Finite Group Databasebeta

← #13140178

Subgroups — #13140178

Order 1728.

671 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#3752⋊C3C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×#88450C212yesmaximal
4#11720574C211yesmaximal, characteristic, complemented
5#11720334C212yesmaximal, complemented
6#10591326—31nomaximal
7#10591315—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13#3752⋊C3—22no—
14C4⋊((C3⋊Dic3)⋊C3)—22no—
15#88450—24no—
16C2×#3752D311yescharacteristic
17C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
18C2×C3×(C2×D3)⋊C4—31no—
19C2×C3×C4⋊Dic3—31no—
20C2×#3722—32no—
21C2×(C2×D3)⋊Dic3—32no—
22C3×C3⋊#95—31no—
23C3×C3⋊#95—31no—
24C3×C3⋊#95—61no—
25#63579—32no—
26#63579—32no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C22×(C3⋊D3)⋊C3—21no—
29C22×(C3⋊D3)⋊C3—21no—
30C22×(C3⋊D3)⋊C3—22no—
31C22×(C3⋊D3)⋊C3—22no—
32C22×(C3⋊D3)⋊C3D412yes—
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)D412yes—
35C2×#1084—22no—
36C2×#1084—22no—
37C2×#1084—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—22no—
43#4992—91no—
44#4953—91no—
45#4674—91no—
46C23×C3⋊D3C2×D311yescharacteristic
47C23×C3×D3—31no—
48C22×C32×C4—21no—
49C22×C32×C4C2×D311yescharacteristic
50C22×C32×C4C2×D311yescharacteristic
51C22×C3⋊Dic3C2×D311yescharacteristic
52C22×C3⋊Dic3C2×D312yes—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—31no—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—62no—
58#3752—22no—
59C4⋊(C3⋊Dic3)—22no—
60C3×(C2×D3)⋊C4—62no—
61C3×C4⋊Dic3—62no—
62#3722—64no—
63(C2×D3)⋊Dic3—64no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C22×H(3,𝔽3)#4111yescharacteristic
67C22×H(3,𝔽3)#4112yes—
68C22×H(3,𝔽3)#4112yes—
69C2×(C3⋊D3)⋊C3—22no—
70C2×(C3⋊D3)⋊C3—24no—
71C2×(C3⋊D3)⋊C3—41no—
72C2×(C3⋊D3)⋊C3—41no—
73C2×(C3⋊D3)⋊C3—42no—
74C4×H(3,𝔽3)—22no—
75#1084—42no—
76(C3⋊Dic3)⋊C3—42no—
77(C3⋊Dic3)⋊C3—41no—
78C2×C3×C4⋊C4—91no—
79C2×C3×C22⋊C4—91no—
80C2×#217—92no—
81C2×(C2×D3)⋊C4—31no—
82C2×(C2×D3)⋊C4—91no—
83C2×(C2×D3)⋊C4—92no—
84C2×(C2×D3)⋊C4—92no—
85C2×C4⋊Dic3—31no—
86C2×C4⋊Dic3—91no—
87C3×#95—91no—
88C3⋊#95—31no—
89C3⋊#95—31no—
90C3⋊#95—61no—
91C3⋊#95—92no—
92C3⋊#95—92no—
93C3⋊#95—92no—
94C23×C32—21no—
95C23×C32C22×D311yescharacteristic
96C23×C32C22×D311yescharacteristic
97C22×C3⋊D3—21no—
98C22×C3⋊D3—22no—
99C22×C3⋊D3—22no—
100C22×C3⋊D3—21no—
101C22×C3⋊D3D1212yes—
102C22×C3×D3—32no—
103C22×C3×D3—62no—
104C22×C3×D3—61no—
105C22×C3×D3—62no—
106C22×C3×D3—61no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—42no—
111C2×C32×C4#6712yes—
112C2×C32×C4#6712yes—
113C2×C3⋊Dic3—22no—
114C2×C3⋊Dic3—22no—
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—21no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—122no—
131C2×C3×Dic3—122no—
132#397—271noSylow
133C2×H(3,𝔽3)#12111yescharacteristic
134C2×H(3,𝔽3)#12312yes—
135C2×H(3,𝔽3)#12311yescharacteristic
136C2×H(3,𝔽3)#12411yescharacteristic
137C2×H(3,𝔽3)#12612yes—
138(C3⋊D3)⋊C3—42no—
139C24×C3—91no—
140C23×D3—31no—
141C23×D3—91no—
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—61no—
145C22×C3×C4—91no—
146C22×C3×C4—92no—
147C22×C3×C4—92no—
148C22×C3×C4D3211yescharacteristic
149C22×Dic3—32no—
150C22×Dic3—32no—
151C22×Dic3—31no—
152C22×Dic3—32no—
153C22×Dic3—62no—
154C22×Dic3—91no—
155C22×Dic3—92no—
156C3×C4⋊C4—182no—
157C3×C22⋊C4—182no—
158#217—184no—
159(C2×D3)⋊C4—62no—
160(C2×D3)⋊C4—182no—
161(C2×D3)⋊C4—184no—
162(C2×D3)⋊C4—184no—
163C4⋊Dic3—62no—
164C4⋊Dic3—182no—
165C22×C32—21no—
166C22×C32—22no—
167C22×C32—21no—
168C22×C32—22no—
169C22×C32—21no—
170C22×C32C2×D1211yescharacteristic
171C22×C32#23511yescharacteristic
172C22×C32#23512yes—
173C22×C32D3×D411yescharacteristic
174C22×C32#23711yescharacteristic
175C22×C32#23712yes—
176C22×C32#23712yes—
177C22×C32#23912yes—
178C22×C32C2×#6711yescharacteristic
179C22×C32C2×#6711yescharacteristic
180C2×C3⋊D3—24no—
181C2×C3⋊D3—22no—
182C2×C3⋊D3—42no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3×D3—64no—
186C2×C3×D3—62no—
187C2×C3×D3—122no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—42no—
193C3⋊Dic3—41no—
194C3⋊Dic3—42no—
195C3×Dic3—124no—
196C3×Dic3—122no—
197C3×Dic3—122no—
198C3×Dic3—122no—
199C3×Dic3—121no—
200C2×C4⋊C4—91no—
201C2×C22⋊C4—91no—
202C2×C22⋊C4—272no—
203#95—91no—
204#95—272no—
205H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
206C23×C3—31no—
207C23×C3—31no—
208C23×C3—61no—
209C23×C3—92no—
210C23×C3—181no—
211C23×C3—181no—
212C23×C3—182no—
213C23×C3—182no—
214C23×C3C2×D3211yescharacteristic
215C22×D3—32no—
216C22×D3—61no—
217C22×D3—62no—
218C22×D3—62no—
219C22×D3—61no—
220C22×D3—92no—
221C22×D3—181no—
222C22×D3—181no—
223C22×D3—182no—
224C22×D3—182no—
225C2×C3×C4—22no—
226C2×C3×C4—32no—
227C2×C3×C4—32no—
228C2×C3×C4—62no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—122no—
232C2×C3×C4—182no—
233C2×C3×C4—181no—
234C2×C3×C4—182no—
235C2×C3×C4—182no—
236C2×C3×C4—182no—
237C2×C3×C4—182no—
238C2×C3×C4—182no—
239C2×C3×C4—182no—
240C2×C3×C4#62512yes—
241C2×Dic3—62no—
242C2×Dic3—62no—
243C2×Dic3—62no—
244C2×Dic3—62no—
245C2×Dic3—62no—
246C2×Dic3—62no—
247C2×Dic3—64no—
248C2×Dic3—62no—
249C2×Dic3—62no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—61no—
253C2×Dic3—122no—
254C2×Dic3—122no—
255C2×Dic3—182no—
256C2×Dic3—181no—
257C2×Dic3—182no—
258C2×Dic3—182no—
259C2×Dic3—182no—
260C2×C32—22no—
261C2×C32—21no—
262C2×C32—22no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32#87912yes—
266C2×C32#88612yes—
267C2×C32#89412yes—
268C2×C32#89812yes—
269C2×C32#92911yescharacteristic
270C2×C32#93011yescharacteristic
271C2×C32#93511yescharacteristic
272C2×C32#93611yescharacteristic
273C2×C32#93811yescharacteristic
274C2×C32#94711yescharacteristic
275C3⋊D3—42no—
276C3×D3—122no—
277C24—91no—
278C22×C4—92no—
279C22×C4—91no—
280C22×C4—92no—
281C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
282C4⋊C4—182no—
283C22⋊C4—182no—
284C22⋊C4—544no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—32no—
288C22×C3—32no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—32no—
292C22×C3—32no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—61no—
296C22×C3—62no—
297C22×C3—61no—
298C22×C3—62no—
299C22×C3—61no—
300C22×C3—184no—
301C22×C3—182no—
302C22×C3—361no—
303C22×C3—362no—
304C22×C3—361no—
305C22×C3#379511yescharacteristic
306C22×C3#379612yes—
307C22×C3D3×D1211yescharacteristic
308C22×C3#380412yes—
309C22×C3C2×#62511yescharacteristic
310C2×D3—64no—
311C2×D3—62no—
312C2×D3—122no—
313C2×D3—121no—
314C2×D3—121no—
315C2×D3—184no—
316C2×D3—182no—
317C2×D3—361no—
318C2×D3—362no—
319C2×D3—361no—
320C3×C4—22no—
321C3×C4—62no—
322C3×C4—62no—
323C3×C4—122no—
324C3×C4—361no—
325C3×C4—362no—
326C3×C4—362no—
327Dic3—122no—
328Dic3—122no—
329Dic3—124no—
330Dic3—121no—
331Dic3—122no—
332Dic3—361no—
333Dic3—362no—
334C32—21no—
335C32#495311yescharacteristic, complemented
336C32#499211yescharacteristic, complemented
337C23—92no—
338C23—181no—
339C23—182no—
340C23—182no—
341C23—181no—
342C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
343C2×C4—22no—
344C2×C4—182no—
345C2×C4—182no—
346C2×C4—182no—
347C2×C4—182no—
348C2×C4—182no—
349C2×C4—182no—
350C2×C4—181no—
351C2×C4—182no—
352C2×C4#619812yes—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—32no—
356C2×C3—32no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—32no—
360C2×C3—32no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—61no—
364C2×C3—62no—
365C2×C3—61no—
366C2×C3—62no—
367C2×C3—61no—
368C2×C3—362no—
369C2×C3#6385812yes—
370C2×C3#6386311yescharacteristic
371C2×C3#6389012yes—
372C2×C3#6390811yescharacteristic
373C2×C3#6396111yescharacteristic
374D3—122no—
375D3—362no—
376C22—184no—
377C22—182no—
378C22—361no—
379C22—362no—
380C22—361no—
381C22#8864811yescharacteristic
382C22#8864912yes—
383C22#8865411yescharacteristic
384C22#8867112yes—
385C22C2×#619811yescharacteristic
386C4—22no—
387C4—361no—
388C4—362no—
389C4—362no—
390C3—31no—
391C3—31no—
392C3—61no—
393C3#1059131411yescharacteristic
394C2—362no—
395C2#1172110812yes—
396C2#1172111311yescharacteristic
397C2#1172113012yes—
398C2#1172114111yescharacteristic
399C2#1172130611yescharacteristic
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)#397
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
2. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
4. #11720574C2semidirect
5. #11720334C2semidirect
17. C2×C4⋊(C3⋊Dic3)D3semidirect
205. H(3,𝔽3)#397semidirect
335. C32#4953semidirect
336. C32#4992semidirect

Maximal subgroups

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

subgroupindex
C2×#3752⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884502
#117205742
#117203342
#105913263
#105913153