Finite Group Databasebeta

← #13140174

Subgroups — #13140174

Order 1728.

783 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 127 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×#634⋊C3C211yesmaximal, characteristic, complemented
2C2×#3752⋊C3C211yesmaximal, characteristic
3C2×#88450C212yesmaximal
4#11720574C211yesmaximal, characteristic, complemented
5#11720334C212yesmaximal, complemented
6#10591322—31nomaximal
7#10591310—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C2×#634⋊C3—22no—
14C2×#634⋊C3—24no—
15#3752⋊C3—22no—
16#88450—24no—
17C22×#634D311yescharacteristic, complemented
18C22×C3×Dic6—31no—
19C2×#3752D311yescharacteristic
20C2×C3×(C2×D3)⋊C4—31no—
21C2×#3722—32no—
22C2×(C2×D3)⋊Dic3—32no—
23C3×C3⋊#95—31no—
24C3×C3⋊#95—31no—
25C3×C3⋊#95—61no—
26#63579—32no—
27#63579—32no—
28C23×H(3,𝔽3)C2311yescharacteristic
29C22×(C3⋊D3)⋊C3—22no—
30C22×(C3⋊D3)⋊C3—21no—
31C22×(C3⋊D3)⋊C3—21no—
32C22×(C3⋊D3)⋊C3—22no—
33C22×(C3⋊D3)⋊C3Q812yescomplemented
34C2×C4×H(3,𝔽3)—22no—
35C2×C4×H(3,𝔽3)D412yes—
36C2×#1084—22no—
37C2×#1084—22no—
38C2×#1084—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3D414yes—
44#634⋊C3—28no—
45#5231—91no—
46#4947—91no—
47#4673—91no—
48C23×C3⋊D3C2×D311yescharacteristic
49C23×C3×D3—31no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C2×D311yescharacteristic
53C22×C3⋊Dic3C2×D311yescharacteristic
54C22×C3⋊Dic3C2×D312yes—
55C2×#634—22no—
56C2×#634—24no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—31no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—62no—
62C2×C3×Dic6—64no—
63C2×C3×Dic6—62no—
64#3752—22no—
65C3×(C2×D3)⋊C4—62no—
66#3722—64no—
67(C2×D3)⋊Dic3—64no—
68C22×H(3,𝔽3)C2×D412yes—
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C22×H(3,𝔽3)C2×Q811yescharacteristic
71C22×H(3,𝔽3)#4111yescharacteristic
72C22×H(3,𝔽3)#4112yes—
73C2×(C3⋊D3)⋊C3—24no—
74C2×(C3⋊D3)⋊C3—22no—
75C2×(C3⋊D3)⋊C3—41no—
76C2×(C3⋊D3)⋊C3—42no—
77C2×(C3⋊D3)⋊C3—41no—
78C4×H(3,𝔽3)—22no—
79#1084—42no—
80(C3⋊Dic3)⋊C3—24no—
81(C3⋊Dic3)⋊C3—41no—
82C22×C3×Q8—91no—
83C22×Dic6—31no—
84C22×Dic6—91no—
85C2×C3×C22⋊C4—91no—
86C2×#217—92no—
87C2×(C2×D3)⋊C4—31no—
88C2×(C2×D3)⋊C4—91no—
89C2×(C2×D3)⋊C4—92no—
90C2×(C2×D3)⋊C4—92no—
91C3×#95—91no—
92C3⋊#95—31no—
93C3⋊#95—31no—
94C3⋊#95—61no—
95C3⋊#95—92no—
96C3⋊#95—92no—
97C3⋊#95—92no—
98C23×C32—21no—
99C23×C32C22×D311yescharacteristic
100C23×C32C22×D311yescharacteristic
101C22×C3⋊D3—21no—
102C22×C3⋊D3—22no—
103C22×C3⋊D3—22no—
104C22×C3⋊D3—21no—
105C22×C3⋊D3Dic612yescomplemented
106C22×C3×D3—32no—
107C22×C3×D3—62no—
108C22×C3×D3—61no—
109C22×C3×D3—62no—
110C22×C3×D3—61no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—42no—
115C2×C32×C4#6712yes—
116C2×C32×C4#6712yes—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3—22no—
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3#6714yes—
122#634—28no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—122no—
136C2×C3×Dic3—122no—
137C3×Dic6—68no—
138#393—271noSylow
139C2×H(3,𝔽3)#12011yescharacteristic
140C2×H(3,𝔽3)C22⋊Q812yes—
141C2×H(3,𝔽3)C22⋊Q811yescharacteristic
142C2×H(3,𝔽3)#12412yes—
143C2×H(3,𝔽3)#12411yescharacteristic
144(C3⋊D3)⋊C3—42no—
145C24×C3—91no—
146C23×D3—31no—
147C23×D3—91no—
148C2×C3×Q8—182no—
149C2×C3×Q8—184no—
150C22×C3×C4—31no—
151C22×C3×C4—31no—
152C22×C3×C4—61no—
153C22×C3×C4—92no—
154C22×C3×C4—91no—
155C22×C3×C4—92no—
156C22×C3×C4D3211yescharacteristic
157C22×Dic3—32no—
158C22×Dic3—32no—
159C22×Dic3—32no—
160C22×Dic3—31no—
161C22×Dic3—62no—
162C22×Dic3—92no—
163C22×Dic3—91no—
164C2×Dic6—64no—
165C2×Dic6—62no—
166C2×Dic6—184no—
167C2×Dic6—182no—
168C3×C22⋊C4—182no—
169#217—184no—
170(C2×D3)⋊C4—62no—
171(C2×D3)⋊C4—182no—
172(C2×D3)⋊C4—184no—
173(C2×D3)⋊C4—184no—
174C22×C32—21no—
175C22×C32—22no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—22no—
179C22×C32C2×Dic611yescharacteristic
180C22×C32#23511yescharacteristic
181C22×C32D3×D412yes—
182C22×C32#23712yes—
183C22×C32D3×Q811yescharacteristic
184C22×C32#23912yes—
185C22×C32#23911yescharacteristic
186C22×C32C2×#6711yescharacteristic
187C22×C32C2×#6711yescharacteristic
188C22×C32C2×#6712yes—
189C2×C3⋊D3—24no—
190C2×C3⋊D3—22no—
191C2×C3⋊D3—42no—
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3×D3—62no—
195C2×C3×D3—64no—
196C2×C3×D3—121no—
197C2×C3×D3—122no—
198C2×C3×D3—121no—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—42no—
202C3⋊Dic3—24no—
203C3⋊Dic3—41no—
204C3×Dic3—64no—
205C3×Dic3—124no—
206C3×Dic3—122no—
207C3×Dic3—121no—
208C3×Dic3—122no—
209C22×Q8—91no—
210C2×C22⋊C4—91no—
211C2×C22⋊C4—272no—
212#95—91no—
213#95—272no—
214H(3,𝔽3)#39311yescharacteristic, complemented, Sylow
215C23×C3—31no—
216C23×C3—31no—
217C23×C3—61no—
218C23×C3—92no—
219C23×C3—182no—
220C23×C3—181no—
221C23×C3—182no—
222C23×C3—181no—
223C23×C3C2×D3211yescharacteristic
224C22×D3—32no—
225C22×D3—62no—
226C22×D3—61no—
227C22×D3—61no—
228C22×D3—62no—
229C22×D3—92no—
230C22×D3—181no—
231C22×D3—182no—
232C22×D3—182no—
233C22×D3—181no—
234C3×Q8—188no—
235C2×C3×C4—22no—
236C2×C3×C4—32no—
237C2×C3×C4—32no—
238C2×C3×C4—62no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—94no—
242C2×C3×C4—122no—
243C2×C3×C4—182no—
244C2×C3×C4—182no—
245C2×C3×C4—181no—
246C2×C3×C4—182no—
247C2×C3×C4—182no—
248C2×C3×C4—182no—
249C2×C3×C4—182no—
250C2×C3×C4#62512yes—
251C2×Dic3—34no—
252C2×Dic3—62no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—62no—
257C2×Dic3—62no—
258C2×Dic3—62no—
259C2×Dic3—62no—
260C2×Dic3—64no—
261C2×Dic3—62no—
262C2×Dic3—61no—
263C2×Dic3—94no—
264C2×Dic3—122no—
265C2×Dic3—122no—
266C2×Dic3—182no—
267C2×Dic3—182no—
268C2×Dic3—181no—
269C2×Dic3—182no—
270Dic6—68no—
271Dic6—188no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—22no—
275C2×C32—22no—
276C2×C32—21no—
277C2×C32#87812yes—
278C2×C32#88512yes—
279C2×C32#92411yescharacteristic
280C2×C32#93611yescharacteristic
281C2×C32#93711yescharacteristic
282C2×C32#94611yescharacteristic
283C2×C32#94612yes—
284C2×C32#94711yescharacteristic
285C2×C32#94712yes—
286C2×C32#95311yescharacteristic
287C3⋊D3—42no—
288C3×D3—122no—
289C24—91no—
290C2×Q8—184no—
291C2×Q8—182no—
292C22×C4—92no—
293C22×C4—92no—
294C22×C4—91no—
295C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
296C22⋊C4—182no—
297C22⋊C4—544no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—32no—
301C22×C3—32no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—31no—
306C22×C3—32no—
307C22×C3—32no—
308C22×C3—61no—
309C22×C3—62no—
310C22×C3—61no—
311C22×C3—61no—
312C22×C3—62no—
313C22×C3—182no—
314C22×C3—184no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3—362no—
318C22×C3D3×Dic611yescharacteristic
319C22×C3#379612yes—
320C22×C3#379911yescharacteristic
321C22×C3C2×#62511yescharacteristic
322C22×C3D3×#6712yes—
323C2×D3—62no—
324C2×D3—64no—
325C2×D3—121no—
326C2×D3—122no—
327C2×D3—121no—
328C2×D3—182no—
329C2×D3—184no—
330C2×D3—361no—
331C2×D3—361no—
332C2×D3—362no—
333C3×C4—22no—
334C3×C4—62no—
335C3×C4—62no—
336C3×C4—122no—
337C3×C4—184no—
338C3×C4—361no—
339C3×C4—362no—
340Dic3—64no—
341Dic3—122no—
342Dic3—122no—
343Dic3—124no—
344Dic3—121no—
345Dic3—184no—
346Dic3—361no—
347C32—21no—
348C32#494711yescharacteristic, complemented
349C32#523111yescharacteristic, complemented
350C23—92no—
351C23—182no—
352C23—181no—
353C23—181no—
354C23—182no—
355C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
356Q8—188no—
357C2×C4—22no—
358C2×C4—94no—
359C2×C4—182no—
360C2×C4—182no—
361C2×C4—182no—
362C2×C4—182no—
363C2×C4—182no—
364C2×C4—181no—
365C2×C4—182no—
366C2×C4#619812yes—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—32no—
374C2×C3—32no—
375C2×C3—32no—
376C2×C3—32no—
377C2×C3—61no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—62no—
381C2×C3—62no—
382C2×C3—362no—
383C2×C3#6385611yescharacteristic
384C2×C3#6386311yescharacteristic
385C2×C3#6388712yes—
386C2×C3#6390712yes—
387C2×C3#6397611yescharacteristic
388D3—122no—
389D3—362no—
390C22—182no—
391C22—184no—
392C22—361no—
393C22—361no—
394C22—362no—
395C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
396C22#8864912yes—
397C22#8865211yescharacteristic
398C22C2×#619811yescharacteristic
399C22#8867612yes—
400C4—22no—
401C4—184no—
402C4—361no—
403C4—362no—
404C3—31no—
405C3—31no—
406C3—61no—
407C3#1059130411yescharacteristic
408C2—362no—
409C2#1172110611yescharacteristic
410C2#1172111311yescharacteristic
411C2#1172112812yes—
412C2#1172115512yes—
413C2#1172131611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#393
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#634⋊C3C2semidirect
4. #11720574C2semidirect
5. #11720334C2semidirect
17. C22×#634D3semidirect
33. C22×(C3⋊D3)⋊C3Q8semidirect
105. C22×C3⋊D3Dic6semidirect
214. H(3,𝔽3)#393semidirect
348. C32#4947semidirect
349. C32#5231semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
C2×#3752⋊C32
C2×#884502
#117205742
#117203342
#105913223
#105913103