Finite Group Databasebeta

← #13148478

Subgroups — #13148478

Order 1728.

797 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 797 conjugacy classes of subgroups — too many to draw. Only its 147 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
1#11721733C211yesmaximal, characteristic
2#64087⋊C3C212yesmaximal
3C2×#88449C211yesmaximal, characteristic
4#11721299C212yesmaximal
5#11721298C211yesmaximal, characteristic
6#11721117C211yesmaximal, characteristic, complemented
7#11721099C211yesmaximal, characteristic, complemented
8#11721098C211yesmaximal, characteristic, complemented
9#11721098C211yesmaximal, characteristic, complemented
10#11721086C212yesmaximal, complemented
11#11721079C212yesmaximal, complemented
12#10594035—32nomaximal
13C22×#1084C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C2×#634⋊C3C2212yes—
16C2×#6196C2212yes—
17C22⋊C4×H(3,𝔽3)C2211yescharacteristic
18#88552C2211yescharacteristic
19#88540C2211yescharacteristic
20#88540C2211yescharacteristic
21C4×#1084C2211yescharacteristic
22C4×#1084C2211yescharacteristic
23#3757⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2212yes—
25#3750⋊C3C2212yes—
26#3750⋊C3C2212yes—
27#88449C2211yescharacteristic
28#88449C2211yescharacteristic
29#88449C2212yes—
30#88449C2212yes—
31#88449C2211yescharacteristic
32#88449C2211yescharacteristic
33#88448C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#64087D312yes—
37C3×#878—32no—
38C3×#877—32no—
39C3×#877—61no—
40#63982—32no—
41C2×Dic3⋊Dic3—32no—
42#63964—32no—
43#63955—32no—
44#63918—32no—
45#63867—32no—
46#63867—32no—
47#63866—32no—
48#63846—32no—
49#63845—32no—
50#63840—32no—
51#63839—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×#1084Q814yescomplemented
56C2×#1084C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58C2×#1084C2311yescharacteristic
59C2×#1084C2311yescharacteristic
60C2×(C3⋊Dic3)⋊C3—42no—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65#634⋊C3—42no—
66#6196—42no—
67#5623—92no—
68#5519—91no—
69C22×C3⋊Dic3C2×D312yes—
70C2×#634C2×D312yes—
71C22×C3×Dic3—32no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—61no—
74C2×C3×Dic6—32no—
75C2×#627—32no—
76C2×#627—32no—
77C32×C22⋊C4—21no—
78C32×C22⋊C4C2×D312yes—
79#3757C2×D312yes—
80C4⋊(C3⋊Dic3)C2×D312yes—
81#3750C2×D312yes—
82#3750C2×D312yes—
83C3×#217—32no—
84C3×#217—32no—
85C3×#217—61no—
86C3×C4⋊Dic3—32no—
87C3×Dic3⋊C4—32no—
88C3×Dic3⋊C4—32no—
89C3×Dic3⋊C4—32no—
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—61no—
93C3×C4×Dic3—32no—
94C3×C4×Dic3—32no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3—61no—
97#3724—32no—
98#3724—32no—
99Dic3⋊Dic3—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—
108Dic3⋊Dic3—32no—
109Dic32—32no—
110Dic32—32no—
111C22×H(3,𝔽3)—21no—
112C22×H(3,𝔽3)—21no—
113C22×H(3,𝔽3)C2×Q812yes—
114C22×H(3,𝔽3)C2411yescharacteristic
115C4×H(3,𝔽3)—21no—
116C4×H(3,𝔽3)—21no—
117#1084—21no—
118#1084—21no—
119#1084C2×Q812yes—
120#1084C2×Q812yes—
121(C3⋊Dic3)⋊C3—22no—
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3—22no—
124(C3⋊Dic3)⋊C3—22no—
125C3×C22⋊Q8—92no—
126C3×#117—91no—
127#944—92no—
128#944—92no—
129#926—91no—
130C2×C4⋊Dic3—91no—
131#924—92no—
132C2×Dic3⋊C4—92no—
133#890—92no—
134#889—92no—
135#889—92no—
136#889—92no—
137#889—92no—
138#888—92no—
139#888—92no—
140#888—92no—
141#878—32no—
142#878—92no—
143#878—92no—
144#878—92no—
145#877—32no—
146#877—61no—
147#877—92no—
148#870—91no—
149#870—91no—
150#869—91no—
151#869—91no—
152C23×C32—21no—
153C23×C32C22×D312yes—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D312yes—
158C2×C3⋊Dic3—42no—
159C2×C3⋊Dic3C22×D312yes—
160C2×C3⋊Dic3C22×D312yes—
161C2×C3⋊Dic3C22×D312yes—
162C2×C3⋊Dic3C22×D312yes—
163#634—42no—
164C2×C3×Dic3—38no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
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—64no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—122no—
179C3×Dic6—122no—
180#627—122no—
181#627—122no—
182#544—271noSylow
183C2×H(3,𝔽3)—22no—
184C2×H(3,𝔽3)C22×Q811yescharacteristic
185C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
186C2×H(3,𝔽3)2−1+411yescharacteristic
187C2×C3×Q8—92no—
188C22×C3×C4—91no—
189C22×C3×C4—92no—
190C22×Dic3—32no—
191C22×Dic3—32no—
192C22×Dic3—61no—
193C22×Dic3—92no—
194C2×Dic6—32no—
195C2×Dic6—92no—
196C2×Dic6—92no—
197C2×Dic6—92no—
198C2×Dic6—92no—
199C3×C4⋊C4—92no—
200C3×C4⋊C4—92no—
201C3×C4⋊C4—92no—
202C3×C4⋊C4—91no—
203C3×C4⋊C4—91no—
204C3×C22⋊C4—32no—
205C3×C22⋊C4—61no—
206C3×C22⋊C4—92no—
207C3×C22⋊C4—91no—
208C3×C22⋊C4D3211yescharacteristic
209C3×C42—91no—
210C3×C42—91no—
211#217—32no—
212#217—32no—
213#217—61no—
214#217—92no—
215C4⋊Dic3—32no—
216C4⋊Dic3—92no—
217C4⋊Dic3—91no—
218C4⋊Dic3—91no—
219C4⋊Dic3—91no—
220C4⋊Dic3—92no—
221C4⋊Dic3—91no—
222C4⋊Dic3—92no—
223C4⋊Dic3—92no—
224C4⋊Dic3—92no—
225Dic3⋊C4—32no—
226Dic3⋊C4—32no—
227Dic3⋊C4—32no—
228Dic3⋊C4—32no—
229Dic3⋊C4—61no—
230Dic3⋊C4—61no—
231Dic3⋊C4—92no—
232Dic3⋊C4—92no—
233Dic3⋊C4—92no—
234Dic3⋊C4—92no—
235Dic3⋊C4—92no—
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—
245C4×Dic3—32no—
246C4×Dic3—32no—
247C4×Dic3—61no—
248C4×Dic3—61no—
249C4×Dic3—92no—
250C4×Dic3—92no—
251C4×Dic3—92no—
252C22×C32—22no—
253C22×C32—22no—
254C22×C32—21no—
255C22×C32—22no—
256C22×C32—41no—
257C22×C32—41no—
258C22×C32D3×Q814yes—
259C22×C32C23×D312yes—
260C32×C4—22no—
261C32×C4—22no—
262C32×C4—41no—
263C32×C4—41no—
264C3⋊Dic3—22no—
265C3⋊Dic3—22no—
266C3⋊Dic3—22no—
267C3⋊Dic3—22no—
268C3×Dic3—34no—
269C3×Dic3—34no—
270C3×Dic3—62no—
271C3×Dic3—62no—
272C3×Dic3—62no—
273C3×Dic3—62no—
274C3×Dic3—62no—
275C3×Dic3—62no—
276C3×Dic3—62no—
277C3×Dic3—62no—
278C3×Dic3—121no—
279C3×Dic3—121no—
280C4⋊Q8—271no—
281C4⋊Q8—271no—
282C4⋊Q8—272no—
283#125—271no—
284#125—271no—
285#125—272no—
286C22⋊Q8—92no—
287C22⋊Q8—272no—
288#117—91no—
289#117—271no—
290C2×C4⋊C4—271no—
291H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
292C23×C3—32no—
293C23×C3—61no—
294C23×C3C2×D3211yescharacteristic
295C3×Q8—362no—
296C2×C3×C4—32no—
297C2×C3×C4—32no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—91no—
301C2×C3×C4—91no—
302C2×C3×C4—94no—
303C2×C3×C4—92no—
304C2×C3×C4—92no—
305C2×C3×C4—92no—
306C2×C3×C4—92no—
307C2×C3×C4—91no—
308C2×C3×C4—91no—
309C2×C3×C4—362no—
310C2×C3×C4C2×D3211yescharacteristic
311C2×C3×C4C2×D3211yescharacteristic
312C2×Dic3—32no—
313C2×Dic3—32no—
314C2×Dic3—38no—
315C2×Dic3—32no—
316C2×Dic3—32no—
317C2×Dic3—32no—
318C2×Dic3—32no—
319C2×Dic3—32no—
320C2×Dic3—32no—
321C2×Dic3—61no—
322C2×Dic3—61no—
323C2×Dic3—64no—
324C2×Dic3—61no—
325C2×Dic3—61no—
326C2×Dic3—92no—
327C2×Dic3—92no—
328C2×Dic3—92no—
329C2×Dic3—92no—
330C2×Dic3—122no—
331C2×Dic3—362no—
332Dic6—122no—
333Dic6—362no—
334Dic6—362no—
335Dic6—362no—
336Dic6—362no—
337C2×C32—24no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—42no—
342C2×C32#100412yes—
343C2×C32C2×D3×Q812yes—
344C2×C32#101012yes—
345C2×Q8—92no—
346C2×Q8—272no—
347C22×C4—91no—
348C22×C4—92no—
349C4⋊C4—91no—
350C4⋊C4—91no—
351C4⋊C4—92no—
352C4⋊C4—92no—
353C4⋊C4—92no—
354C4⋊C4—271no—
355C4⋊C4—271no—
356C4⋊C4—272no—
357C4⋊C4—271no—
358C4⋊C4—271no—
359C4⋊C4—272no—
360C4⋊C4—272no—
361C4⋊C4—272no—
362C22⋊C4—91no—
363C22⋊C4—92no—
364C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
365C42—91no—
366C42—91no—
367C42—272no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—32no—
371C22×C3—34no—
372C22×C3—62no—
373C22×C3—62no—
374C22×C3—61no—
375C22×C3—62no—
376C22×C3—121no—
377C22×C3—121no—
378C22×C3#379712yes—
379C22×C3C22×D3211yescharacteristic
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—62no—
383C3×C4—62no—
384C3×C4—92no—
385C3×C4—92no—
386C3×C4—121no—
387C3×C4—121no—
388C3×C4—182no—
389C3×C4—182no—
390C3×C4—182no—
391C3×C4—182no—
392C3×C4—181no—
393C3×C4—181no—
394Dic3—34no—
395Dic3—34no—
396Dic3—62no—
397Dic3—62no—
398Dic3—62no—
399Dic3—62no—
400Dic3—62no—
401Dic3—62no—
402Dic3—62no—
403Dic3—62no—
404Dic3—121no—
405Dic3—121no—
406Dic3—182no—
407Dic3—182no—
408Dic3—182no—
409Dic3—182no—
410C32—21no—
411C32#562312yescomplemented
412C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
413Q8—362no—
414Q8—1082no—
415C2×C4—91no—
416C2×C4—91no—
417C2×C4—94no—
418C2×C4—91no—
419C2×C4—91no—
420C2×C4—92no—
421C2×C4—92no—
422C2×C4—92no—
423C2×C4—92no—
424C2×C4—362no—
425C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
426C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
427C2×C3—22no—
428C2×C3—32no—
429C2×C3—32no—
430C2×C3—32no—
431C2×C3—64no—
432C2×C3—61no—
433C2×C3—61no—
434C2×C3—61no—
435C2×C3—122no—
436C2×C3C2×#379711yescharacteristic
437C2×C3#6430911yescharacteristic
438C2×C3#6431311yescharacteristic
439C22—21no—
440C22—21no—
441C22#8865112yes—
442C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
443C4—21no—
444C4—21no—
445C4—92no—
446C4—92no—
447C4—181no—
448C4—182no—
449C4—182no—
450C4—182no—
451C4—182no—
452C4—181no—
453C3—32no—
454C3—61no—
455C3#1059403311yescharacteristic
456C2—22no—
457C2C2×#8865111yescharacteristic
458C2#1172250211yescharacteristic
459C2#1172250611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22⋊C4×H(3,𝔽3)C22
prime radicalC23×H(3,𝔽3)C23
hypercenterC22⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#544
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
6. #11721117C2semidirect
7. #11721099C2semidirect
8. #11721098C2semidirect
9. #11721098C2semidirect
10. #11721086C2semidirect
11. #11721079C2semidirect
55. C2×#1084Q8semidirect
291. H(3,𝔽3)#544semidirect
411. C32#5623semidirect

Maximal subgroups

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

subgroupindex
#117217332
#64087⋊C32
C2×#884492
#117212992
#117212982
#117211172
#117210992
#117210982
#117210862
#117210792
#105940353