Finite Group Databasebeta

← #13149581

Subgroups — #13149581

Order 1728.

883 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 883 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C4⋊C4×#272C211yesmaximal, characteristic
2#64098⋊C3C212yesmaximal, complemented
3#11721152C211yesmaximal, characteristic
4#11721147C211yesmaximal, characteristic
5#11721147C211yesmaximal, characteristic
6#11721147C211yesmaximal, characteristic
7#11721147C211yesmaximal, characteristic
8C4×#6199C211yesmaximal, characteristic
9#11721091C212yesmaximal, complemented
10#11721087C212yesmaximal, complemented
11#11721085C212yesmaximal, complemented
12D3×#889—32nomaximal
13C2×C4×#272C2211yescharacteristic
14C2×C4×#272C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×#6199C2211yescharacteristic
17C2×#6199C2211yescharacteristic
18C2×#6199C2211yescharacteristic
19C2×#6199C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊#1084C2211yescharacteristic
22#88540C2211yescharacteristic
23#88540C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2212yes—
25#3750⋊C3C2212yes—
26#3750⋊C3C2212yes—
27#3750⋊C3C2212yes—
28#3750⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2212yes—
30#88449C2212yes—
31#88448C2212yes—
32#88448C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#64098D312yescomplemented
37C3×D3×C4⋊C4—32no—
38C3×D3×C4⋊C4—61no—
39C3×#889—32no—
40D3×C4⋊Dic3—32no—
41D3×Dic3⋊C4—32no—
42D3×Dic3⋊C4—32no—
43D3×Dic3⋊C4—32no—
44D3×Dic3⋊C4—32no—
45C4×D3×Dic3—32no—
46#63870—32no—
47#63867—32no—
48#63847—32no—
49#63847—32no—
50#63846—32no—
51#63845—32no—
52C22×#272C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×#1084C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58C2×#1084C2311yescharacteristic
59C4×#272—22no—
60C4×#272—22no—
61C4×#272—22no—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66#6199—22no—
67#6199—22no—
68#6199—22no—
69#6199Q814yes—
70D3×#125—92no—
71C2×#889—91no—
72C2×C3×C4×D3—32no—
73C2×C3×C4×D3—32no—
74C2×C3×C4×D3—32no—
75C2×C3×C4×D3—61no—
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3—61no—
78C2×D3×Dic3—32no—
79C2×D3×Dic3—32no—
80C2×D3×Dic3—32no—
81C2×D3×Dic3—32no—
82C32×C4⋊C4—21no—
83C32×C4⋊C4C2×D312yes—
84C4⋊(C3⋊Dic3)C2×D312yes—
85#3750C2×D312yes—
86#3750C2×D312yes—
87#3750C2×D312yes—
88#3750C2×D312yes—
89C4×C3⋊Dic3C2×D312yes—
90C3×C4⋊Dic3—32no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—61no—
93C3×Dic3⋊C4—32no—
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—32no—
99C3×Dic3⋊C4—61no—
100C3×Dic3⋊C4—61no—
101C3×C4×Dic3—32no—
102#3724—32no—
103#3724—32no—
104#3724—32no—
105#3724—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108Dic3⋊Dic3—32no—
109Dic3⋊Dic3—32no—
110Dic3⋊Dic3—32no—
111Dic3⋊Dic3—32no—
112Dic32—32no—
113Dic32—32no—
114C22×H(3,𝔽3)C2411yescharacteristic
115C2×#272C2×Q812yes—
116C2×#272#4112yes—
117C2×#272#4112yes—
118C4×H(3,𝔽3)—21no—
119C4×H(3,𝔽3)—21no—
120C4×H(3,𝔽3)—21no—
121#1084—21no—
122#1084—21no—
123#1084—21no—
124(C3⋊Dic3)⋊C3—22no—
125(C3⋊Dic3)⋊C3—22no—
126(C3⋊Dic3)⋊C3—22no—
127(C3⋊Dic3)⋊C3C2×Q814yes—
128C3×#125—92no—
129C2×C3×C4⋊C4—91no—
130C2×C4⋊Dic3—91no—
131C2×Dic3⋊C4—91no—
132C2×Dic3⋊C4—91no—
133C2×Dic3⋊C4—91no—
134C2×Dic3⋊C4—91no—
135C2×C4×Dic3—91no—
136D3×C4⋊C4—32no—
137D3×C4⋊C4—61no—
138D3×C4⋊C4—92no—
139D3×C4⋊C4—92no—
140D3×C4⋊C4—92no—
141D3×C4⋊C4—92no—
142D3×C4⋊C4—92no—
143#890—92no—
144#890—92no—
145#889—32no—
146#889—92no—
147#889—92no—
148#889—92no—
149#889—92no—
150#889—92no—
151#889—92no—
152#889—92no—
153C42×D3—92no—
154#870—92no—
155C22×C3×D3—32no—
156C22×C3×D3—61no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4C22×D312yes—
161C2×C32×C4C22×D312yes—
162C2×C32×C4C22×D312yes—
163C2×C3⋊Dic3C22×D312yes—
164C2×C3⋊Dic3C22×D312yes—
165C2×C3⋊Dic3C22×D312yes—
166C2×C3⋊Dic3C22×D312yes—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C3×C4×D3—64no—
178C3×C4×D3—64no—
179C3×C4×D3—64no—
180C3×C4×D3—64no—
181C3×C4×D3—122no—
182C3×C4×D3—122no—
183D3×Dic3—38no—
184D3×Dic3—64no—
185D3×Dic3—64no—
186D3×Dic3—64no—
187C2×#125—271noSylow
188C2×H(3,𝔽3)C22×Q811yescharacteristic
189C2×H(3,𝔽3)C2×#4111yescharacteristic
190C2×H(3,𝔽3)C2×#4111yescharacteristic
191#272#12514yescomplemented
192C22×C3×C4—91no—
193C22×C3×C4—91no—
194C22×C3×C4—91no—
195C22×Dic3—91no—
196C22×Dic3—91no—
197C22×Dic3—91no—
198C22×Dic3—91no—
199C2×C4×D3—32no—
200C2×C4×D3—32no—
201C2×C4×D3—32no—
202C2×C4×D3—61no—
203C2×C4×D3—61no—
204C2×C4×D3—61no—
205C2×C4×D3—92no—
206C2×C4×D3—92no—
207C2×C4×D3—92no—
208C2×C4×D3—92no—
209C3×C4⋊C4—32no—
210C3×C4⋊C4—61no—
211C3×C4⋊C4—92no—
212C3×C4⋊C4—92no—
213C3×C4⋊C4—92no—
214C3×C4⋊C4—92no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4—91no—
217C3×C4⋊C4—91no—
218C3×C4⋊C4—91no—
219C3×C4⋊C4D3211yescharacteristic
220C3×C42—92no—
221C4⋊Dic3—32no—
222C4⋊Dic3—32no—
223C4⋊Dic3—61no—
224C4⋊Dic3—92no—
225C4⋊Dic3—92no—
226C4⋊Dic3—92no—
227C4⋊Dic3—92no—
228C4⋊Dic3—92no—
229C4⋊Dic3—92no—
230Dic3⋊C4—32no—
231Dic3⋊C4—32no—
232Dic3⋊C4—32no—
233Dic3⋊C4—32no—
234Dic3⋊C4—32no—
235Dic3⋊C4—32no—
236Dic3⋊C4—61no—
237Dic3⋊C4—61no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252C4×Dic3—32no—
253C4×Dic3—92no—
254C4×Dic3—92no—
255C4×Dic3—92no—
256C4×Dic3—92no—
257C22×C32—21no—
258C22×C32C23×D312yes—
259C2×C3×D3—34no—
260C2×C3×D3—34no—
261C2×C3×D3—34no—
262C2×C3×D3—62no—
263C2×C3×D3—62no—
264C2×C3×D3—62no—
265C32×C4—22no—
266C32×C4—22no—
267C32×C4—22no—
268C32×C4—22no—
269C32×C4—41no—
270C32×C4—41no—
271C3⋊Dic3—22no—
272C3⋊Dic3—22no—
273C3⋊Dic3—22no—
274C3⋊Dic3D3×Q814yes—
275C3×Dic3—34no—
276C3×Dic3—62no—
277C3×Dic3—62no—
278C3×Dic3—62no—
279C3×Dic3—62no—
280C3×Dic3—62no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—121no—
284C3×Dic3—121no—
285#125—92no—
286#125—272no—
287#125—272no—
288#125—272no—
289C2×C4⋊C4—91no—
290C2×C4⋊C4—271no—
291C2×C4⋊C4—271no—
292C2×C4⋊C4—271no—
293C2×C4⋊C4—271no—
294C2×C4⋊C4—271no—
295C2×C42—271no—
296H(3,𝔽3)C2×#12511yescharacteristic, complemented, Sylow
297C23×C3—91no—
298C22×D3—32no—
299C22×D3—61no—
300C2×C3×C4—32no—
301C2×C3×C4—32no—
302C2×C3×C4—32no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—92no—
307C2×C3×C4—92no—
308C2×C3×C4—92no—
309C2×C3×C4—92no—
310C2×C3×C4—91no—
311C2×C3×C4—91no—
312C2×C3×C4—91no—
313C2×C3×C4—182no—
314C2×C3×C4—182no—
315C2×C3×C4—182no—
316C2×C3×C4C2×D3211yescharacteristic
317C2×C3×C4C2×D3211yescharacteristic
318C2×C3×C4C2×D3211yescharacteristic
319C2×Dic3—32no—
320C2×Dic3—32no—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—61no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—92no—
330C2×Dic3—92no—
331C2×Dic3—92no—
332C2×Dic3—92no—
333C2×Dic3—94no—
334C2×Dic3—182no—
335C2×Dic3—182no—
336C2×Dic3—182no—
337C4×D3—64no—
338C4×D3—64no—
339C4×D3—64no—
340C4×D3—64no—
341C4×D3—98no—
342C4×D3—122no—
343C4×D3—122no—
344C4×D3—184no—
345C4×D3—184no—
346C4×D3—184no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32C2×D3×Q812yes—
351C2×C32D3×#4112yes—
352C2×C32D3×#4112yes—
353C3×D3—38no—
354C3×D3—64no—
355C22×C4—91no—
356C22×C4—91no—
357C22×C4—91no—
358C22×C4—271no—
359C22×C4—271no—
360C22×C4—271no—
361C22×C4—271no—
362C4⋊C4—91no—
363C4⋊C4—91no—
364C4⋊C4—91no—
365C4⋊C4—92no—
366C4⋊C4—92no—
367C4⋊C4—92no—
368C4⋊C4—92no—
369C4⋊C4—92no—
370C4⋊C4—272no—
371C4⋊C4—272no—
372C4⋊C4—272no—
373C4⋊C4—272no—
374C4⋊C4—272no—
375C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
376C42—92no—
377C42—272no—
378C22×C3—32no—
379C22×C3—61no—
380C22×C3—92no—
381C22×C3—92no—
382C22×C3—92no—
383C22×C3C22×D3211yescharacteristic
384C2×D3—34no—
385C2×D3—34no—
386C2×D3—34no—
387C2×D3—62no—
388C2×D3—62no—
389C2×D3—62no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—62no—
394C3×C4—62no—
395C3×C4—62no—
396C3×C4—62no—
397C3×C4—94no—
398C3×C4—121no—
399C3×C4—121no—
400C3×C4—182no—
401C3×C4—182no—
402C3×C4—182no—
403C3×C4—181no—
404C3×C4—181no—
405C3×C4—181no—
406Dic3—34no—
407Dic3—62no—
408Dic3—62no—
409Dic3—62no—
410Dic3—62no—
411Dic3—62no—
412Dic3—62no—
413Dic3—62no—
414Dic3—94no—
415Dic3—121no—
416Dic3—121no—
417Dic3—182no—
418Dic3—182no—
419Dic3—182no—
420C32—21no—
421C32D3×#12512yescomplemented
422C23—91no—
423C2×C4—92no—
424C2×C4—92no—
425C2×C4—91no—
426C2×C4—92no—
427C2×C4—92no—
428C2×C4—91no—
429C2×C4—91no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4—182no—
433C2×C4—274no—
434C2×C4—542no—
435C2×C4—542no—
436C2×C4—542no—
437C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
438C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
439C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
440C2×C3—32no—
441C2×C3—32no—
442C2×C3—32no—
443C2×C3—61no—
444C2×C3—61no—
445C2×C3—61no—
446C2×C3—94no—
447C2×C3#6430611yescharacteristic
448C2×C3#6431211yescharacteristic
449C2×C3D32×Q811yescharacteristic
450D3—38no—
451D3—64no—
452C22—92no—
453C22—92no—
454C22—92no—
455C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—94no—
460C4—181no—
461C4—181no—
462C4—182no—
463C4—181no—
464C4—182no—
465C4—182no—
466C3—32no—
467C3—61no—
468C3#1059439911yescharacteristic
469C2—94no—
470C2#1172249111yescharacteristic
471C2#1172250511yescharacteristic
472C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#125
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #64098⋊C3C2semidirect
9. #11721091C2semidirect
10. #11721087C2semidirect
11. #11721085C2semidirect
36. #64098D3semidirect
191. #272#125semidirect
296. H(3,𝔽3)C2×#125semidirect
421. C32D3×#125semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#2722
#64098⋊C32
#117211522
#117211472
C4×#61992
#117210912
#117210872
#117210852
D3×#8893