Finite Group Databasebeta

← #13144305

Subgroups — #13144305

Order 1728.

1309 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1309 conjugacy classes of subgroups — too many to draw. Only its 171 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×#6235C211yesmaximal, characteristic
2C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#88552C211yesmaximal, characteristic, complemented
4C2×#88542C211yesmaximal, characteristic, complemented
5C2×#88542C211yesmaximal, characteristic, complemented
6C2×#88540C211yesmaximal, characteristic, complemented
7#11720574C211yesmaximal, characteristic, complemented
8C3×#4959—34nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×#272C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#6235—22no—
14C2×#6235—22no—
15C2×#6235—22no—
16C22×#1084C2211yescharacteristic
17C22×#1084C2211yescharacteristic
18C22×#1084C2211yescharacteristic
19C22⋊C4×H(3,𝔽3)—22no—
20#88552—22no—
21#88542—22no—
22#88542—22no—
23#88540—22no—
24C22×C3×#67—34no—
25C2×C32×C22⋊C4D314yescomplemented
26C2×C3×#217—34no—
27C2×C3×(C2×D3)⋊C4—34no—
28C2×C3×(C2×D3)⋊C4—34no—
29C2×C3×Dic3⋊C4—34no—
30C3×C3⋊#95—34no—
31C23×H(3,𝔽3)—21no—
32C23×H(3,𝔽3)—21no—
33C23×H(3,𝔽3)—21no—
34C23×H(3,𝔽3)—21no—
35C23×H(3,𝔽3)—21no—
36C23×H(3,𝔽3)—21no—
37C23×H(3,𝔽3)D412yes—
38C23×H(3,𝔽3)C2311yescharacteristic
39C22×#272—21no—
40C22×#272—21no—
41C22×#272—21no—
42C22×#272—21no—
43C22×#272—21no—
44C22×#272—21no—
45C22×#272D412yes—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)—21no—
49C2×C4×H(3,𝔽3)—21no—
50C2×C4×H(3,𝔽3)—21no—
51C2×C4×H(3,𝔽3)—21no—
52#6235—28no—
53C2×#1084—21no—
54C2×#1084—21no—
55C2×#1084—21no—
56C2×#1084—21no—
57C2×#1084—21no—
58C2×#1084—21no—
59C2×#1084—21no—
60C2×#1084D412yes—
61C2×#1084D412yescomplemented
62C3×#394—91no—
63#4959—34no—
64C24×C32C2×D314yes—
65C23×C3×D3—34no—
66C22×C32×C4C2×D314yes—
67C22×C32×C4C2×D314yes—
68C2×C3×#67—68no—
69C2×C3×#67—68no—
70C2×C3×#67—68no—
71C22×C3×Dic3—34no—
72C22×C3×Dic3—34no—
73C22×C3×Dic3—34no—
74C32×C22⋊C4—28no—
75C3×#217—68no—
76C3×(C2×D3)⋊C4—68no—
77C3×(C2×D3)⋊C4—68no—
78C3×Dic3⋊C4—68no—
79C22×H(3,𝔽3)—22no—
80C22×H(3,𝔽3)—22no—
81C22×H(3,𝔽3)—22no—
82C22×H(3,𝔽3)—41no—
83C22×H(3,𝔽3)—41no—
84C22×H(3,𝔽3)—41no—
85C22×H(3,𝔽3)—41no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C22×H(3,𝔽3)#4111yescharacteristic
91C22×H(3,𝔽3)#4111yescharacteristic
92C22×H(3,𝔽3)#4111yescharacteristic
93C2×#272—22no—
94C2×#272—22no—
95C2×#272—22no—
96C2×#272—41no—
97C2×#272—41no—
98C2×#272—41no—
99C2×#272—41no—
100C4×H(3,𝔽3)—41no—
101C4×H(3,𝔽3)—41no—
102#1084—22no—
103#1084—22no—
104#1084—41no—
105C22×C3×D4—91no—
106C22×#67—34no—
107C2×C3×C4⋊C4—91no—
108C2×C3×C22⋊C4—34no—
109C2×C3×C22⋊C4—91no—
110C2×C3×C22⋊C4—91no—
111C2×C3×C22⋊C4—91no—
112C2×C3×C22⋊C4C3⋊D311yescharacteristic
113C2×#217—34no—
114C2×(C2×D3)⋊C4—34no—
115C2×(C2×D3)⋊C4—34no—
116C2×Dic3⋊C4—34no—
117C3×#95—91no—
118C3⋊#95—34no—
119C23×C32—24no—
120C23×C32—24no—
121C23×C32—24no—
122C23×C32—24no—
123C23×C32—24no—
124C23×C32—24no—
125C23×C32#6718yes—
126C23×C32C22×D314yes—
127C22×C3×D3—38no—
128C22×C3×D3—64no—
129C22×C3×D3—64no—
130C22×C3×D3—64no—
131C22×C3×D3—64no—
132C22×C3×D3—64no—
133C22×C3×D3—64no—
134C2×C32×C4—24no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—24no—
140C3×#67—632no—
141C2×C3×Dic3—38no—
142C2×C3×Dic3—38no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—64no—
149C2×C3×Dic3—64no—
150#394—91noSylow
151C2×H(3,𝔽3)—42no—
152C2×H(3,𝔽3)#12011yescharacteristic
153C2×H(3,𝔽3)#12111yescharacteristic
154C2×H(3,𝔽3)#12111yescharacteristic
155C2×H(3,𝔽3)#12111yescharacteristic
156C2×H(3,𝔽3)#12311yescharacteristic
157C2×H(3,𝔽3)#12311yescharacteristic
158C2×H(3,𝔽3)#12411yescharacteristic
159#272—42no—
160C24×C3—34no—
161C24×C3—91no—
162C24×C3C2×C3⋊D311yescharacteristic
163C23×D3—34no—
164C2×C3×D4—182no—
165C2×C3×D4—182no—
166C2×C3×D4—182no—
167C22×C3×C4—34no—
168C22×C3×C4—34no—
169C22×C3×C4—91no—
170C22×C3×C4—91no—
171C22×C3×C4—91no—
172C22×C3×C4C2×C3⋊D311yescharacteristic
173C22×C3×C4C2×C3⋊D311yescharacteristic
174C2×#67—68no—
175C2×#67—68no—
176C2×#67—68no—
177C22×Dic3—34no—
178C22×Dic3—34no—
179C22×Dic3—34no—
180C3×C4⋊C4—182no—
181C3×C22⋊C4—22no—
182C3×C22⋊C4—68no—
183C3×C22⋊C4—182no—
184C3×C22⋊C4—182no—
185C3×C22⋊C4—182no—
186#217—68no—
187(C2×D3)⋊C4—68no—
188(C2×D3)⋊C4—68no—
189Dic3⋊C4—68no—
190C22×C32—28no—
191C22×C32—28no—
192C22×C32—28no—
193C22×C32—44no—
194C22×C32—44no—
195C22×C32—44no—
196C22×C32—44no—
197C22×C32#23514yes—
198C22×C32#23514yes—
199C22×C32D3×D414yes—
200C22×C32D3×D414yes—
201C22×C32D3×D414yes—
202C22×C32#23714yes—
203C22×C32C2×#6714yes—
204C2×C3×D3—68no—
205C2×C3×D3—68no—
206C2×C3×D3—68no—
207C2×C3×D3—124no—
208C2×C3×D3—124no—
209C2×C3×D3—124no—
210C2×C3×D3—124no—
211C32×C4—44no—
212C32×C4—44no—
213C3×Dic3—68no—
214C3×Dic3—68no—
215C3×Dic3—124no—
216C22×D4—91no—
217C2×C4⋊C4—91no—
218C2×C22⋊C4—91no—
219C2×C22⋊C4—91no—
220C2×C22⋊C4—91no—
221C2×C22⋊C4#27211yescharacteristic, complemented
222#95—91no—
223H(3,𝔽3)#39411yescharacteristic, complemented, Sylow
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—38no—
231C23×C3—34no—
232C23×C3—64no—
233C23×C3—64no—
234C23×C3—64no—
235C23×C3—64no—
236C23×C3—64no—
237C23×C3—64no—
238C23×C3—92no—
239C23×C3—181no—
240C23×C3—181no—
241C23×C3—181no—
242C23×C3—181no—
243C23×C3—181no—
244C23×C3—181no—
245C23×C3#63812yes—
246C23×C3C22×C3⋊D311yescharacteristic
247C22×D3—38no—
248C22×D3—64no—
249C22×D3—64no—
250C22×D3—64no—
251C22×D3—64no—
252C22×D3—64no—
253C22×D3—64no—
254C3×D4—188no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—64no—
262C2×C3×C4—64no—
263C2×C3×C4—64no—
264C2×C3×C4—64no—
265C2×C3×C4—64no—
266C2×C3×C4—64no—
267C2×C3×C4—92no—
268C2×C3×C4—92no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276#67—632no—
277C2×Dic3—38no—
278C2×Dic3—38no—
279C2×Dic3—64no—
280C2×Dic3—64no—
281C2×Dic3—64no—
282C2×Dic3—64no—
283C2×Dic3—64no—
284C2×Dic3—64no—
285C2×Dic3—64no—
286C2×C32—48no—
287C2×C32#88314yes—
288C2×C32#88414yes—
289C2×C32#88414yes—
290C2×C32#88514yes—
291C2×C32#92914yes—
292C2×C32#93714yes—
293C2×C32#93914yes—
294C3×D3—128no—
295C24—91no—
296C24C2×#27211yescharacteristic
297C2×D4—182no—
298C2×D4—182no—
299C2×D4—182no—
300C22×C4—91no—
301C22×C4—91no—
302C22×C4—91no—
303C22×C4C2×#27211yescharacteristic
304C22×C4C2×#27211yescharacteristic
305C4⋊C4—182no—
306C22⋊C4—22no—
307C22⋊C4—182no—
308C22⋊C4—182no—
309C22⋊C4—182no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—34no—
314C22×C3—34no—
315C22×C3—34no—
316C22×C3—34no—
317C22×C3—34no—
318C22×C3—34no—
319C22×C3—34no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—68no—
325C22×C3—68no—
326C22×C3—68no—
327C22×C3—124no—
328C22×C3—124no—
329C22×C3—124no—
330C22×C3—124no—
331C22×C3—182no—
332C22×C3—182no—
333C22×C3—182no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3—361no—
337C22×C3—361no—
338C22×C3#382811yescharacteristic
339C22×C3#382811yescharacteristic
340C22×C3D4×C3⋊D311yescharacteristic
341C22×C3D4×C3⋊D311yescharacteristic
342C22×C3D4×C3⋊D311yescharacteristic
343C22×C3#383011yescharacteristic
344C22×C3C2×#63811yescharacteristic
345C2×D3—68no—
346C2×D3—68no—
347C2×D3—68no—
348C2×D3—124no—
349C2×D3—124no—
350C2×D3—124no—
351C2×D3—124no—
352C3×C4—41no—
353C3×C4—41no—
354C3×C4—124no—
355C3×C4—124no—
356C3×C4—182no—
357C3×C4—182no—
358C3×C4—361no—
359Dic3—68no—
360Dic3—68no—
361Dic3—124no—
362C32#495914yescomplemented
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—92no—
370C23—181no—
371C23—181no—
372C23—181no—
373C23—181no—
374C23—181no—
375C23—181no—
376C23#623512yes—
377C23C22×#27211yescharacteristic
378D4—188no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—92no—
386C2×C4—92no—
387C2×C4—181no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—181no—
391C2×C4—181no—
392C2×C4—181no—
393C2×C4—181no—
394C2×C3—34no—
395C2×C3—34no—
396C2×C3—34no—
397C2×C3—34no—
398C2×C3—34no—
399C2×C3—34no—
400C2×C3—34no—
401C2×C3—42no—
402C2×C3—128no—
403C2×C3—362no—
404C2×C3#6409211yescharacteristic
405C2×C3#6409311yescharacteristic
406C2×C3#6409311yescharacteristic
407C2×C3#6409411yescharacteristic
408C2×C3#6413811yescharacteristic
409C2×C3#6414611yescharacteristic
410C2×C3#6414811yescharacteristic
411D3—128no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—182no—
420C22—182no—
421C22—182no—
422C22—361no—
423C22—361no—
424C22—361no—
425C22—361no—
426C22#8874011yescharacteristic
427C22#8874011yescharacteristic
428C22D4×#27211yescharacteristic
429C22D4×#27211yescharacteristic
430C22D4×#27211yescharacteristic
431C22#8874411yescharacteristic
432C22C2×#623511yescharacteristic
433C4—41no—
434C4—41no—
435C4—182no—
436C4—182no—
437C4—361no—
438C3—34no—
439C3#1059265811yescharacteristic
440C2—42no—
441C2—362no—
442C2#1172173911yescharacteristic
443C2#1172174011yescharacteristic
444C2#1172174011yescharacteristic
445C2#1172174111yescharacteristic
446C2#1172182911yescharacteristic
447C2#1172184711yescharacteristic
448C2#1172184911yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC22×#6235C2
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#394
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×H(3,𝔽3)C2semidirect
3. C2×#88552C2semidirect
4. C2×#88542C2semidirect
5. C2×#88542C2semidirect
6. C2×#88540C2semidirect
7. #11720574C2semidirect
25. C2×C32×C22⋊C4D3semidirect
61. C2×#1084D4semidirect
221. C2×C22⋊C4#272semidirect
223. H(3,𝔽3)#394semidirect
362. C32#4959semidirect

Maximal subgroups

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

subgroupindex
C22×#62352
C2×C22⋊C4×H(3,𝔽3)2
C2×#885522
C2×#885422
C2×#885402
#117205742
C3×#49593