Finite Group Databasebeta

← #13140638

Subgroups — #13140638

Order 1728.

971 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 971 conjugacy classes of subgroups — too many to draw. Only its 83 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×D4×#272C211yesmaximal, characteristic
2#11721320C212yesmaximal, complemented
3#63661⋊C3C212yesmaximal, complemented
4#11720335C212yesmaximal, complemented
5D3×#834—32nomaximal
6C23×#272C411yescharacteristic, complemented
7C23×#272C2211yescharacteristic
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#6235C411yescharacteristic, complemented
10C2×#6235C2211yescharacteristic
11D4×#272—22no—
12D4×#272—22no—
13C2×C4⋊#272C411yescharacteristic, complemented
14C2×C4×#272C411yescharacteristic, complemented
15C2×#6199—22no—
16#3757⋊C3C2214yescomplemented
17#88451—22no—
18C2×C3×D3×D4—32no—
19C2×C3×D3×D4—61no—
20D3×#217—34no—
21#63661D312yescomplemented
22C3×#834—32no—
23#63580—34no—
24C23×H(3,𝔽3)C2×C411yescharacteristic, complemented
25C23×H(3,𝔽3)C2311yescharacteristic
26C22×#272—21no—
27C22×#272—21no—
28C22×#272—42no—
29C22×#272—42no—
30C22×#272C2×C411yescharacteristic
31C22×#272C2×C411yescharacteristic
32C22×#272C2×C411yescharacteristic, complemented
33C22×#272D412yes—
34C22×#272D412yes—
35D4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)C2×C411yescharacteristic, complemented
37#6235—22no—
38#6235—22no—
39C2×#1084C2×C411yescharacteristic, complemented
40C4⋊#272—22no—
41C4×#272—21no—
42C4×#272—21no—
43C2×(C3⋊Dic3)⋊C3—24no—
44#6199—44no—
45C2×#834—91no—
46D3×#99—92no—
47C23×C3×D3—32no—
48C23×C3×D3—32no—
49C23×C3×D3—61no—
50C23×C3×D3—61no—
51C2×C32×D4—21no—
52C2×C32×D4C2×D312yes—
53C2×C3×#67—32no—
54C2×C3×#67—32no—
55C2×C3×#67—61no—
56C2×C3×#67—61no—
57C3×D3×D4—64no—
58C3×D3×D4—64no—
59C3×D3×D4—64no—
60C3×D3×D4—64no—
61C2×C3×D12—32no—
62C2×C3×D12—61no—
63C2×C3×C4×D3—32no—
64C2×C3×C4×D3—61no—
65C2×D3×Dic3—64no—
66#3757C2×D314yescomplemented
67C3×#217—34no—
68(C2×D3)⋊Dic3—64no—
69C22×H(3,𝔽3)—21no—
70C22×H(3,𝔽3)—41no—
71C22×H(3,𝔽3)—41no—
72C22×H(3,𝔽3)C22×C411yescharacteristic
73C22×H(3,𝔽3)C2×D412yes—
74C2×#272—22no—
75C2×#272—24no—
76C2×#272—21no—
77C2×#272—41no—
78C2×#272—42no—
79C2×#272—42no—
80C2×#272—42no—
81C2×#272—41no—
82C2×#272C22⋊C411yescharacteristic
83C2×#272C22⋊C412yes—
84C2×#272C22⋊C411yescharacteristic
85C4×H(3,𝔽3)—21no—
86#1084—21no—
87(C3⋊Dic3)⋊C3—44no—
88C22×C3×D4—91no—
89C2×D3×D4—32no—
90C2×D3×D4—61no—
91C2×#217—92no—
92D3×C22⋊C4—94no—
93C3×#99—92no—
94#834—32no—
95#834—92no—
96#834—92no—
97#806—94no—
98C23×C32—21no—
99C23×C32—21no—
100C23×C32C4×D312yescomplemented
101C23×C32C22×D312yes—
102C22×C3×D3—32no—
103C22×C3×D3—34no—
104C22×C3×D3—32no—
105C22×C3×D3—34no—
106C22×C3×D3—32no—
107C22×C3×D3—61no—
108C22×C3×D3—61no—
109C22×C3×D3—61no—
110C22×C3×D3—62no—
111C22×C3×D3—62no—
112C22×C3×D3—62no—
113C22×C3×D3—62no—
114C22×C3×D3—62no—
115C22×C3×D3—62no—
116C22×C3×D3—124no—
117C22×C3×D3—124no—
118C22×C3×D3—124no—
119C22×C3×D3—124no—
120C32×D4—24no—
121C32×D4—24no—
122C2×C32×C4—21no—
123C2×C32×C4C4×D312yescomplemented
124C2×C3⋊Dic3—24no—
125C3×#67—64no—
126C3×#67—64no—
127C3×#67—64no—
128C3×#67—64no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—64no—
132C3×D12—64no—
133C3×D12—64no—
134C3×C4×D3—62no—
135C3×C4×D3—62no—
136C3×C4×D3—62no—
137C3×C4×D3—62no—
138D3×Dic3—128no—
139C2×#99—271noSylow
140C2×H(3,𝔽3)—21no—
141C2×H(3,𝔽3)—22no—
142C2×H(3,𝔽3)—41no—
143C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
144#272—21no—
145#272—22no—
146#272—41no—
147#272#9912yescomplemented
148C24×C3—91no—
149C24×C3—91no—
150C23×D3—32no—
151C23×D3—32no—
152C23×D3—61no—
153C23×D3—61no—
154C2×C3×D4—32no—
155C2×C3×D4—61no—
156C2×C3×D4—91no—
157C2×C3×D4—91no—
158C2×C3×D4—91no—
159C2×C3×D4—182no—
160C2×C3×D4—182no—
161C2×C3×D4D3211yescharacteristic
162C22×C3×C4—91no—
163C2×#67—32no—
164C2×#67—32no—
165C2×#67—61no—
166C2×#67—61no—
167C22×Dic3—182no—
168D3×D4—64no—
169D3×D4—64no—
170D3×D4—64no—
171D3×D4—64no—
172C2×D12—32no—
173C2×D12—61no—
174C2×C4×D3—32no—
175C2×C4×D3—61no—
176C2×C4×D3—184no—
177C3×C22⋊C4—94no—
178#217—34no—
179#217—94no—
180#217—182no—
181(C2×D3)⋊C4—184no—
182C22×C32—22no—
183C22×C32—22no—
184C22×C32—21no—
185C22×C32—22no—
186C22×C32—42no—
187C22×C32—42no—
188C22×C32—42no—
189C22×C32—42no—
190C22×C32C2×C4×D312yes—
191C22×C32D3×D414yes—
192C2×C3×D3—32no—
193C2×C3×D3—32no—
194C2×C3×D3—34no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C2×C3×D3—62no—
198C2×C3×D3—68no—
199C2×C3×D3—64no—
200C2×C3×D3—64no—
201C2×C3×D3—62no—
202C2×C3×D3—62no—
203C2×C3×D3—124no—
204C2×C3×D3—124no—
205C2×C3×D3—122no—
206C2×C3×D3—124no—
207C2×C3×D3—124no—
208C2×C3×D3—124no—
209C2×C3×D3—124no—
210C2×C3×D3—124no—
211C2×C3×D3—122no—
212C2×C3×D3—122no—
213C2×C3×D3—122no—
214C32×C4—22no—
215C32×C4—22no—
216C3⋊Dic3—44no—
217C3×Dic3—62no—
218C3×Dic3—62no—
219C3×Dic3—124no—
220C22×D4—91no—
221C2×C22⋊C4—272no—
222#99—92no—
223#99—272no—
224H(3,𝔽3)C2×#9911yescharacteristic, complemented, Sylow
225C23×C3—32no—
226C23×C3—32no—
227C23×C3—61no—
228C23×C3—61no—
229C23×C3—91no—
230C23×C3—92no—
231C23×C3—91no—
232C23×C3—92no—
233C23×C3—91no—
234C23×C3—181no—
235C23×C3—181no—
236C23×C3—362no—
237C23×C3—362no—
238C23×C3#62411yescharacteristic
239C23×C3C2×D3211yescharacteristic
240C22×D3—32no—
241C22×D3—34no—
242C22×D3—34no—
243C22×D3—32no—
244C22×D3—32no—
245C22×D3—61no—
246C22×D3—61no—
247C22×D3—61no—
248C22×D3—62no—
249C22×D3—62no—
250C22×D3—62no—
251C22×D3—62no—
252C22×D3—62no—
253C22×D3—62no—
254C22×D3—124no—
255C22×D3—124no—
256C22×D3—124no—
257C22×D3—124no—
258C3×D4—22no—
259C3×D4—64no—
260C3×D4—64no—
261C3×D4—182no—
262C3×D4—182no—
263C3×D4—182no—
264C2×C3×C4—32no—
265C2×C3×C4—61no—
266C2×C3×C4—91no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4—184no—
270C2×C3×C4#62411yescharacteristic
271#67—64no—
272#67—64no—
273#67—64no—
274#67—64no—
275C2×Dic3—32no—
276C2×Dic3—61no—
277C2×Dic3—64no—
278C2×Dic3—184no—
279C2×Dic3—364no—
280D12—64no—
281D12—64no—
282C4×D3—62no—
283C4×D3—62no—
284C4×D3—62no—
285C4×D3—62no—
286C4×D3—368no—
287C2×C32—22no—
288C2×C32—24no—
289C2×C32—22no—
290C2×C32—21no—
291C2×C32—42no—
292C2×C32—42no—
293C2×C32—42no—
294C2×C32D3×C22⋊C412yes—
295C3×D3—34no—
296C3×D3—62no—
297C3×D3—62no—
298C3×D3—64no—
299C3×D3—62no—
300C3×D3—122no—
301C3×D3—122no—
302C3×D3—122no—
303C24—91no—
304C24—91no—
305C2×D4—91no—
306C2×D4—91no—
307C2×D4—91no—
308C2×D4—182no—
309C2×D4—182no—
310C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C22×C4—91no—
312C22×C4—542no—
313C22⋊C4—94no—
314C22⋊C4—542no—
315C22×C3—21no—
316C22×C3—32no—
317C22×C3—34no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—62no—
321C22×C3—62no—
322C22×C3—61no—
323C22×C3—62no—
324C22×C3—91no—
325C22×C3—91no—
326C22×C3—92no—
327C22×C3—122no—
328C22×C3—122no—
329C22×C3—122no—
330C22×C3—122no—
331C22×C3—184no—
332C22×C3—182no—
333C22×C3—181no—
334C22×C3—362no—
335C22×C3—361no—
336C22×C3—361no—
337C22×C3—362no—
338C22×C3—362no—
339C22×C3C2×#62411yescharacteristic
340C22×C3#381112yes—
341C2×D3—32no—
342C2×D3—32no—
343C2×D3—34no—
344C2×D3—61no—
345C2×D3—61no—
346C2×D3—62no—
347C2×D3—68no—
348C2×D3—64no—
349C2×D3—64no—
350C2×D3—62no—
351C2×D3—62no—
352C2×D3—124no—
353C2×D3—124no—
354C2×D3—124no—
355C2×D3—122no—
356C2×D3—124no—
357C2×D3—124no—
358C2×D3—124no—
359C2×D3—124no—
360C2×D3—122no—
361C2×D3—122no—
362C2×D3—122no—
363C3×C4—21no—
364C3×C4—62no—
365C3×C4—62no—
366C3×C4—181no—
367C3×C4—364no—
368Dic3—62no—
369Dic3—62no—
370Dic3—124no—
371Dic3—364no—
372C32—21no—
373C32D3×#9912yescomplemented
374C23—91no—
375C23—92no—
376C23—91no—
377C23—92no—
378C23—91no—
379C23—181no—
380C23—181no—
381C23—362no—
382C23—362no—
383C23#619911yescharacteristic, complemented
384C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
385D4—22no—
386D4—182no—
387D4—182no—
388D4—182no—
389C2×C4—91no—
390C2×C4—181no—
391C2×C4—181no—
392C2×C4—184no—
393C2×C4—1084no—
394C2×C4#619911yescharacteristic, complemented
395C2×C3—22no—
396C2×C3—21no—
397C2×C3—32no—
398C2×C3—41no—
399C2×C3—64no—
400C2×C3—62no—
401C2×C3—62no—
402C2×C3—61no—
403C2×C3—92no—
404C2×C3—122no—
405C2×C3—122no—
406C2×C3—122no—
407C2×C3—181no—
408C2×C3—182no—
409C2×C3—361no—
410C2×C3#6397411yescharacteristic
411D3—34no—
412D3—62no—
413D3—62no—
414D3—64no—
415D3—62no—
416D3—122no—
417D3—122no—
418D3—122no—
419C22—21no—
420C22—41no—
421C22—41no—
422C22—91no—
423C22—91no—
424C22—92no—
425C22—184no—
426C22—182no—
427C22—181no—
428C22—362no—
429C22—361no—
430C22—361no—
431C22—362no—
432C22—362no—
433C22C2×#619911yescharacteristic
434C22#8867712yes—
435C4—21no—
436C4—181no—
437C4—364no—
438C3—32no—
439C3—61no—
440C3#1059152711yescharacteristic
441C2—22no—
442C2—21no—
443C2—41no—
444C2—92no—
445C2—181no—
446C2—182no—
447C2—361no—
448C2#1172132011yescharacteristic
subgroupG/H
centerC2#11721320
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC23×C3C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC2×C3#63974
prime radicalC2×D4×#272C2
cosocleC23×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#99
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721320C2semidirect
3. #63661⋊C3C2semidirect
4. #11720335C2semidirect
6. C23×#272C4semidirect
9. C2×#6235C4semidirect
13. C2×C4⋊#272C4semidirect
14. C2×C4×#272C4semidirect
16. #3757⋊C3C22semidirect
21. #63661D3semidirect
24. C23×H(3,𝔽3)C2×C4semidirect
32. C22×#272C2×C4semidirect
36. C2×C4×H(3,𝔽3)C2×C4semidirect
39. C2×#1084C2×C4semidirect
66. #3757C2×D3semidirect
100. C23×C32C4×D3semidirect
123. C2×C32×C4C4×D3semidirect
147. #272#99semidirect
224. H(3,𝔽3)C2×#99semidirect
373. C32D3×#99semidirect
383. C23#6199semidirect
394. C2×C4#6199semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
#117213202
#63661⋊C32
#117203352
D3×#8343