Finite Group Databasebeta

← #13139938

Subgroups — #13139938

Order 1728.

410 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 75 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#11721824C211yesmaximal, characteristic
2#64135⋊C3C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#11720326C211yesmaximal, characteristic, complemented
5#11720326C211yesmaximal, characteristic, complemented
6#11720307C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591127—31nomaximal
9#10591102—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12#88552—21no—
13C4⋊#1084C2211yescharacteristic
14#88540—21no—
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64135D311yescharacteristic
22C3×#926—31no—
23C3×#924—31no—
24C3×#924—31no—
25C3×#924—61no—
26#63657D311yescharacteristic
27C3×#830—31no—
28#63576—31no—
29#63575—31no—
30#63574—31no—
31#63573—31no—
32#63566—31no—
33#63566—31no—
34#63565—31no—
35#63565—31no—
36C23×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C2×#1084—21no—
43#6229—21no—
44#6229—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#5045—91no—
50#4996—91no—
51#4723—91no—
52C22×C32×C4—21no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C2×D311yescharacteristic
55C2×C3×Dic6—31no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6—61no—
58#3757—21no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4×C3⋊Dic3—21no—
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C3×#217—62no—
65C3×#217—61no—
66C3×#217—61no—
67C3×#217—61no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—61no—
73C3×Dic3⋊C4—62no—
74C3×Dic3⋊C4—61no—
75C3×Dic3⋊C4—61no—
76C3×C4×Dic3—61no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C22×H(3,𝔽3)—21no—
80C22×H(3,𝔽3)—21no—
81C22×H(3,𝔽3)—21no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)#4111yescharacteristic
85C4×H(3,𝔽3)#4111yescharacteristic
86#1084—41no—
87#1084—41no—
88(C3⋊Dic3)⋊C3—41no—
89(C3⋊Dic3)⋊C3—41no—
90C3×C22⋊Q8—91no—
91C3×#117—91no—
92#926—31no—
93#926—91no—
94#924—31no—
95#924—31no—
96#924—61no—
97C3×C22⋊C8—91no—
98#835—91no—
99#835—91no—
100#830—31no—
101#830—91no—
102#818—91no—
103#817—91no—
104#816—91no—
105#816—91no—
106#810—91no—
107#810—91no—
108#808—91no—
109#808—91no—
110#807—91no—
111#807—91no—
112C23×C32—21no—
113C23×C32D1211yescharacteristic
114C23×C32#6711yescharacteristic
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—41no—
122C2×C32×C4—41no—
123C2×C32×C4D1211yescharacteristic
124C2×C32×C4#6711yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C3×Dic6—61no—
138C3×Dic6—61no—
139C3×Dic6—61no—
140C3×Dic6—61no—
141C3×Dic6—121no—
142C3×Dic6—121no—
143C3⋊(C3⋊C8)—21no—
144C3⋊(C3⋊C8)—21no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147#485—271noSylow
148C2×H(3,𝔽3)—41no—
149C2×H(3,𝔽3)#12311yescharacteristic
150C2×H(3,𝔽3)#13511yescharacteristic
151C2×H(3,𝔽3)#13711yescharacteristic
152C2×C3×Q8—91no—
153C22×C3×C4—31no—
154C22×C3×C4—31no—
155C22×C3×C4—61no—
156C22×C3×C4D3211yescharacteristic
157C2×Dic6—31no—
158C2×Dic6—31no—
159C2×Dic6—61no—
160C2×C3×C8—91no—
161C2×C3×C8—91no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—181no—
166C3×C22⋊C4—181no—
167C3×C22⋊C4—181no—
168C3×C42—181no—
169#217—61no—
170#217—61no—
171#217—62no—
172#217—61no—
173#217—181no—
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—61no—
179C4⋊Dic3—91no—
180C4⋊Dic3—91no—
181Dic3⋊C4—61no—
182Dic3⋊C4—61no—
183Dic3⋊C4—62no—
184C4×Dic3—61no—
185C4×Dic3—181no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—91no—
190C22×C32—22no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32C2×D1211yescharacteristic
201C22×C32C2×#6711yescharacteristic
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—41no—
207C32×C4#23511yescharacteristic
208C32×C4#23511yescharacteristic
209C32×C4#23711yescharacteristic
210C32×C4#23711yescharacteristic
211C3⋊Dic3—41no—
212C3⋊Dic3—41no—
213C3×Dic3—122no—
214C3×Dic3—122no—
215C3×Dic3—121no—
216C3×Dic3—121no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C3×Dic3—121no—
220C3×Dic3—121no—
221C22⋊Q8—91no—
222#117—91no—
223#107—271no—
224#106—271no—
225#103—271no—
226#103—271no—
227C22⋊C8—91no—
228H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
229C23×C3—31no—
230C23×C3—31no—
231C23×C3—61no—
232C23×C3#62611yescharacteristic
233C3×Q8—181no—
234C3×Q8—181no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—121no—
248C2×C3×C4—121no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4#62611yescharacteristic
254C2×C3×C4C2×D3211yescharacteristic
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—62no—
260C2×Dic3—62no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—181no—
264C2×Dic3—181no—
265Dic6—61no—
266Dic6—61no—
267Dic6—61no—
268Dic6—61no—
269Dic6—121no—
270Dic6—121no—
271C3×C8—181no—
272C3×C8—181no—
273C3⋊C8—61no—
274C3⋊C8—61no—
275C3⋊C8—181no—
276C3⋊C8—181no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—42no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32#88611yescharacteristic
284C2×C32#90411yescharacteristic
285C2×C32#90911yescharacteristic
286C2×C32#92911yescharacteristic
287C2×C32#95011yescharacteristic
288C2×C32#95111yescharacteristic
289C2×Q8—91no—
290C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
291C2×C8—91no—
292C2×C8—91no—
293C4⋊C4—91no—
294C4⋊C4—91no—
295C4⋊C4—91no—
296C4⋊C4—181no—
297C22⋊C4—181no—
298C22⋊C4—181no—
299C42—181no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—61no—
306C22×C3—61no—
307C22×C3—61no—
308C22×C3—61no—
309C22×C3—61no—
310C22×C3—61no—
311C22×C3—62no—
312C22×C3—61no—
313C22×C3—121no—
314C22×C3—121no—
315C22×C3C2×#62611yescharacteristic
316C3×C4—21no—
317C3×C4—31no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—61no—
322C3×C4—61no—
323C3×C4—61no—
324C3×C4—61no—
325C3×C4—121no—
326C3×C4—361no—
327C3×C4—361no—
328C3×C4—361no—
329C3×C4—361no—
330C3×C4#380411yescharacteristic
331C3×C4#380411yescharacteristic
332Dic3—121no—
333Dic3—121no—
334Dic3—121no—
335Dic3—121no—
336Dic3—122no—
337Dic3—122no—
338Dic3—121no—
339Dic3—121no—
340Dic3—361no—
341Dic3—361no—
342C32—21no—
343C32#472311yescharacteristic, complemented
344C32#499611yescharacteristic, complemented
345C23#620011yescharacteristic
346Q8—181no—
347Q8—181no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—181no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4#620011yescharacteristic
355C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C8—181no—
357C8—181no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—41no—
365C2×C3—61no—
366C2×C3—61no—
367C2×C3—61no—
368C2×C3—121no—
369C2×C3—121no—
370C2×C3—122no—
371C2×C3#6382911yescharacteristic
372C2×C3#6383111yescharacteristic
373C2×C3#6396611yescharacteristic
374C22—21no—
375C22—21no—
376C22—21no—
377C22C2×#620011yescharacteristic
378C4—21no—
379C4—361no—
380C4—361no—
381C4—361no—
382C4—361no—
383C4#8867111yescharacteristic
384C4#8867111yescharacteristic
385C3—31no—
386C3—31no—
387C3—61no—
388C3#1059113411yescharacteristic
389C2—41no—
390C2#1172107311yescharacteristic
391C2#1172107411yescharacteristic
392C2#1172130111yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#485
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720326C2semidirect
5. #11720326C2semidirect
6. #11720307C2semidirect
7. #11720306C2semidirect
228. H(3,𝔽3)#485semidirect
343. C32#4723semidirect
344. C32#4996semidirect

Maximal subgroups

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

subgroupindex
#117218242
#64135⋊C32
#63657⋊C32
#117203262
#117203072
#117203062
#105911273
#105911023