Finite Group Databasebeta

← #13151320

Subgroups — #13151320

Order 1728.

843 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 843 conjugacy classes of subgroups — too many to draw. Only its 141 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
1D4×#1084C211yesmaximal, characteristic
2#64144⋊C3C212yesmaximal
3#11721309C211yesmaximal, characteristic, complemented
4#11721309C211yesmaximal, characteristic, complemented
5C2×#88449C211yesmaximal, characteristic, complemented
6#11721298C211yesmaximal, characteristic, complemented
7#11721297C212yesmaximal, complemented
8#11721296C212yesmaximal, complemented
9#11721098C211yesmaximal, characteristic, complemented
10#11721085C212yesmaximal, complemented
11#11721078C211yesmaximal, characteristic, complemented
12#10595117—32nomaximal
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C2×#6196C2211yescharacteristic
18#88552C2211yescharacteristic
19#88552C2211yescharacteristic
20C4⋊#1084C2211yescharacteristic
21C4×#1084C2211yescharacteristic
22#3757⋊C3C2212yes—
23#3757⋊C3C2212yes—
24#3757⋊C3C2212yes—
25#3750⋊C3C2212yes—
26#3750⋊C3C2212yes—
27#88449C2212yes—
28#88449C2211yescharacteristic
29#88449C2211yescharacteristic
30#88449C2211yescharacteristic
31#88448C2212yes—
32#88448C2212yes—
33#88448C2212yes—
34#88447C2212yes—
35#88447C2212yes—
36#64144D312yes—
37C3×#935—32no—
38C3×D4×Dic3—32no—
39C3×D4×Dic3—61no—
40C2×Dic3⋊Dic3—32no—
41#63964—32no—
42#63964—32no—
43#63957—32no—
44#63956—32no—
45#63956—32no—
46#63955—32no—
47#63955—32no—
48#63867—32no—
49#63847—32no—
50#63846—32no—
51#63837—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C23×H(3,𝔽3)C2311yescharacteristic
54D4×H(3,𝔽3)—22no—
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×#1084—22no—
57C2×#1084—22no—
58C2×#1084C2311yescharacteristic
59C2×#1084C2311yescharacteristic
60C2×#1084C2311yescharacteristic
61C2×#1084C2311yescharacteristic
62C2×(C3⋊Dic3)⋊C3—24no—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67#6196—22no—
68#5650—92no—
69#5541—91no—
70C2×C32×D4—21no—
71C2×C32×D4C2×D312yes—
72C22×C3⋊Dic3C2×D312yes—
73C22×C3×Dic3—32no—
74C22×C3×Dic3—32no—
75C22×C3×Dic3—32no—
76C22×C3×Dic3—61no—
77C22×C3×Dic3—61no—
78C2×#627—32no—
79#3757C2×D312yes—
80#3757C2×D312yes—
81#3757C2×D312yes—
82#3750C2×D312yes—
83#3750C2×D312yes—
84C3×#217—32no—
85C3×#217—32no—
86C3×#217—32no—
87C3×#217—32no—
88C3×#217—32no—
89C3×#217—61no—
90C3×#217—61no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—61no—
93C3×Dic3⋊C4—32no—
94C3×Dic3⋊C4—32no—
95C3×C4×Dic3—32no—
96C3×C4×Dic3—61no—
97#3724—32no—
98#3724—32no—
99#3724—32no—
100Dic3⋊Dic3—32no—
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—32no—
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108Dic32—32no—
109Dic32—32no—
110Dic32—32no—
111Dic32—32no—
112C22×H(3,𝔽3)—21no—
113C22×H(3,𝔽3)—21no—
114C22×H(3,𝔽3)—21no—
115C22×H(3,𝔽3)—21no—
116C22×H(3,𝔽3)—21no—
117C22×H(3,𝔽3)#4112yes—
118C22×H(3,𝔽3)C2411yescharacteristic
119C4×H(3,𝔽3)—21no—
120#1084—21no—
121#1084—21no—
122#1084—21no—
123#1084#4112yes—
124(C3⋊Dic3)⋊C3—22no—
125(C3⋊Dic3)⋊C3—22no—
126(C3⋊Dic3)⋊C3—22no—
127(C3⋊Dic3)⋊C3—22no—
128C3×#123—92no—
129C3×C4×D4—91no—
130#935—32no—
131#935—92no—
132D4×Dic3—32no—
133D4×Dic3—61no—
134#926—91no—
135#926—92no—
136C2×C4⋊Dic3—91no—
137#924—91no—
138#924—91no—
139C2×Dic3⋊C4—92no—
140#890—92no—
141#889—92no—
142#889—92no—
143#889—92no—
144#887—92no—
145#879—92no—
146#879—92no—
147#879—92no—
148#878—92no—
149#878—92no—
150#877—92no—
151#877—92no—
152#877—92no—
153#870—91no—
154C4×Dic6—91no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32C22×D312yes—
158C23×C32C22×D312yes—
159C32×D4—24no—
160C32×D4—24no—
161C2×C32×C4—21no—
162C2×C32×C4C22×D312yes—
163C2×C3⋊Dic3—24no—
164C2×C3⋊Dic3C22×D312yes—
165C2×C3⋊Dic3C22×D312yes—
166C2×C3⋊Dic3C22×D312yes—
167C2×C3⋊Dic3C22×D312yes—
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—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—64no—
181C2×C3×Dic3—64no—
182C2×C3×Dic3—64no—
183C2×C3×Dic3—122no—
184C2×C3×Dic3—122no—
185#627—64no—
186#552—271noSylow
187C2×H(3,𝔽3)—22no—
188C2×H(3,𝔽3)—41no—
189C2×H(3,𝔽3)C2×#4111yescharacteristic
190C2×H(3,𝔽3)C2×#4111yescharacteristic
191C2×H(3,𝔽3)2−1+411yescharacteristic
192C2×C3×D4—32no—
193C2×C3×D4—61no—
194C2×C3×D4D3211yescharacteristic
195C22×C3×C4—92no—
196C22×C3×C4—91no—
197C22×C3×C4—91no—
198C22×Dic3—32no—
199C22×Dic3—32no—
200C22×Dic3—32no—
201C22×Dic3—61no—
202C22×Dic3—61no—
203C22×Dic3—92no—
204C2×Dic6—91no—
205C2×Dic6—92no—
206C3×C4⋊C4—92no—
207C3×C4⋊C4—92no—
208C3×C4⋊C4—91no—
209C3×C22⋊C4—92no—
210C3×C22⋊C4—92no—
211C3×C22⋊C4—92no—
212C3×C22⋊C4—91no—
213C3×C22⋊C4—91no—
214C3×C42—91no—
215#217—32no—
216#217—32no—
217#217—32no—
218#217—32no—
219#217—32no—
220#217—61no—
221#217—61no—
222#217—92no—
223#217—92no—
224#217—92no—
225C4⋊Dic3—32no—
226C4⋊Dic3—61no—
227C4⋊Dic3—91no—
228C4⋊Dic3—91no—
229C4⋊Dic3—91no—
230C4⋊Dic3—92no—
231C4⋊Dic3—92no—
232C4⋊Dic3—92no—
233C4⋊Dic3—92no—
234Dic3⋊C4—32no—
235Dic3⋊C4—32no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
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—
249C4×Dic3—32no—
250C4×Dic3—61no—
251C4×Dic3—92no—
252C4×Dic3—92no—
253C4×Dic3—92no—
254C4×Dic3—92no—
255C4×Dic3—92no—
256C4×Dic3—92no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—21no—
264C22×C32—22no—
265C22×C32—41no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32—41no—
269C22×C32#23714yes—
270C22×C32C23×D312yes—
271C32×C4—22no—
272C32×C4—22no—
273C3⋊Dic3—22no—
274C3⋊Dic3—22no—
275C3⋊Dic3—22no—
276C3⋊Dic3—22no—
277C3×Dic3—34no—
278C3×Dic3—62no—
279C3×Dic3—62no—
280C3×Dic3—62no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3—62no—
286C3×Dic3—121no—
287C3×Dic3—121no—
288C3×Dic3—121no—
289#126—272no—
290#125—271no—
291#125—272no—
292#123—92no—
293C22⋊Q8—271no—
294C22⋊Q8—271no—
295C4×Q8—271no—
296C4×D4—91no—
297#117—271no—
298#117—272no—
299C2×C4⋊C4—271no—
300H(3,𝔽3)#55211yescharacteristic, complemented, Sylow
301C23×C3—32no—
302C23×C3—32no—
303C23×C3—61no—
304C23×C3—61no—
305C23×C3C2×D3211yescharacteristic
306C23×C3C2×D3211yescharacteristic
307C3×D4—22no—
308C3×D4—64no—
309C3×D4—64no—
310C2×C3×C4—32no—
311C2×C3×C4—61no—
312C2×C3×C4—91no—
313C2×C3×C4—92no—
314C2×C3×C4—92no—
315C2×C3×C4—92no—
316C2×C3×C4—92no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4—91no—
320C2×C3×C4—182no—
321C2×C3×C4—182no—
322C2×C3×C4—184no—
323C2×C3×C4C2×D3211yescharacteristic
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3—32no—
328C2×Dic3—32no—
329C2×Dic3—32no—
330C2×Dic3—32no—
331C2×Dic3—32no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—64no—
337C2×Dic3—64no—
338C2×Dic3—64no—
339C2×Dic3—92no—
340C2×Dic3—92no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—122no—
344C2×Dic3—122no—
345C2×Dic3—184no—
346Dic6—184no—
347Dic6—182no—
348C2×C32—24no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—42no—
353C2×C32—42no—
354C2×C32—42no—
355C2×C32C2×#23712yes—
356C2×C32D3×#4112yes—
357C2×C32#101012yes—
358C2×Q8—271no—
359C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
360C22×C4—92no—
361C22×C4—91no—
362C22×C4—91no—
363C4⋊C4—91no—
364C4⋊C4—92no—
365C4⋊C4—92no—
366C4⋊C4—271no—
367C4⋊C4—271no—
368C4⋊C4—272no—
369C4⋊C4—271no—
370C4⋊C4—272no—
371C4⋊C4—272no—
372C4⋊C4—272no—
373C22⋊C4—91no—
374C22⋊C4—92no—
375C22⋊C4—92no—
376C22⋊C4—91no—
377C22⋊C4—92no—
378C42—91no—
379C42—272no—
380C42—272no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—32no—
387C22×C3—34no—
388C22×C3—62no—
389C22×C3—62no—
390C22×C3—62no—
391C22×C3—62no—
392C22×C3—62no—
393C22×C3—62no—
394C22×C3—61no—
395C22×C3—62no—
396C22×C3—121no—
397C22×C3—121no—
398C22×C3—121no—
399C22×C3—121no—
400C22×C3#380512yes—
401C22×C3C22×D3211yescharacteristic
402C3×C4—21no—
403C3×C4—62no—
404C3×C4—62no—
405C3×C4—92no—
406C3×C4—182no—
407C3×C4—182no—
408C3×C4—182no—
409C3×C4—182no—
410C3×C4—181no—
411C3×C4—181no—
412C3×C4—181no—
413Dic3—34no—
414Dic3—62no—
415Dic3—62no—
416Dic3—62no—
417Dic3—62no—
418Dic3—62no—
419Dic3—62no—
420Dic3—62no—
421Dic3—62no—
422Dic3—121no—
423Dic3—121no—
424Dic3—121no—
425Dic3—182no—
426Dic3—182no—
427Dic3—182no—
428Dic3—182no—
429C32—21no—
430C32#565012yescomplemented
431C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
432C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
433Q8—542no—
434D4—22no—
435C2×C4—91no—
436C2×C4—91no—
437C2×C4—91no—
438C2×C4—92no—
439C2×C4—92no—
440C2×C4—92no—
441C2×C4—91no—
442C2×C4—92no—
443C2×C4—182no—
444C2×C4—182no—
445C2×C4—184no—
446C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
447C2×C3—22no—
448C2×C3—32no—
449C2×C3—32no—
450C2×C3—32no—
451C2×C3—41no—
452C2×C3—64no—
453C2×C3—61no—
454C2×C3—61no—
455C2×C3—61no—
456C2×C3—122no—
457C2×C3—122no—
458C2×C3—122no—
459C2×C3#6430911yescharacteristic
460C2×C3#6431211yescharacteristic
461C2×C3C2×#380511yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22#8867212yes—
468C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
469C4—21no—
470C4—92no—
471C4—181no—
472C4—182no—
473C4—181no—
474C4—181no—
475C4—182no—
476C4—182no—
477C4—182no—
478C3—32no—
479C3—61no—
480C3#1059512011yescharacteristic
481C2—22no—
482C2—41no—
483C2#1172250211yescharacteristic
484C2#1172250511yescharacteristic
485C2C2×#8867211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#552
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #11721309C2semidirect
4. #11721309C2semidirect
5. C2×#88449C2semidirect
6. #11721298C2semidirect
7. #11721297C2semidirect
8. #11721296C2semidirect
9. #11721098C2semidirect
10. #11721085C2semidirect
11. #11721078C2semidirect
300. H(3,𝔽3)#552semidirect
430. C32#5650semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
#64144⋊C32
#117213092
C2×#884492
#117212982
#117212972
#117212962
#117210982
#117210852
#117210782
#105951173