Finite Group Databasebeta

← #13140097

Subgroups — #13140097

Order 1728.

617 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 125 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×#88540C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
3C2×#88448C212yesmaximal
4#63658⋊C3C211yesmaximal, characteristic
5#11720334C212yesmaximal
6#10591353—31nomaximal
7#10591214—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13#88540—22no—
14C4⋊((C3⋊Dic3)⋊C3)—22no—
15#88448—24no—
16C2×C4⋊(C3⋊Dic3)D311yescharacteristic
17C2×C3×C4⋊Dic3—31no—
18C2×C3×Dic3⋊C4—31no—
19C2×C3×Dic3⋊C4—31no—
20C2×C3×Dic3⋊C4—61no—
21C2×#3724—32no—
22C2×Dic3⋊Dic3—32no—
23#63658D311yescharacteristic
24C3×C3⋊#95—31no—
25#63579—32no—
26#63579—32no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)D412yes—
30C2×#1084—21no—
31C2×#1084—21no—
32C2×#1084—21no—
33C2×#1084—22no—
34C2×#1084D412yes—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3Q814yes—
41#4983—91no—
42#4981—91no—
43#4661—91no—
44C22×C32×C4—21no—
45C22×C32×C4C2×D311yescharacteristic
46C22×C32×C4C2×D311yescharacteristic
47C22×C3⋊Dic3C2×D312yes—
48C22×C3⋊Dic3C2×D312yes—
49C22×C3×Dic3—31no—
50C22×C3×Dic3—31no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—31no—
53C22×C3×Dic3—31no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—61no—
56C22×C3×Dic3—61no—
57C4⋊(C3⋊Dic3)—22no—
58C3×C4⋊Dic3—62no—
59C3×Dic3⋊C4—64no—
60C3×Dic3⋊C4—62no—
61C3×Dic3⋊C4—62no—
62#3724—64no—
63Dic3⋊Dic3—64no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C22×H(3,𝔽3)C2×Q812yes—
67C22×H(3,𝔽3)#4111yescharacteristic
68C22×H(3,𝔽3)#4112yes—
69C4×H(3,𝔽3)—22no—
70#1084—22no—
71#1084—41no—
72(C3⋊Dic3)⋊C3—24no—
73(C3⋊Dic3)⋊C3—42no—
74C2×C3×C4⋊C4—91no—
75C2×C3×C4⋊C4—91no—
76C2×C4⋊Dic3—31no—
77C2×C4⋊Dic3—91no—
78C2×C4⋊Dic3—92no—
79C2×Dic3⋊C4—31no—
80C2×Dic3⋊C4—31no—
81C2×Dic3⋊C4—61no—
82C2×Dic3⋊C4—92no—
83C2×Dic3⋊C4—92no—
84C3×#95—91no—
85C3⋊#95—31no—
86C3⋊#95—91no—
87C3⋊#95—92no—
88C3⋊#95—92no—
89C3⋊#95—92no—
90C23×C32—21no—
91C23×C32C22×D311yescharacteristic
92C23×C32C22×D311yescharacteristic
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—42no—
97C2×C32×C4D1212yes—
98C2×C32×C4#6712yes—
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3Dic614yes—
105C2×C3×Dic3—32no—
106C2×C3×Dic3—32no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—122no—
126#398—271noSylow
127C2×H(3,𝔽3)#12111yescharacteristic
128C2×H(3,𝔽3)C22⋊Q812yes—
129C2×H(3,𝔽3)#12311yescharacteristic
130C2×H(3,𝔽3)#12512yes—
131C2×H(3,𝔽3)C4⋊Q811yescharacteristic
132C22×C3×C4—31no—
133C22×C3×C4—31no—
134C22×C3×C4—61no—
135C22×C3×C4—92no—
136C22×C3×C4—91no—
137C22×C3×C4—92no—
138C22×C3×C4—91no—
139C22×C3×C4D3211yescharacteristic
140C22×Dic3—31no—
141C22×Dic3—31no—
142C22×Dic3—32no—
143C22×Dic3—31no—
144C22×Dic3—31no—
145C22×Dic3—32no—
146C22×Dic3—61no—
147C22×Dic3—61no—
148C22×Dic3—92no—
149C22×Dic3—92no—
150C3×C4⋊C4—182no—
151C3×C4⋊C4—182no—
152C4⋊Dic3—62no—
153C4⋊Dic3—182no—
154C4⋊Dic3—184no—
155Dic3⋊C4—62no—
156Dic3⋊C4—62no—
157Dic3⋊C4—64no—
158Dic3⋊C4—184no—
159Dic3⋊C4—184no—
160C22×C32—22no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—22no—
164C22×C32—21no—
165C22×C32C2×Dic612yes—
166C22×C32C2×D1211yescharacteristic
167C22×C32#23512yes—
168C22×C32D3×D411yescharacteristic
169C22×C32D3×D411yescharacteristic
170C22×C32#23711yescharacteristic
171C22×C32#23712yes—
172C22×C32D3×Q812yes—
173C22×C32#23911yescharacteristic
174C22×C32C2×#6711yescharacteristic
175C32×C4—22no—
176C32×C4—22no—
177C32×C4—42no—
178C3⋊Dic3—24no—
179C3⋊Dic3—42no—
180C3×Dic3—64no—
181C3×Dic3—62no—
182C3×Dic3—64no—
183C3×Dic3—62no—
184C3×Dic3—122no—
185C3×Dic3—121no—
186C3×Dic3—122no—
187C3×Dic3—121no—
188C2×C4⋊C4—91no—
189C2×C4⋊C4—91no—
190C2×C4⋊C4—272no—
191#95—91no—
192#95—272no—
193H(3,𝔽3)#39811yescharacteristic, complemented, Sylow
194C23×C3—31no—
195C23×C3—31no—
196C23×C3—61no—
197C23×C3C2×D3211yescharacteristic
198C2×C3×C4—22no—
199C2×C3×C4—32no—
200C2×C3×C4—32no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4—62no—
204C2×C3×C4—94no—
205C2×C3×C4—92no—
206C2×C3×C4—122no—
207C2×C3×C4—182no—
208C2×C3×C4—182no—
209C2×C3×C4—181no—
210C2×C3×C4—181no—
211C2×C3×C4—182no—
212C2×C3×C4—182no—
213C2×C3×C4—182no—
214C2×C3×C4—182no—
215C2×C3×C4—181no—
216C2×C3×C4#62612yes—
217C2×Dic3—32no—
218C2×Dic3—32no—
219C2×Dic3—34no—
220C2×Dic3—62no—
221C2×Dic3—61no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—62no—
226C2×Dic3—62no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—94no—
238C2×Dic3—122no—
239C2×Dic3—182no—
240C2×Dic3—182no—
241C2×Dic3—182no—
242C2×Dic3—182no—
243C2×Dic3—182no—
244C2×C32—22no—
245C2×C32—21no—
246C2×C32—22no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32#86911yescharacteristic
250C2×C32#87812yes—
251C2×C32#88611yescharacteristic
252C2×C32#88912yes—
253C2×C32#89012yes—
254C2×C32#89511yescharacteristic
255C2×C32#89612yes—
256C2×C32#92911yescharacteristic
257C2×C32#93811yescharacteristic
258C2×C32#94411yescharacteristic
259C22×C4—92no—
260C22×C4—91no—
261C22×C4—91no—
262C22×C4—92no—
263C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
264C4⋊C4—182no—
265C4⋊C4—182no—
266C4⋊C4—544no—
267C22×C3—32no—
268C22×C3—32no—
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—31no—
273C22×C3—32no—
274C22×C3—32no—
275C22×C3—31no—
276C22×C3—31no—
277C22×C3—62no—
278C22×C3—61no—
279C22×C3—61no—
280C22×C3—62no—
281C22×C3—61no—
282C22×C3D3×Dic612yes—
283C22×C3#379611yescharacteristic
284C22×C3#380211yescharacteristic
285C22×C3#380412yes—
286C22×C3C2×#62611yescharacteristic
287C3×C4—22no—
288C3×C4—62no—
289C3×C4—62no—
290C3×C4—122no—
291C3×C4—184no—
292C3×C4—182no—
293C3×C4—362no—
294C3×C4—361no—
295Dic3—62no—
296Dic3—62no—
297Dic3—64no—
298Dic3—64no—
299Dic3—121no—
300Dic3—121no—
301Dic3—122no—
302Dic3—122no—
303Dic3—184no—
304Dic3—362no—
305C32—21no—
306C32#466111yescharacteristic, complemented
307C32#498311yescharacteristic, complemented
308C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
309C2×C4—22no—
310C2×C4—94no—
311C2×C4—92no—
312C2×C4—181no—
313C2×C4—182no—
314C2×C4—182no—
315C2×C4—181no—
316C2×C4—182no—
317C2×C4—182no—
318C2×C4—182no—
319C2×C4—181no—
320C2×C4—182no—
321C2×C4#620012yes—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—32no—
325C2×C3—32no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—32no—
329C2×C3—32no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—61no—
333C2×C3—62no—
334C2×C3—61no—
335C2×C3—62no—
336C2×C3—61no—
337C2×C3#6384612yes—
338C2×C3#6386611yescharacteristic
339C2×C3#6389912yes—
340C2×C3#6391611yescharacteristic
341C2×C3#6396611yescharacteristic
342C22((C3⋊D3)⋊C3)⋊Q812yes—
343C22#8864911yescharacteristic
344C22#8865511yescharacteristic
345C22#8867112yes—
346C22C2×#620011yescharacteristic
347C4—22no—
348C4—184no—
349C4—182no—
350C4—361no—
351C4—362no—
352C3—31no—
353C3—31no—
354C3—61no—
355C3#1059135811yescharacteristic
356C2#1172108712yes—
357C2#1172109011yescharacteristic
358C2#1172113412yes—
359C2#1172115911yescharacteristic
360C2#1172130111yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#398
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
193. H(3,𝔽3)#398semidirect
306. C32#4661semidirect
307. C32#4983semidirect

Maximal subgroups

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

subgroupindex
C2×#885402
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884482
#63658⋊C32
#117203342
#105913533
#105912143