Finite Group Databasebeta

← #13140050

Subgroups — #13140050

Order 1728.

717 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 91 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×#88740C211yesmaximal, characteristic
2C2×#88435C211yesmaximal, characteristic, complemented
3C2×#88434C211yesmaximal, characteristic, complemented
4#63657⋊C3C212yesmaximal, complemented
5#11720323C212yesmaximal, complemented
6#10591145—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#6235C411yescharacteristic
9C2×#6235C411yescharacteristic
10#88740—22no—
11#88740—22no—
12C2×C4⋊#272C411yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×C4×#272C2211yescharacteristic
15C2×#6229C411yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18#88435—22no—
19#88434—22no—
20C2×C3×#235—32no—
21C2×C3×#235—61no—
22C2×#3711—32no—
23C2×D3×C3⋊C8—32no—
24#63657D312yescomplemented
25C3×#830—32no—
26(C2×D3)⋊(C3⋊C8)—32no—
27(C2×D3)⋊(C3⋊C8)—32no—
28C23×H(3,𝔽3)C2×C411yescharacteristic
29C22×#272C2×C411yescharacteristic
30C22×#272C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)C2×C411yescharacteristic
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35#6235—22no—
36#6235—22no—
37C2×#1084C2×C411yescharacteristic
38C2×#1084C2×C411yescharacteristic
39C4⋊#272—22no—
40C4×#272—21no—
41C4×#272—21no—
42C4×#272D412yes—
43C4×#272D412yes—
44#6229—22no—
45(C3⋊(C3⋊C8))⋊C3—22no—
46(C3⋊(C3⋊C8))⋊C3—22no—
47#5234—91no—
48#4727—92no—
49C22×C32×C4—21no—
50C22×C32×C4C2×D312yes—
51C2×C3×#67—32no—
52C2×C3×#67—32no—
53C2×C3×#67—61no—
54C2×C3×#67—61no—
55C3×#235—64no—
56C3×#235—64no—
57C3×#235—64no—
58C3×#235—64no—
59C2×C3×D12—32no—
60C2×C3×D12—61no—
61C2×C3×C4×D3—32no—
62C2×C3×C4×D3—32no—
63C2×C3×C4×D3—61no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—32no—
66C2×C3×Dic6—61no—
67C2×C3⋊(C3⋊C8)C2×D312yes—
68C2×C3⋊(C3⋊C8)C2×D312yes—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—32no—
71#3711—64no—
72D3×C3⋊C8—64no—
73C22×H(3,𝔽3)—21no—
74C22×H(3,𝔽3)—22no—
75C22×H(3,𝔽3)C22×C411yescharacteristic
76C2×#272—22no—
77C2×#272—21no—
78C2×#272—22no—
79C2×#272C22⋊C412yes—
80C4×H(3,𝔽3)—21no—
81C4×H(3,𝔽3)C2×D411yescharacteristic
82C4×H(3,𝔽3)C2×D411yescharacteristic
83#1084—21no—
84#1084C22⋊C412yes—
85C2×C3×#41—91no—
86C2×#235—32no—
87C2×#235—61no—
88C2×#208—91no—
89C22×C3⋊C8—91no—
90C2×ΓL(1,𝔽25)—92no—
91C2×D3×C8—92no—
92C3×C22⋊C8—92no—
93#830—32no—
94#830—92no—
95#830—92no—
96(C2×D3)⋊C8—92no—
97(C2×D3)⋊C8—92no—
98C23×C32—21no—
99C23×C32C4×D312yes—
100C22×C3×D3—32no—
101C22×C3×D3—32no—
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4C4×D312yes—
111C2×C32×C4C22×D312yes—
112C3×#67—64no—
113C3×#67—64no—
114C3×#67—64no—
115C3×#67—64no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C3×D12—64no—
121C3×D12—64no—
122C3×C4×D3—34no—
123C3×C4×D3—34no—
124C3×C4×D3—62no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128C3×C4×D3—62no—
129C3×C4×D3—62no—
130C3×Dic6—64no—
131C3×Dic6—64no—
132C3⋊(C3⋊C8)—22no—
133C3⋊(C3⋊C8)—22no—
134C3×C3⋊C8—62no—
135C3×C3⋊C8—62no—
136#408—271noSylow
137C2×H(3,𝔽3)—41no—
138C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
139C2×H(3,𝔽3)#13112yes—
140#272—22no—
141#272—41no—
142C3×#41—182no—
143C3×#41—182no—
144C2×C3×Q8—91no—
145C2×C3×D4—91no—
146C2×C3×D4—91no—
147C2×C3×D4—91no—
148C22×C3×C4—32no—
149C22×C3×C4—61no—
150C22×C3×C4—91no—
151C22×C3×C4—91no—
152C22×C3×C4D3211yescharacteristic
153C2×#67—32no—
154C2×#67—32no—
155C2×#67—61no—
156C2×#67—61no—
157#235—64no—
158#235—64no—
159#235—64no—
160#235—64no—
161C2×D12—32no—
162C2×D12—61no—
163C2×C4×D3—32no—
164C2×C4×D3—32no—
165C2×C4×D3—61no—
166C2×C4×D3—61no—
167C2×Dic6—32no—
168C2×Dic6—61no—
169C2×C3×C8—92no—
170C2×C3×C8—92no—
171#208—182no—
172C2×C3⋊C8—32no—
173C2×C3⋊C8—32no—
174C2×C3⋊C8—92no—
175C2×C3⋊C8—92no—
176C2×C3⋊C8—182no—
177ΓL(1,𝔽25)—184no—
178D3×C8—184no—
179C22×C32—22no—
180C22×C32—22no—
181C22×C32—22no—
182C22×C32—22no—
183C22×C32—21no—
184C22×C32—42no—
185C22×C32C2×C4×D312yes—
186C2×C3×D3—34no—
187C2×C3×D3—62no—
188C2×C3×D3—62no—
189C2×C3×D3—62no—
190C2×C3×D3—62no—
191C2×C3×D3—62no—
192C2×C3×D3—62no—
193C2×C3×D3—62no—
194C2×C3×D3—122no—
195C2×C3×D3—122no—
196C32×C4—22no—
197C32×C4—22no—
198C32×C4—21no—
199C32×C4—21no—
200C32×C4D3×D412yes—
201C32×C4D3×D412yes—
202C3×Dic3—34no—
203C3×Dic3—62no—
204C3×Dic3—62no—
205C3×Dic3—62no—
206C2×#41—91no—
207C2×M16—271no—
208C22×C8—271no—
209C22⋊C8—92no—
210C22⋊C8—272no—
211H(3,𝔽3)#40811yescharacteristic, complemented, Sylow
212C23×C3—32no—
213C23×C3—61no—
214C23×C3—91no—
215C23×C3—91no—
216C23×C3#62411yescharacteristic
217C22×D3—32no—
218C22×D3—32no—
219C22×D3—61no—
220C22×D3—61no—
221C3×Q8—182no—
222C3×D4—182no—
223C3×D4—182no—
224C3×D4—182no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—32no—
228C2×C3×C4—32no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—91no—
236C2×C3×C4—92no—
237C2×C3×C4—92no—
238C2×C3×C4—91no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4#62411yescharacteristic
242C2×C3×C4C2×D3211yescharacteristic
243#67—64no—
244#67—64no—
245#67—64no—
246#67—64no—
247C2×Dic3—32no—
248C2×Dic3—32no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251D12—64no—
252D12—64no—
253C4×D3—34no—
254C4×D3—34no—
255C4×D3—62no—
256C4×D3—62no—
257C4×D3—62no—
258C4×D3—62no—
259C4×D3—62no—
260C4×D3—62no—
261Dic6—64no—
262Dic6—64no—
263C3×C8—182no—
264C3×C8—182no—
265C3⋊C8—62no—
266C3⋊C8—62no—
267C3⋊C8—182no—
268C3⋊C8—182no—
269C2×C32—22no—
270C2×C32—21no—
271C2×C32—42no—
272C2×C32—42no—
273C2×C32D3×C22⋊C412yes—
274C2×C32#90112yes—
275C2×C32#90712yes—
276C3×D3—64no—
277C3×D3—122no—
278C3×D3—122no—
279C3×D3—122no—
280#41—182no—
281#41—182no—
282C2×Q8—91no—
283C2×D4—91no—
284C2×D4—91no—
285C2×D4—91no—
286C22×C4—91no—
287C22×C4—91no—
288C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
289M16—542no—
290C2×C8—92no—
291C2×C8—92no—
292C2×C8—542no—
293C22×C3—22no—
294C22×C3—21no—
295C22×C3—32no—
296C22×C3—62no—
297C22×C3—62no—
298C22×C3—62no—
299C22×C3—62no—
300C22×C3—61no—
301C22×C3—92no—
302C22×C3—122no—
303C22×C3—182no—
304C22×C3—182no—
305C22×C3—181no—
306C22×C3C2×#62411yescharacteristic
307C2×D3—34no—
308C2×D3—62no—
309C2×D3—62no—
310C2×D3—62no—
311C2×D3—62no—
312C2×D3—62no—
313C2×D3—62no—
314C2×D3—62no—
315C2×D3—122no—
316C2×D3—122no—
317C3×C4—21no—
318C3×C4—32no—
319C3×C4—32no—
320C3×C4—62no—
321C3×C4—62no—
322C3×C4—61no—
323C3×C4—61no—
324C3×C4—92no—
325C3×C4—181no—
326C3×C4#381111yescharacteristic
327C3×C4#381111yescharacteristic
328Dic3—34no—
329Dic3—62no—
330Dic3—62no—
331Dic3—62no—
332C32—21no—
333C32#472712yescomplemented
334C23—91no—
335C23—91no—
336C23#619911yescharacteristic
337Q8—182no—
338D4—182no—
339D4—182no—
340D4—182no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—91no—
344C2×C4—92no—
345C2×C4—92no—
346C2×C4—91no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4#619911yescharacteristic
350C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
351C8—182no—
352C8—182no—
353C2×C3—32no—
354C2×C3—32no—
355C2×C3—32no—
356C2×C3—41no—
357C2×C3—62no—
358C2×C3—61no—
359C2×C3—122no—
360C2×C3—122no—
361C2×C3—182no—
362C2×C3—361no—
363C2×C3#6381712yes—
364C2×C3#6397411yescharacteristic
365D3—64no—
366D3—122no—
367D3—122no—
368D3—122no—
369C22—22no—
370C22—21no—
371C22—92no—
372C22—182no—
373C22—182no—
374C22—181no—
375C22C2×#619911yescharacteristic
376C4—21no—
377C4—92no—
378C4—181no—
379C4#8867711yescharacteristic
380C4#8867711yescharacteristic
381C3—32no—
382C3—61no—
383C3#1059115311yescharacteristic
384C2—41no—
385C2—182no—
386C2—361no—
387C2#1172106412yes—
388C2#1172132011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC2×#88740C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#408
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88435C2semidirect
3. C2×#88434C2semidirect
4. #63657⋊C3C2semidirect
5. #11720323C2semidirect
24. #63657D3semidirect
211. H(3,𝔽3)#408semidirect
333. C32#4727semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
C2×#884352
C2×#884342
#63657⋊C32
#117203232
#105911453