Finite Group Databasebeta

← #13139953

Subgroups — #13139953

Order 1728.

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

normal subgroups only

This group has 412 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#11721826C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#11720309C211yesmaximal, characteristic, complemented
5#11720309C211yesmaximal, characteristic, complemented
6#11720307C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591134—31nomaximal
9#10591102—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12#88552—21no—
13C4⋊#1084C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4×#1084—21no—
16#3757⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64133D311yescharacteristic
22C3×#926—31no—
23C3×#926—31no—
24C3×#926—61no—
25C3×#924—31no—
26#63657D311yescharacteristic
27C3×#830—31no—
28#63576—31no—
29#63575—31no—
30#63574—31no—
31#63573—31no—
32#63567—31no—
33#63567—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—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45#634⋊C3—21no—
46#634⋊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×#634C2×D311yescharacteristic
56C2×C3×Dic6—31no—
57#3757—21no—
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59#3750—21no—
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C3×#217—62no—
63C3×#217—61no—
64C3×#217—61no—
65C3×#217—61no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—61no—
72C3×C4⋊Dic3—61no—
73C3×Dic3⋊C4—61no—
74C3×C4×Dic3—62no—
75C3×C4×Dic3—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—31no—
94#926—61no—
95#924—31no—
96#924—91no—
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—22no—
116C2×C32×C4—22no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4D1211yescharacteristic
124C2×C32×C4#6711yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129#634—21no—
130#634—21no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C3×Dic6—61no—
140C3×Dic6—61no—
141C3⋊(C3⋊C8)—21no—
142C3⋊(C3⋊C8)—21no—
143C3×C3⋊C8—61no—
144C3×C3⋊C8—61no—
145#485—271noSylow
146C2×H(3,𝔽3)—41no—
147C2×H(3,𝔽3)#12311yescharacteristic
148C2×H(3,𝔽3)#13511yescharacteristic
149C2×H(3,𝔽3)#13711yescharacteristic
150C2×C3×Q8—91no—
151C22×C3×C4—31no—
152C22×C3×C4—31no—
153C22×C3×C4—61no—
154C22×C3×C4D3211yescharacteristic
155C2×Dic6—31no—
156C2×Dic6—91no—
157C2×C3×C8—91no—
158C2×C3×C8—91no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—181no—
163C3×C22⋊C4—181no—
164C3×C22⋊C4—181no—
165C3×C42—181no—
166#217—61no—
167#217—61no—
168#217—62no—
169#217—61no—
170#217—181no—
171C4⋊Dic3—31no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—61no—
177C4⋊Dic3—61no—
178C4⋊Dic3—91no—
179Dic3⋊C4—61no—
180Dic3⋊C4—181no—
181C4×Dic3—61no—
182C4×Dic3—61no—
183C4×Dic3—62no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—91no—
188C22×C32—22no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32C2×D1211yescharacteristic
199C22×C32C2×#6711yescharacteristic
200C32×C4—22no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4#23711yescharacteristic
206C32×C4#23711yescharacteristic
207C32×C4#23711yescharacteristic
208C32×C4#23711yescharacteristic
209C3⋊Dic3—41no—
210C3⋊Dic3—41no—
211C3×Dic3—122no—
212C3×Dic3—122no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C3×Dic3—121no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C22⋊Q8—91no—
220#117—91no—
221#107—271no—
222#106—271no—
223#103—271no—
224#103—271no—
225C22⋊C8—91no—
226H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
227C23×C3—31no—
228C23×C3—31no—
229C23×C3—61no—
230C23×C3#62611yescharacteristic
231C3×Q8—181no—
232C3×Q8—181no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4#62611yescharacteristic
252C2×C3×C4C2×D3211yescharacteristic
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—62no—
258C2×Dic3—62no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—181no—
262C2×Dic3—181no—
263Dic6—61no—
264Dic6—61no—
265Dic6—181no—
266Dic6—181no—
267C3×C8—181no—
268C3×C8—181no—
269C3⋊C8—61no—
270C3⋊C8—61no—
271C3⋊C8—181no—
272C3⋊C8—181no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—42no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32#88611yescharacteristic
280C2×C32#90411yescharacteristic
281C2×C32#90911yescharacteristic
282C2×C32#93511yescharacteristic
283C2×C32#94211yescharacteristic
284C2×C32#95011yescharacteristic
285C2×Q8—91no—
286C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
287C2×C8—91no—
288C2×C8—91no—
289C4⋊C4—91no—
290C4⋊C4—91no—
291C4⋊C4—91no—
292C4⋊C4—181no—
293C22⋊C4—181no—
294C22⋊C4—181no—
295C42—181no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—31no—
300C22×C3—31no—
301C22×C3—61no—
302C22×C3—61no—
303C22×C3—61no—
304C22×C3—61no—
305C22×C3—61no—
306C22×C3—61no—
307C22×C3—62no—
308C22×C3—61no—
309C22×C3—121no—
310C22×C3—121no—
311C22×C3C2×#62611yescharacteristic
312C3×C4—21no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—31no—
316C3×C4—31no—
317C3×C4—61no—
318C3×C4—61no—
319C3×C4—62no—
320C3×C4—61no—
321C3×C4—61no—
322C3×C4—361no—
323C3×C4—361no—
324C3×C4—361no—
325C3×C4—361no—
326C3×C4#380511yescharacteristic
327C3×C4#380511yescharacteristic
328Dic3—121no—
329Dic3—121no—
330Dic3—121no—
331Dic3—121no—
332Dic3—122no—
333Dic3—122no—
334Dic3—121no—
335Dic3—121no—
336Dic3—361no—
337Dic3—361no—
338C32—21no—
339C32#472311yescharacteristic, complemented
340C32#504511yescharacteristic, complemented
341C23#620011yescharacteristic
342Q8—181no—
343Q8—181no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—181no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4—181no—
350C2×C4#620011yescharacteristic
351C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
352C8—181no—
353C8—181no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—41no—
361C2×C3—61no—
362C2×C3—61no—
363C2×C3—61no—
364C2×C3—121no—
365C2×C3—121no—
366C2×C3—122no—
367C2×C3#6382911yescharacteristic
368C2×C3#6383311yescharacteristic
369C2×C3#6396211yescharacteristic
370C22—21no—
371C22—21no—
372C22—21no—
373C22C2×#620011yescharacteristic
374C4—21no—
375C4—361no—
376C4—361no—
377C4—361no—
378C4—361no—
379C4#8867211yescharacteristic
380C4#8867211yescharacteristic
381C3—31no—
382C3—31no—
383C3—61no—
384C3#1059112711yescharacteristic
385C2—41no—
386C2#1172106911yescharacteristic
387C2#1172107311yescharacteristic
388C2#1172130711yescharacteristic
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. #11720309C2semidirect
5. #11720309C2semidirect
6. #11720307C2semidirect
7. #11720306C2semidirect
226. H(3,𝔽3)#485semidirect
339. C32#4723semidirect
340. C32#5045semidirect

Maximal subgroups

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

subgroupindex
#117218262
#64133⋊C32
#63657⋊C32
#117203092
#117203072
#117203062
#105911343
#105911023