Finite Group Databasebeta

← #13140155

Subgroups — #13140155

Order 1728.

863 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 863 conjugacy classes of subgroups — too many to draw. Only its 123 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×#6229C211yesmaximal, characteristic, complemented
2C2×#3752⋊C3C211yesmaximal, characteristic
3C2×#88450C212yesmaximal
4#63658⋊C3C211yesmaximal, characteristic, complemented
5#11720334C212yesmaximal, complemented
6#10591310—31nomaximal
7#10591304—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C2×#6229—22no—
12C2×#6229—24no—
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15#3752⋊C3—22no—
16#88450—24no—
17C22×C3×Dic6—31no—
18C22×C3×Dic6—31no—
19C22×C3×Dic6—61no—
20C2×#3752D311yescharacteristic
21C2×C3×(C2×D3)⋊C4—31no—
22C2×#3722—32no—
23C2×(C2×D3)⋊Dic3—32no—
24#63658D311yescharacteristic, complemented
25C3×C3⋊#95—31no—
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—22no—
32C22×(C3⋊D3)⋊C3—21no—
33C22×(C3⋊D3)⋊C3Q812yescomplemented
34C2×C4×H(3,𝔽3)—22no—
35C2×C4×H(3,𝔽3)D412yes—
36C2×#1084—22no—
37C2×#1084—22no—
38C2×#1084D414yes—
39#6229—28no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3—22no—
44C2×(C3⋊Dic3)⋊C3—22no—
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—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—31no—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—62no—
60C2×C3×Dic6—64no—
61C2×C3×Dic6—64no—
62C2×C3×Dic6—68no—
63C2×C3×Dic6—64no—
64C2×C3×Dic6—62no—
65C2×C3×Dic6—62no—
66#3752—22no—
67C3×(C2×D3)⋊C4—62no—
68#3722—64no—
69(C2×D3)⋊Dic3—64no—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C22×H(3,𝔽3)C2×D412yes—
72C22×H(3,𝔽3)C2×Q811yescharacteristic
73C22×H(3,𝔽3)#4112yes—
74C22×H(3,𝔽3)#4111yescharacteristic
75C2×(C3⋊D3)⋊C3—24no—
76C2×(C3⋊D3)⋊C3—22no—
77C2×(C3⋊D3)⋊C3—41no—
78C2×(C3⋊D3)⋊C3—41no—
79C2×(C3⋊D3)⋊C3—42no—
80C4×H(3,𝔽3)—22no—
81#1084—24no—
82(C3⋊Dic3)⋊C3—41no—
83(C3⋊Dic3)⋊C3—42no—
84C22×C3×Q8—91no—
85C22×Dic6—31no—
86C22×Dic6—31no—
87C22×Dic6—61no—
88C2×C3×C22⋊C4—91no—
89C2×#217—92no—
90C2×(C2×D3)⋊C4—31no—
91C2×(C2×D3)⋊C4—91no—
92C2×(C2×D3)⋊C4—92no—
93C2×(C2×D3)⋊C4—92no—
94C3×#95—91no—
95C3⋊#95—31no—
96C3⋊#95—91no—
97C3⋊#95—92no—
98C3⋊#95—92no—
99C3⋊#95—92no—
100C23×C32—21no—
101C23×C32C22×D311yescharacteristic
102C23×C32C22×D311yescharacteristic
103C22×C3⋊D3—21no—
104C22×C3⋊D3—22no—
105C22×C3⋊D3—22no—
106C22×C3⋊D3—21no—
107C22×C3⋊D3Dic612yescomplemented
108C22×C3×D3—32no—
109C22×C3×D3—61no—
110C22×C3×D3—62no—
111C22×C3×D3—62no—
112C22×C3×D3—61no—
113C2×C32×C4—24no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4D1212yes—
118C2×C32×C4#6712yes—
119C2×C3⋊Dic3—22no—
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3—22no—
123C2×C3⋊Dic3—21no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—64no—
128C2×C3×Dic3—64no—
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—62no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C3×Dic6—616no—
139C3×Dic6—68no—
140C3×Dic6—68no—
141#393—271noSylow
142C2×H(3,𝔽3)#12011yescharacteristic
143C2×H(3,𝔽3)C22⋊Q812yes—
144C2×H(3,𝔽3)C22⋊Q811yescharacteristic
145C2×H(3,𝔽3)#12411yescharacteristic
146C2×H(3,𝔽3)#12412yes—
147(C3⋊D3)⋊C3—42no—
148C24×C3—91no—
149C23×D3—31no—
150C23×D3—91no—
151C2×C3×Q8—184no—
152C2×C3×Q8—182no—
153C22×C3×C4—31no—
154C22×C3×C4—31no—
155C22×C3×C4—61no—
156C22×C3×C4—92no—
157C22×C3×C4—91no—
158C22×C3×C4—92no—
159C22×C3×C4D3211yescharacteristic
160C22×Dic3—32no—
161C22×Dic3—32no—
162C22×Dic3—32no—
163C22×Dic3—31no—
164C22×Dic3—62no—
165C22×Dic3—91no—
166C22×Dic3—92no—
167C2×Dic6—64no—
168C2×Dic6—64no—
169C2×Dic6—62no—
170C2×Dic6—62no—
171C2×Dic6—68no—
172C2×Dic6—64no—
173C3×C22⋊C4—182no—
174#217—184no—
175(C2×D3)⋊C4—62no—
176(C2×D3)⋊C4—182no—
177(C2×D3)⋊C4—184no—
178(C2×D3)⋊C4—184no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—22no—
182C22×C32—21no—
183C22×C32—22no—
184C22×C32C2×Dic611yescharacteristic
185C22×C32C2×D1211yescharacteristic
186C22×C32#23512yes—
187C22×C32#23511yescharacteristic
188C22×C32D3×D412yes—
189C22×C32D3×D412yes—
190C22×C32#23712yes—
191C22×C32#23711yescharacteristic
192C22×C32D3×Q811yescharacteristic
193C22×C32C2×#6711yescharacteristic
194C2×C3⋊D3—24no—
195C2×C3⋊D3—22no—
196C2×C3⋊D3—42no—
197C2×C3⋊D3—41no—
198C2×C3⋊D3—41no—
199C2×C3×D3—64no—
200C2×C3×D3—62no—
201C2×C3×D3—121no—
202C2×C3×D3—121no—
203C2×C3×D3—122no—
204C32×C4—24no—
205C32×C4—22no—
206C32×C4—22no—
207C3⋊Dic3—42no—
208C3⋊Dic3—41no—
209C3×Dic3—68no—
210C3×Dic3—64no—
211C3×Dic3—64no—
212C3×Dic3—121no—
213C3×Dic3—122no—
214C22×Q8—91no—
215C2×C22⋊C4—91no—
216C2×C22⋊C4—272no—
217#95—91no—
218#95—272no—
219H(3,𝔽3)#39311yescharacteristic, complemented, Sylow
220C23×C3—31no—
221C23×C3—31no—
222C23×C3—61no—
223C23×C3—92no—
224C23×C3—182no—
225C23×C3—181no—
226C23×C3—181no—
227C23×C3—182no—
228C23×C3C2×D3211yescharacteristic
229C22×D3—32no—
230C22×D3—61no—
231C22×D3—62no—
232C22×D3—61no—
233C22×D3—62no—
234C22×D3—92no—
235C22×D3—181no—
236C22×D3—182no—
237C22×D3—182no—
238C22×D3—181no—
239C3×Q8—188no—
240C2×C3×C4—22no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—64no—
246C2×C3×C4—62no—
247C2×C3×C4—94no—
248C2×C3×C4—182no—
249C2×C3×C4—182no—
250C2×C3×C4—182no—
251C2×C3×C4—181no—
252C2×C3×C4—182no—
253C2×C3×C4—182no—
254C2×C3×C4—182no—
255C2×C3×C4#62612yes—
256C2×Dic3—34no—
257C2×Dic3—34no—
258C2×Dic3—64no—
259C2×Dic3—62no—
260C2×Dic3—62no—
261C2×Dic3—62no—
262C2×Dic3—62no—
263C2×Dic3—64no—
264C2×Dic3—64no—
265C2×Dic3—62no—
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—61no—
269C2×Dic3—62no—
270C2×Dic3—181no—
271C2×Dic3—182no—
272C2×Dic3—182no—
273C2×Dic3—182no—
274C2×Dic3—182no—
275Dic6—68no—
276Dic6—68no—
277Dic6—616no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—22no—
281C2×C32—22no—
282C2×C32—21no—
283C2×C32#87511yescharacteristic
284C2×C32#87812yes—
285C2×C32#88211yescharacteristic
286C2×C32#88512yes—
287C2×C32#88512yes—
288C2×C32#89612yes—
289C2×C32#89711yescharacteristic
290C2×C32#92411yescharacteristic
291C2×C32#93611yescharacteristic
292C2×C32#93711yescharacteristic
293C3⋊D3—42no—
294C3×D3—122no—
295C24—91no—
296C2×Q8—184no—
297C2×Q8—182no—
298C22×C4—92no—
299C22×C4—92no—
300C22×C4—91no—
301C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
302C22⋊C4—182no—
303C22⋊C4—544no—
304C22×C3—31no—
305C22×C3—31no—
306C22×C3—31no—
307C22×C3—32no—
308C22×C3—32no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—31no—
312C22×C3—32no—
313C22×C3—32no—
314C22×C3—61no—
315C22×C3—61no—
316C22×C3—62no—
317C22×C3—61no—
318C22×C3—62no—
319C22×C3—182no—
320C22×C3—184no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3—362no—
324C22×C3D3×Dic611yescharacteristic
325C22×C3#379511yescharacteristic
326C22×C3#379512yes—
327C22×C3C2×#62611yescharacteristic
328C22×C3#381112yes—
329C2×D3—62no—
330C2×D3—64no—
331C2×D3—121no—
332C2×D3—122no—
333C2×D3—121no—
334C2×D3—182no—
335C2×D3—184no—
336C2×D3—361no—
337C2×D3—361no—
338C2×D3—362no—
339C3×C4—22no—
340C3×C4—62no—
341C3×C4—62no—
342C3×C4—64no—
343C3×C4—184no—
344C3×C4—361no—
345C3×C4—362no—
346Dic3—64no—
347Dic3—64no—
348Dic3—68no—
349Dic3—121no—
350Dic3—122no—
351Dic3—361no—
352Dic3—362no—
353C32—21no—
354C32#467311yescharacteristic, complemented
355C32#494711yescharacteristic, complemented
356C23—92no—
357C23—181no—
358C23—182no—
359C23—181no—
360C23—182no—
361C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
362Q8—188no—
363C2×C4—22no—
364C2×C4—94no—
365C2×C4—182no—
366C2×C4—182no—
367C2×C4—182no—
368C2×C4—182no—
369C2×C4—182no—
370C2×C4—181no—
371C2×C4—182no—
372C2×C4#620012yes—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—32no—
380C2×C3—32no—
381C2×C3—32no—
382C2×C3—32no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—62no—
387C2×C3—62no—
388C2×C3—362no—
389C2×C3#6385711yescharacteristic
390C2×C3#6386011yescharacteristic
391C2×C3#6389612yes—
392C2×C3#6391312yes—
393C2×C3#6398111yescharacteristic
394D3—122no—
395D3—362no—
396C22—182no—
397C22—184no—
398C22—361no—
399C22—361no—
400C22—362no—
401C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
402C22#8864811yescharacteristic
403C22#8864812yes—
404C22C2×#620011yescharacteristic
405C22#8867712yes—
406C4—22no—
407C4—184no—
408C4—361no—
409C4—362no—
410C3—31no—
411C3—31no—
412C3—61no—
413C3#1059132211yescharacteristic
414C2—362no—
415C2#1172110711yescharacteristic
416C2#1172111011yescharacteristic
417C2#1172113512yes—
418C2#1172114312yes—
419C2#1172132311yescharacteristic
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×#6229C2semidirect
4. #63658⋊C3C2semidirect
5. #11720334C2semidirect
24. #63658D3semidirect
33. C22×(C3⋊D3)⋊C3Q8semidirect
107. C22×C3⋊D3Dic6semidirect
219. H(3,𝔽3)#393semidirect
354. C32#4673semidirect
355. C32#4947semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
C2×#3752⋊C32
C2×#884502
#63658⋊C32
#117203342
#105913103
#105913043