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—62no—
153C3×D12—616no—
154C3×D12—68no—
155C3×D12—68no—
156#394—271noSylow
157C2×H(3,𝔽3)#12011yescharacteristic
158C2×H(3,𝔽3)#12112yes—
159C2×H(3,𝔽3)#12111yescharacteristic
160C2×H(3,𝔽3)#12312yes—
161C2×H(3,𝔽3)#12411yescharacteristic
162#272—44no—
163C24×C3—92no—
164C23×D3—32no—
165C23×D3—32no—
166C23×D3—62no—
167C2×C3×D4—184no—
168C2×C3×D4—182no—
169C22×C3×C4—31no—
170C22×C3×C4—31no—
171C22×C3×C4—61no—
172C22×C3×C4—91no—
173C22×C3×C4—92no—
174C22×C3×C4—91no—
175C22×C3×C4D3211yescharacteristic
176C22×Dic3—31no—
177C22×Dic3—32no—
178C22×Dic3—31no—
179C22×Dic3—91no—
180C22×Dic3—92no—
181C22×Dic3—91no—
182C2×D12—62no—
183C2×D12—62no—
184C2×D12—64no—
185C2×D12—64no—
186C2×D12—68no—
187C2×D12—64no—
188C3×C4⋊C4—182no—
189#217—184no—
190#217—184no—
191(C2×D3)⋊C4—184no—
192(C2×D3)⋊C4—184no—
193(C2×D3)⋊C4—184no—
194(C2×D3)⋊C4—184no—
195Dic3⋊C4—62no—
196Dic3⋊C4—182no—
197C22×C32—22no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—22no—
202C22×C32C2×D1211yescharacteristic
203C22×C32C2×D1211yescharacteristic
204C22×C32#23512yes—
205C22×C32#23511yescharacteristic
206C22×C32D3×D411yescharacteristic
207C22×C32D3×D412yes—
208C22×C32D3×D412yes—
209C22×C32#23911yescharacteristic
210C22×C32#23912yes—
211C22×C32C2×#6711yescharacteristic
212C2×C3×D3—68no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C2×C3×D3—64no—
216C2×C3×D3—64no—
217C2×C3×D3—64no—
218C2×C3×D3—64no—
219C2×C3×D3—124no—
220C2×C3×D3—124no—
221C2×C3×D3—124no—
222C2×C3×D3—124no—
223C2×C3×D3—124no—
224C2×C3×D3—124no—
225C2×C3×D3—122no—
226C2×C3×D3—122no—
227C2×C3×D3—122no—
228C2×C3×D3—122no—
229C2×C3×D3—122no—
230C2×C3×D3—122no—
231C2×C3×D3—122no—
232C2×C3×D3—122no—
233C32×C4—24no—
234C32×C4—22no—
235C32×C4—22no—
236C3⋊Dic3—22no—
237C3⋊Dic3—42no—
238C3⋊Dic3—41no—
239C3×Dic3—62no—
240C3×Dic3—121no—
241C3×Dic3—122no—
242C22×D4—91no—
243C2×C4⋊C4—91no—
244C2×C22⋊C4—272no—
245C2×C22⋊C4—272no—
246#95—91no—
247H(3,𝔽3)#39411yescharacteristic, complemented, Sylow
248C23×C3—31no—
249C23×C3—31no—
250C23×C3—61no—
251C23×C3—94no—
252C23×C3—182no—
253C23×C3—182no—
254C23×C3—182no—
255C23×C3—182no—
256C23×C3—182no—
257C23×C3—182no—
258C23×C3C2×D3211yescharacteristic
259C22×D3—34no—
260C22×D3—34no—
261C22×D3—64no—
262C22×D3—62no—
263C22×D3—62no—
264C22×D3—62no—
265C22×D3—62no—
266C22×D3—62no—
267C22×D3—62no—
268C22×D3—62no—
269C22×D3—62no—
270C22×D3—62no—
271C22×D3—62no—
272C22×D3—62no—
273C22×D3—62no—
274C22×D3—64no—
275C22×D3—64no—
276C22×D3—122no—
277C22×D3—122no—
278C22×D3—122no—
279C22×D3—122no—
280C3×D4—188no—
281C2×C3×C4—22no—
282C2×C3×C4—32no—
283C2×C3×C4—32no—
284C2×C3×C4—62no—
285C2×C3×C4—62no—
286C2×C3×C4—64no—
287C2×C3×C4—62no—
288C2×C3×C4—92no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4—182no—
292C2×C3×C4—182no—
293C2×C3×C4—182no—
294C2×C3×C4—181no—
295C2×C3×C4—182no—
296C2×C3×C4#62612yes—
297C2×Dic3—32no—
298C2×Dic3—61no—
299C2×Dic3—61no—
300C2×Dic3—62no—
301C2×Dic3—62no—
302C2×Dic3—61no—
303C2×Dic3—62no—
304C2×Dic3—62no—
305C2×Dic3—92no—
306C2×Dic3—181no—
307C2×Dic3—181no—
308C2×Dic3—181no—
309C2×Dic3—182no—
310C2×Dic3—182no—
311C2×Dic3—182no—
312C2×Dic3—182no—
313D12—68no—
314D12—68no—
315D12—616no—
316C2×C32—21no—
317C2×C32—22no—
318C2×C32—21no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32#87511yescharacteristic
322C2×C32#88211yescharacteristic
323C2×C32#88412yes—
324C2×C32#89412yes—
325C2×C32#89412yes—
326C2×C32#89511yescharacteristic
327C2×C32#89512yes—
328C2×C32#93011yescharacteristic
329C2×C32#93711yescharacteristic
330C2×C32#94711yescharacteristic
331C3×D3—128no—
332C3×D3—124no—
333C3×D3—124no—
334C24—92no—
335C2×D4—184no—
336C2×D4—182no—
337C22×C4—91no—
338C22×C4—92no—
339C22×C4—91no—
340C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
341C4⋊C4—182no—
342C22⋊C4—544no—
343C22⋊C4—544no—
344C22×C3—32no—
345C22×C3—31no—
346C22×C3—32no—
347C22×C3—31no—
348C22×C3—31no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—32no—
353C22×C3—32no—
354C22×C3—62no—
355C22×C3—61no—
356C22×C3—61no—
357C22×C3—61no—
358C22×C3—62no—
359C22×C3—184no—
360C22×C3—184no—
361C22×C3—184no—
362C22×C3—362no—
363C22×C3—362no—
364C22×C3—362no—
365C22×C3—362no—
366C22×C3#379912yes—
367C22×C3#379911yescharacteristic
368C22×C3D3×D1211yescharacteristic
369C22×C3C2×#62611yescharacteristic
370C22×C3#381112yes—
371C2×D3—64no—
372C2×D3—64no—
373C2×D3—64no—
374C2×D3—64no—
375C2×D3—64no—
376C2×D3—64no—
377C2×D3—68no—
378C2×D3—124no—
379C2×D3—124no—
380C2×D3—122no—
381C2×D3—122no—
382C2×D3—122no—
383C2×D3—122no—
384C2×D3—122no—
385C2×D3—122no—
386C2×D3—122no—
387C2×D3—122no—
388C2×D3—124no—
389C2×D3—124no—
390C2×D3—124no—
391C2×D3—124no—
392C3×C4—22no—
393C3×C4—62no—
394C3×C4—62no—
395C3×C4—64no—
396C3×C4—182no—
397C3×C4—361no—
398C3×C4—362no—
399Dic3—62no—
400Dic3—122no—
401Dic3—121no—
402Dic3—182no—
403Dic3—361no—
404Dic3—362no—
405C32—21no—
406C32#467211yescharacteristic, complemented
407C32#499111yescharacteristic, complemented
408C23—94no—
409C23—182no—
410C23—182no—
411C23—182no—
412C23—182no—
413C23—182no—
414C23—182no—
415C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
416D4—188no—
417C2×C4—22no—
418C2×C4—92no—
419C2×C4—181no—
420C2×C4—181no—
421C2×C4—182no—
422C2×C4—182no—
423C2×C4—181no—
424C2×C4—182no—
425C2×C4—182no—
426C2×C4#620012yes—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—32no—
432C2×C3—32no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—32no—
436C2×C3—32no—
437C2×C3—61no—
438C2×C3—61no—
439C2×C3—62no—
440C2×C3—61no—
441C2×C3—62no—
442C2×C3—364no—
443C2×C3#6386411yescharacteristic
444C2×C3#6389712yes—
445C2×C3#6390911yescharacteristic
446C2×C3#6391012yes—
447C2×C3#6398111yescharacteristic
448D3—124no—
449D3—124no—
450D3—128no—
451C22—184no—
452C22—184no—
453C22—184no—
454C22—362no—
455C22—362no—
456C22—362no—
457C22—362no—
458C22#8865212yes—
459C22#8865211yescharacteristic
460C22#8865411yescharacteristic
461C22C2×#620011yescharacteristic
462C22#8867712yes—
463C4—22no—
464C4—182no—
465C4—361no—
466C4—362no—
467C3—31no—
468C3—31no—
469C3—61no—
470C3#1059132111yescharacteristic
471C2—364no—
472C2#1172111411yescharacteristic
473C2#1172115112yes—
474C2#1172115611yescharacteristic
475C2#1172115712yes—
476C2#1172132311yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC22×C4⋊#272C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#394
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#3750⋊C3C2semidirect
3. C2×#88451C2semidirect
4. C2×#88451C2semidirect
5. #63658⋊C3C2semidirect
21. C2×#3750D3semidirect
27. #63658D3semidirect
47. C2×(C3⋊Dic3)⋊C3D4semidirect
144. C2×C3⋊Dic3D12semidirect
247. H(3,𝔽3)#394semidirect
406. C32#4672semidirect
407. C32#4991semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C2×#3750⋊C32
C2×#884512
#63658⋊C32
#105913023
#105913003