Finite Group Databasebeta

← #13140186

Subgroups — #13140186

Order 1728.

647 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 93 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×#634⋊C3C211yesmaximal, characteristic, complemented
2#11721830C211yesmaximal, characteristic
3C2×#88437C212yesmaximal, complemented
4#63657⋊C3C211yesmaximal, characteristic, complemented
5#11720309C212yesmaximal, complemented
6#10591181—31nomaximal
7#10591171—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#6235—21no—
10C2×C4⋊#272C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3—21no—
12C2×#634⋊C3—21no—
13C2×#634⋊C3—24no—
14C2×#634⋊C3C2212yes—
15#88542—21no—
16C4⋊#1084C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18#88437—24no—
19C22×#634D311yescharacteristic, complemented
20C22×C3×Dic6—31no—
21C3×#930—31no—
22C3×#930—31no—
23C3×#930—61no—
24C2×#3716—32no—
25C2×#3715—32no—
26#63657D311yescharacteristic, complemented
27C3×#830—31no—
28#63567—32no—
29#63565—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×#272—21no—
32C2×C4×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35#6235—42no—
36C2×#1084—21no—
37C4⋊#272—22no—
38C2×(C3⋊Dic3)⋊C3—24no—
39C2×(C3⋊Dic3)⋊C3—22no—
40#634⋊C3—24no—
41#634⋊C3—22no—
42#634⋊C3D414yescomplemented
43(C3⋊(C3⋊C8))⋊C3—22no—
44#5227—91no—
45#5040—91no—
46#4738—91no—
47C22×C32×C4—21no—
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C22×C3⋊Dic3—21no—
51C2×#634—21no—
52C2×#634—24no—
53C2×#634C2×D312yes—
54C2×C3×#67—62no—
55C2×C3×#67—61no—
56C2×C3×#67—61no—
57C22×C3×Dic3—61no—
58C2×C3×D12—31no—
59C2×C3×D12—31no—
60C2×C3×D12—61no—
61C2×C3×Dic6—32no—
62C2×C3×Dic6—64no—
63C2×C3×Dic6—61no—
64C2×C3⋊(C3⋊C8)C2×D312yes—
65C3×(C2×D3)⋊C4—62no—
66C3×(C2×D3)⋊C4—61no—
67C3×(C2×D3)⋊C4—61no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—61no—
71C2×C3×C3⋊C8—32no—
72#3716—64no—
73#3715—64no—
74C22×H(3,𝔽3)—21no—
75C22×H(3,𝔽3)—21no—
76C22×H(3,𝔽3)SD1612yescomplemented
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×#272—41no—
79C2×#272—41no—
80C2×#272—41no—
81C4×H(3,𝔽3)—21no—
82C4×H(3,𝔽3)C2×D412yes—
83#1084—41no—
84(C3⋊Dic3)⋊C3—24no—
85C22×C3×Q8—91no—
86C22×Dic6—31no—
87C22×Dic6—91no—
88C3×#121—91no—
89C2×#215—92no—
90C2×#214—92no—
91#930—31no—
92#930—31no—
93#930—61no—
94C2×#204—92no—
95C3×C22⋊C8—91no—
96#835—92no—
97#830—31no—
98#830—91no—
99#816—92no—
100#810—92no—
101C23×C32—21no—
102C23×C32D1211yescharacteristic
103C23×C32#6711yescharacteristic
104C22×C3×D3—62no—
105C22×C3×D3—61no—
106C22×C3×D3—61no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—42no—
112C2×C32×C4D1211yescharacteristic
113C2×C32×C4#6711yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—24no—
117C2×C3⋊Dic3—22no—
118#634—24no—
119#634—22no—
120#634#6714yescomplemented
121C3×#67—124no—
122C3×#67—122no—
123C3×#67—122no—
124C2×C3×Dic3—64no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C3×D12—62no—
130C3×D12—62no—
131C3×D12—122no—
132C3×Dic6—34no—
133C3×Dic6—64no—
134C3×Dic6—62no—
135C3⋊(C3⋊C8)—22no—
136C3×C3⋊C8—62no—
137#448—271noSylow
138C2×H(3,𝔽3)—22no—
139C2×H(3,𝔽3)#12011yescharacteristic
140C2×H(3,𝔽3)C2×SD1611yescharacteristic
141C2×H(3,𝔽3)#13711yescharacteristic
142#272—81no—
143C2×C3×Q8—92no—
144C2×C3×Q8—181no—
145C2×C3×Q8—184no—
146C2×C3×D4—91no—
147C2×C3×D4—181no—
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—61no—
151C22×C3×C4—181no—
152C22×C3×C4D3211yescharacteristic
153C2×#67—61no—
154C2×#67—61no—
155C2×#67—62no—
156C22×Dic3—61no—
157C22×Dic3—181no—
158C2×D12—31no—
159C2×D12—31no—
160C2×D12—61no—
161C2×Dic6—32no—
162C2×Dic6—64no—
163C2×Dic6—61no—
164C2×Dic6—92no—
165C2×Dic6—184no—
166C2×Dic6—181no—
167C2×C3×C8—92no—
168C3×C4⋊C4—91no—
169C3×C22⋊C4—181no—
170#215—184no—
171#214—184no—
172(C2×D3)⋊C4—61no—
173(C2×D3)⋊C4—61no—
174(C2×D3)⋊C4—62no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—61no—
178C2×C3⋊C8—32no—
179C2×C3⋊C8—92no—
180#204—184no—
181C22×C32—21no—
182C22×C32—21no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—22no—
186C22×C32—21no—
187C22×C32—41no—
188C22×C32—41no—
189C22×C32#20412yescomplemented
190C22×C32#21512yescomplemented
191C22×C32C2×D1211yescharacteristic
192C22×C32C2×#6711yescharacteristic
193C2×C3×D3—122no—
194C2×C3×D3—122no—
195C2×C3×D3—122no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C2×C3×D3—121no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—22no—
205C32×C4—41no—
206C32×C4D3×D412yes—
207C32×C4C2×#6712yes—
208C3⋊Dic3—24no—
209C3×Dic3—64no—
210C3×Dic3—122no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C22×Q8—91no—
214C2×SD16—272no—
215#121—91no—
216#103—272no—
217C22⋊C8—91no—
218H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
219C23×C3—31no—
220C23×C3—31no—
221C23×C3—61no—
222C23×C3—181no—
223C23×C3#62611yescharacteristic
224C22×D3—61no—
225C22×D3—61no—
226C22×D3—62no—
227C3×Q8—94no—
228C3×Q8—182no—
229C3×Q8—184no—
230C3×D4—182no—
231C3×D4—362no—
232C2×C3×C4—22no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—122no—
242C2×C3×C4—182no—
243C2×C3×C4—184no—
244C2×C3×C4—181no—
245C2×C3×C4#62611yescharacteristic
246C2×C3×C4C2×D3211yescharacteristic
247#67—122no—
248#67—122no—
249#67—124no—
250C2×Dic3—64no—
251C2×Dic3—62no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—62no—
255C2×Dic3—182no—
256C2×Dic3—184no—
257D12—62no—
258D12—62no—
259D12—122no—
260Dic6—34no—
261Dic6—64no—
262Dic6—62no—
263Dic6—94no—
264Dic6—182no—
265Dic6—184no—
266C3×C8—182no—
267C3⋊C8—62no—
268C3⋊C8—182no—
269C2×C32—22no—
270C2×C32—22no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—42no—
275C2×C32#88211yescharacteristic
276C2×C32C2×#20411yescharacteristic
277C2×C32#90911yescharacteristic
278C2×C32C2×#21511yescharacteristic
279C2×C32#94211yescharacteristic
280C2×C32#95311yescharacteristic
281C3×D3—242no—
282C3×D3—241no—
283C3×D3—241no—
284C2×Q8—92no—
285C2×Q8—184no—
286C2×Q8—181no—
287C2×D4—91no—
288C2×D4—181no—
289C22×C4—181no—
290C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
291SD16—544no—
292C2×C8—92no—
293C4⋊C4—91no—
294C22⋊C4—181no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—32no—
300C22×C3—32no—
301C22×C3—61no—
302C22×C3—61no—
303C22×C3—61no—
304C22×C3—61no—
305C22×C3—61no—
306C22×C3—62no—
307C22×C3—121no—
308C22×C3—121no—
309C22×C3—361no—
310C22×C3—361no—
311C22×C3—361no—
312C22×C3#371712yes—
313C22×C3C2×#62611yescharacteristic
314C2×D3—121no—
315C2×D3—121no—
316C2×D3—121no—
317C2×D3—121no—
318C2×D3—121no—
319C2×D3—121no—
320C2×D3—122no—
321C2×D3—122no—
322C2×D3—122no—
323C3×C4—21no—
324C3×C4—32no—
325C3×C4—32no—
326C3×C4—61no—
327C3×C4—61no—
328C3×C4—62no—
329C3×C4—121no—
330C3×C4—184no—
331C3×C4—361no—
332C3×C4D3×#6712yes—
333Dic3—64no—
334Dic3—121no—
335Dic3—121no—
336Dic3—122no—
337Dic3—184no—
338C32—21no—
339C32#473811yescharacteristic, complemented
340C32#522711yescharacteristic, complemented
341C23—181no—
342C23#620011yescharacteristic
343Q8—94no—
344Q8—184no—
345Q8—182no—
346D4—182no—
347D4—362no—
348C2×C4—22no—
349C2×C4—184no—
350C2×C4—182no—
351C2×C4—181no—
352C2×C4#620011yescharacteristic
353C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
354C8—182no—
355C2×C3—22no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—62no—
363C2×C3—62no—
364C2×C3—61no—
365C2×C3—61no—
366C2×C3—61no—
367C2×C3—122no—
368C2×C3—721no—
369C2×C3C2×#371711yescharacteristic
370C2×C3#6383311yescharacteristic
371C2×C3#6397711yescharacteristic
372D3—241no—
373D3—241no—
374D3—242no—
375C22—21no—
376C22—21no—
377C22—361no—
378C22—361no—
379C22—361no—
380C22#8843712yescomplemented
381C22C2×#620011yescharacteristic
382C4—21no—
383C4—184no—
384C4—361no—
385C4#8867612yes—
386C3—31no—
387C3—31no—
388C3—61no—
389C3#1059117411yescharacteristic
390C2—22no—
391C2—721no—
392C2#1172106911yescharacteristic
393C2C2×#8843711yescharacteristic
394C2#1172132111yescharacteristic
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 radical#11721830C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#448
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
1. C22×#634⋊C3C2semidirect
3. C2×#88437C2semidirect
4. #63657⋊C3C2semidirect
5. #11720309C2semidirect
19. C22×#634D3semidirect
26. #63657D3semidirect
42. #634⋊C3D4semidirect
76. C22×H(3,𝔽3)SD16semidirect
120. #634#67semidirect
189. C22×C32#204semidirect
190. C22×C32#215semidirect
218. H(3,𝔽3)#448semidirect
339. C32#4738semidirect
340. C32#5227semidirect
380. C22#88437semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
#117218302
C2×#884372
#63657⋊C32
#117203092
#105911813
#105911713