Finite Group Databasebeta

← #13140565

Subgroups — #13140565

Order 1728.

697 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 697 conjugacy classes of subgroups — too many to draw. Only its 101 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×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721053C212yesmaximal, complemented
3#11720579C211yesmaximal, characteristic, complemented
4#63660⋊C3C211yesmaximal, characteristic, complemented
5#11720320C212yesmaximal, complemented
6#10591503—31nomaximal
7D3×#833—31nomaximal
8C23×(C3⋊D3)⋊C3C412yes—
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#638⋊C3C412yes—
11D4×(C3⋊D3)⋊C3—22no—
12D4×(C3⋊D3)⋊C3—22no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15#88538C2212yescomplemented
16#3748⋊C3C2212yescomplemented
17#88430—22no—
18((C3⋊D3)⋊C3)⋊C8—22no—
19C2×D4×C3⋊D3D311yescharacteristic
20C2×C3×D3×D4—31no—
21#63818—32no—
22D3×#208—32no—
23#63660D311yescharacteristic, complemented
24C3×#833—31no—
25C3×#833—31no—
26C3×#833—31no—
27C3×#833—61no—
28#63559—32no—
29#63558—32no—
30C23×H(3,𝔽3)C2×C412yes—
31C22×(C3⋊D3)⋊C3—22no—
32C22×(C3⋊D3)⋊C3—22no—
33C22×(C3⋊D3)⋊C3—44no—
34C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
35C22×(C3⋊D3)⋊C3C2×C412yes—
36D4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38#638⋊C3—24no—
39C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
40C4⋊((C3⋊D3)⋊C3)—22no—
41C4×(C3⋊D3)⋊C3D412yes—
42C4×(C3⋊D3)⋊C3D412yes—
43#6180—22no—
44(C3⋊(C3⋊C8))⋊C3—22no—
45C2×#833—91no—
46D3×#100—91no—
47D3×#100—91no—
48C23×C3⋊D3Dic312yes—
49C23×C3×D3—32no—
50C2×C32×D4—21no—
51C2×C32×D4C2×D311yescharacteristic
52C2×C32×D4C2×D311yescharacteristic
53C2×#638Dic312yes—
54D4×C3⋊D3—22no—
55D4×C3⋊D3—22no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×C3×#67—32no—
59C3×D3×D4—62no—
60C3×D3×D4—62no—
61C2×C3×D12—31no—
62C2×C3×C4×D3—31no—
63#3748C2×D312yescomplemented
64C3×#208—32no—
65C3×#208—32no—
66C3×#208—32no—
67C3×#208—62no—
68#3712—62no—
69#3711—62no—
70#3710—62no—
71D3×C3⋊C8—62no—
72C22×H(3,𝔽3)—22no—
73C22×H(3,𝔽3)—42no—
74C22×H(3,𝔽3)C22×C411yescharacteristic
75C2×(C3⋊D3)⋊C3—22no—
76C2×(C3⋊D3)⋊C3—22no—
77C2×(C3⋊D3)⋊C3—44no—
78C2×(C3⋊D3)⋊C3—44no—
79C2×(C3⋊D3)⋊C3—42no—
80C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
81C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
82C4×H(3,𝔽3)C2×D412yes—
83(C3⋊Dic3)⋊C3C22⋊C412yes—
84C22×C3×D4—91no—
85C2×D3×D4—31no—
86C2×D3×D4—91no—
87C2×#208—92no—
88D3×M16—92no—
89D3×M16—92no—
90C3×#100—91no—
91C3×#100—91no—
92#833—31no—
93#833—31no—
94#833—31no—
95#833—61no—
96#833—91no—
97#833—92no—
98#823—92no—
99#823—92no—
100C23×C32—22no—
101C23×C32C4×D312yes—
102C23×C32C2×Dic312yes—
103C22×C3⋊D3—22no—
104C22×C3⋊D3—22no—
105C22×C3⋊D3—44no—
106C22×C3⋊D3C2×Dic311yescharacteristic
107C22×C3⋊D3C2×Dic312yes—
108C22×C3×D3—31no—
109C22×C3×D3—32no—
110C22×C3×D3—62no—
111C22×C3×D3—62no—
112C22×C3×D3—124no—
113C32×D4—22no—
114C32×D4—22no—
115C32×D4—42no—
116C2×C32×C4—21no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119#638—24no—
120C2×C3⋊Dic3C2×Dic311yescharacteristic
121C4⋊(C3⋊D3)—22no—
122C4×C3⋊D3#6712yes—
123C4×C3⋊D3#6712yes—
124C3×#67—64no—
125C2×C3×Dic3—31no—
126C3×D12—62no—
127C3×C4×D3—32no—
128C3×C4×D3—32no—
129C3⋊(C3⋊C8)—22no—
130C3×C3⋊C8—62no—
131C3×C3⋊C8—62no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8—122no—
134C2×#100—271noSylow
135C2×H(3,𝔽3)—21no—
136C2×H(3,𝔽3)—42no—
137C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
138(C3⋊D3)⋊C3—21no—
139(C3⋊D3)⋊C3—42no—
140(C3⋊D3)⋊C3#10012yescomplemented
141C24×C3—92no—
142C23×D3—32no—
143C23×D3—92no—
144C2×C3×D4—31no—
145C2×C3×D4—31no—
146C2×C3×D4—61no—
147C2×C3×D4—91no—
148C2×C3×D4—92no—
149C2×C3×D4—182no—
150C2×C3×D4—182no—
151C2×C3×D4D3211yescharacteristic
152C22×C3×C4—91no—
153C2×#67—32no—
154C2×#67—92no—
155D3×D4—62no—
156D3×D4—62no—
157D3×D4—182no—
158D3×D4—182no—
159C2×D12—31no—
160C2×D12—91no—
161C2×C4×D3—31no—
162C2×C4×D3—91no—
163C3×M16—92no—
164C3×M16—92no—
165#208—32no—
166#208—32no—
167#208—32no—
168#208—62no—
169#208—92no—
170#208—182no—
171C2×C3⋊C8—182no—
172ΓL(1,𝔽25)—182no—
173ΓL(1,𝔽25)—182no—
174D3×C8—182no—
175D3×C8—182no—
176C22×C32—22no—
177C22×C32—22no—
178C22×C32—21no—
179C22×C32—42no—
180C22×C32—42no—
181C22×C32—42no—
182C22×C32—82no—
183C22×C32C2×C4×D311yescharacteristic
184C22×C32C22×Dic311yescharacteristic
185C2×C3⋊D3—22no—
186C2×C3⋊D3—22no—
187C2×C3⋊D3—44no—
188C2×C3⋊D3—44no—
189C2×C3⋊D3—42no—
190C2×C3⋊D3#21711yescharacteristic
191C2×C3⋊D3#21711yescharacteristic
192C2×C3×D3—31no—
193C2×C3×D3—31no—
194C2×C3×D3—62no—
195C2×C3×D3—62no—
196C2×C3×D3—124no—
197C2×C3×D3—124no—
198C2×C3×D3—122no—
199C32×C4—22no—
200C32×C4D3×D412yes—
201C32×C4C2×#6712yes—
202C3⋊Dic3#21712yes—
203C3×Dic3—32no—
204C22×D4—91no—
205C2×M16—272no—
206#100—91no—
207#100—91no—
208#100—272no—
209H(3,𝔽3)C2×#10011yescharacteristic, complemented, Sylow
210C23×C3—32no—
211C23×C3—32no—
212C23×C3—62no—
213C23×C3—91no—
214C23×C3—92no—
215C23×C3—182no—
216C23×C3—182no—
217C23×C3—364no—
218C23×C3D3×Dic312yes—
219C22×D3—31no—
220C22×D3—32no—
221C22×D3—62no—
222C22×D3—62no—
223C22×D3—91no—
224C22×D3—92no—
225C22×D3—124no—
226C22×D3—182no—
227C22×D3—182no—
228C22×D3—364no—
229C3×D4—22no—
230C3×D4—62no—
231C3×D4—62no—
232C3×D4—122no—
233C3×D4—182no—
234C3×D4—184no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—61no—
238C2×C3×C4—92no—
239C2×C3×C4—92no—
240C2×C3×C4—91no—
241C2×C3×C4C2×D3211yescharacteristic
242#67—64no—
243#67—184no—
244C2×Dic3—31no—
245C2×Dic3—91no—
246D12—62no—
247D12—182no—
248C4×D3—32no—
249C4×D3—32no—
250C4×D3—92no—
251C4×D3—92no—
252C3×C8—182no—
253C3×C8—182no—
254C3⋊C8—62no—
255C3⋊C8—62no—
256C3⋊C8—62no—
257C3⋊C8—122no—
258C3⋊C8—182no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—41no—
263C2×C32—42no—
264C2×C32—42no—
265C2×C32—82no—
266C2×C32D3×C22⋊C411yescharacteristic
267C2×C32C2×#21711yescharacteristic
268C3⋊D3—21no—
269C3⋊D3—42no—
270C3⋊D3#83312yescomplemented
271C3×D3—32no—
272C3×D3—61no—
273C3×D3—122no—
274C24—92no—
275C2×D4—92no—
276C2×D4—91no—
277C2×D4—182no—
278C2×D4—182no—
279C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
280C22×C4—91no—
281M16—92no—
282M16—92no—
283M16—542no—
284C2×C8—542no—
285C22×C3—22no—
286C22×C3—31no—
287C22×C3—31no—
288C22×C3—42no—
289C22×C3—62no—
290C22×C3—62no—
291C22×C3—61no—
292C22×C3—91no—
293C22×C3—91no—
294C22×C3—122no—
295C22×C3—122no—
296C22×C3—122no—
297C22×C3—182no—
298C22×C3—182no—
299C22×C3—242no—
300C22×C3—362no—
301C22×C3—364no—
302C22×C3—364no—
303C22×C3C2×D3×Dic311yescharacteristic
304C2×D3—31no—
305C2×D3—31no—
306C2×D3—62no—
307C2×D3—62no—
308C2×D3—91no—
309C2×D3—91no—
310C2×D3—124no—
311C2×D3—124no—
312C2×D3—122no—
313C2×D3—182no—
314C2×D3—182no—
315C2×D3—364no—
316C2×D3—364no—
317C2×D3—362no—
318C3×C4—32no—
319C3×C4—32no—
320C3×C4—62no—
321C3×C4—92no—
322C3×C4D3×#6712yes—
323Dic3—32no—
324Dic3—92no—
325C32—21no—
326C32D3×#10011yescharacteristic, complemented
327C32C2×#83311yescharacteristic, complemented
328C23—91no—
329C23—92no—
330C23—182no—
331C23—182no—
332C23—364no—
333C23((C3⋊D3)⋊C3)⋊C412yes—
334D4—22no—
335D4—184no—
336D4—182no—
337C2×C4—91no—
338C2×C4—92no—
339C2×C4—92no—
340C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
341C8—182no—
342C8—182no—
343C2×C3—21no—
344C2×C3—31no—
345C2×C3—31no—
346C2×C3—42no—
347C2×C3—61no—
348C2×C3—61no—
349C2×C3—61no—
350C2×C3—92no—
351C2×C3—121no—
352C2×C3—122no—
353C2×C3—122no—
354C2×C3—181no—
355C2×C3—242no—
356C2×C3—362no—
357C2×C3D3×#21711yescharacteristic
358D3—32no—
359D3—61no—
360D3—92no—
361D3—122no—
362D3—181no—
363D3—362no—
364C22—22no—
365C22—42no—
366C22—91no—
367C22—91no—
368C22—182no—
369C22—182no—
370C22—362no—
371C22—364no—
372C22—364no—
373C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
374C4—92no—
375C4#8867612yes—
376C3—31no—
377C3—31no—
378C3—61no—
379C3D3×#83311yescharacteristic
380C2—21no—
381C2—42no—
382C2—92no—
383C2—181no—
384C2—362no—
385C2#1172131011yescharacteristic
subgroupG/H
centerC2#11721310
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC2×C3D3×#217
prime radicalC2×D4×(C3⋊D3)⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#100
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721053C2semidirect
3. #11720579C2semidirect
4. #63660⋊C3C2semidirect
5. #11720320C2semidirect
15. #88538C22semidirect
16. #3748⋊C3C22semidirect
23. #63660D3semidirect
63. #3748C2×D3semidirect
140. (C3⋊D3)⋊C3#100semidirect
209. H(3,𝔽3)C2×#100semidirect
270. C3⋊D3#833semidirect
326. C32D3×#100semidirect
327. C32C2×#833semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
#117210532
#117205792
#63660⋊C32
#117203202
#105915033
D3×#8333