Finite Group Databasebeta

← #13155518

Subgroups — #13155518

Order 1728.

734 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C4×#6289C211yesmaximal, characteristic
2#11722070C211yesmaximal, characteristic, complemented
3#11722024C212yesmaximal
4C3×Dic3⋊(C3⋊C8)C212yesmaximal, complemented
5C3×#63553C211yesmaximal, characteristic, complemented
6#10590643—32nomaximal
7#10590256—31nomaximal
8C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
9C2×#6289C411yescharacteristic
10C2×#6289C411yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4×Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15Dic3×C3⋊Dic3C411yescharacteristic
16Dic3×C3⋊Dic3C411yescharacteristic
17C42×C3⋊D3—31no—
18C4×#624—32no—
19C4×#624—61no—
20#63647—32no—
21C3×Dic3⋊C8—32no—
22C3×C8×Dic3—32no—
23C3×C4⋊(C3⋊C8)—32no—
24Dic3⋊(C3⋊C8)D312yescomplemented
25#63557—32no—
26#63557—32no—
27#63557—62no—
28#63553—31no—
29#63553D311yescharacteristic, complemented
30Dic3×C3⋊C8—32no—
31C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
34C4×C3⋊(C3⋊D3)—22no—
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C2×C32×Dic3C2×C411yescharacteristic
38C2×C32×Dic3C2×C411yescharacteristic
39#6289—22no—
40#6289C814yescomplemented
41C3×C3⋊(C3⋊C8)—22no—
42C32×C3⋊C8—22no—
43#4833—92no—
44#4685—91no—
45C2×C4×C3⋊D3—31no—
46C2×C4×C3⋊D3—31no—
47C2×C4×C3⋊D3—32no—
48C2×C4×C3⋊D3—31no—
49C2×C4×C3⋊D3—61no—
50C2×C4×C3⋊D3—62no—
51C2×C4×C3⋊D3—121no—
52C2×#624—32no—
53C2×#624—32no—
54C2×#624—61no—
55C2×#624—61no—
56C2×C32×C8—32no—
57C32×C42—31no—
58C4×C3⋊Dic3—31no—
59C4×C3⋊Dic3—31no—
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D312yes—
62C3×C4×Dic3—21no—
63C3×C4×Dic3—32no—
64C3×C4×Dic3—61no—
65C3×C4×Dic3C2×D312yes—
66C2×C3×C3⋊C8—22no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—62no—
70C2×C3×C3⋊C8C2×D312yes—
71C2×C3×C3⋊C8C2×D312yes—
72Dic32—32no—
73Dic32—32no—
74Dic32—61no—
75Dic32—61no—
76C22×C33C22×C411yescharacteristic
77C2×C3⋊(C3⋊D3)—21no—
78C2×C3⋊(C3⋊D3)—21no—
79C2×C3⋊(C3⋊D3)C2×C812yes—
80C33×C4#4112yes—
81C3⋊(C3⋊Dic3)—21no—
82C3×C3⋊Dic3C2×C812yes—
83C3×C3⋊Dic3M1612yes—
84C32×Dic3—21no—
85C32×Dic3C2×C812yes—
86C42×D3—92no—
87C42×D3—91no—
88C42×D3—181no—
89C3×C4⋊C8—92no—
90C3×C4×C8—91no—
91(C2×D3)⋊C8—92no—
92(C2×D3)⋊C8—92no—
93(C2×D3)⋊C8—92no—
94(C2×D3)⋊C8—182no—
95Dic3⋊C8—32no—
96C8×Dic3—32no—
97C8×Dic3—92no—
98C4⋊(C3⋊C8)—32no—
99C4×C3⋊C8—91no—
100C22×C3⋊D3—32no—
101C22×C3⋊D3—31no—
102C22×C3⋊D3—61no—
103C22×C3⋊D3—62no—
104C22×C3⋊D3—121no—
105C2×C32×C4—22no—
106C2×C32×C4—21no—
107C2×C32×C4—31no—
108C2×C32×C4—31no—
109C2×C32×C4—41no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C32×C4C22×D312yes—
112C2×C3⋊Dic3—32no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—62no—
116C2×C3⋊Dic3—121no—
117C2×C3⋊Dic3C4×D311yescharacteristic
118C2×C3⋊Dic3C4×D311yescharacteristic
119C4×C3⋊D3—34no—
120C4×C3⋊D3—62no—
121C4×C3⋊D3—62no—
122C4×C3⋊D3—64no—
123C4×C3⋊D3—62no—
124C4×C3⋊D3—62no—
125C4×C3⋊D3—122no—
126C4×C3⋊D3—122no—
127C4×C3⋊D3—124no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3C4×D312yes—
135C2×C3×Dic3C4×D312yes—
136#624—38no—
137#624—64no—
138#624—64no—
139#624—64no—
140C32×C8—62no—
141C3⋊(C3⋊C8)—22no—
142C3×C3⋊C8—22no—
143C3×C3⋊C8—22no—
144C3×C3⋊C8—42no—
145C3×C3⋊C8—62no—
146C3×C3⋊C8—62no—
147C3×C3⋊C8—122no—
148#431—271noSylow
149C2×C33#11711yescharacteristic
150C2×C33C22×C811yescharacteristic
151C2×C33C2×M1611yescharacteristic
152C3⋊(C3⋊D3)—22no—
153C2×C4×D3—92no—
154C2×C4×D3—91no—
155C2×C4×D3—92no—
156C2×C4×D3—91no—
157C2×C4×D3—92no—
158C2×C4×D3—91no—
159C2×C4×D3—181no—
160C2×C4×D3—181no—
161C2×C4×D3—181no—
162C2×C4×D3—182no—
163C2×C4×D3—361no—
164C2×C3×C8—32no—
165C2×C3×C8—32no—
166C2×C3×C8—62no—
167C2×C3×C8—92no—
168C3×C42—32no—
169C3×C42—61no—
170C3×C42—91no—
171C4×Dic3—32no—
172C4×Dic3—61no—
173C4×Dic3—92no—
174C4×Dic3—92no—
175C4×Dic3—91no—
176C4×Dic3—91no—
177C4×Dic3—181no—
178C4×Dic3—181no—
179C4×Dic3D3211yescharacteristic
180C2×C3⋊C8—32no—
181C2×C3⋊C8—32no—
182C2×C3⋊C8—62no—
183C2×C3⋊C8D3212yes—
184C22×C32—22no—
185C22×C32—21no—
186C22×C32—41no—
187C22×C32C2×C4×D311yescharacteristic
188C22×C32C2×C4×D312yes—
189C2×C3⋊D3—34no—
190C2×C3⋊D3—32no—
191C2×C3⋊D3—62no—
192C2×C3⋊D3—64no—
193C2×C3⋊D3—61no—
194C2×C3⋊D3—61no—
195C2×C3⋊D3—62no—
196C2×C3⋊D3—62no—
197C2×C3⋊D3—62no—
198C2×C3⋊D3—62no—
199C2×C3⋊D3—122no—
200C2×C3⋊D3—122no—
201C2×C3⋊D3—122no—
202C2×C3⋊D3—122no—
203C2×C3⋊D3—122no—
204C32×C4—22no—
205C32×C4—22no—
206C32×C4—22no—
207C32×C4—32no—
208C32×C4—42no—
209C32×C4—61no—
210C32×C4#23512yes—
211C32×C4#23712yes—
212C32×C4#23912yes—
213C3⋊Dic3—61no—
214C3⋊Dic3—62no—
215C3⋊Dic3—62no—
216C3⋊Dic3—122no—
217C3⋊Dic3—122no—
218C3⋊Dic3D3×C812yes—
219C3⋊Dic3ΓL(1,𝔽25)12yes—
220C3×Dic3—22no—
221C3×Dic3—22no—
222C3×Dic3—22no—
223C3×Dic3—34no—
224C3×Dic3—34no—
225C3×Dic3—62no—
226C3×Dic3—62no—
227C3×Dic3D3×C814yes—
228C2×C42—271no—
229C4⋊C8—92no—
230C22⋊C8—272no—
231C4×C8—91no—
232C4×C8—271no—
233C33#43111yescharacteristic, complemented, Sylow
234C22×D3—92no—
235C22×D3—91no—
236C22×D3—181no—
237C22×D3—182no—
238C22×D3—361no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—41no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—91no—
247C2×C3×C4—91no—
248C2×C3×C4C2×D3211yescharacteristic
249C2×C3×C4C2×D3212yes—
250C2×Dic3—32no—
251C2×Dic3—32no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—92no—
255C2×Dic3—91no—
256C2×Dic3—181no—
257C2×Dic3—182no—
258C2×Dic3—361no—
259C2×Dic3#62411yescharacteristic
260C2×Dic3#62411yescharacteristic
261C4×D3—98no—
262C4×D3—94no—
263C4×D3—184no—
264C4×D3—182no—
265C4×D3—184no—
266C4×D3—184no—
267C4×D3—184no—
268C4×D3—182no—
269C4×D3—182no—
270C4×D3—182no—
271C4×D3—362no—
272C4×D3—362no—
273C4×D3—364no—
274C3×C8—62no—
275C3×C8—62no—
276C3×C8—122no—
277C3×C8—182no—
278C3⋊C8—22no—
279C3⋊C8—62no—
280C3⋊C8—62no—
281C3⋊C8—122no—
282C2×C32—22no—
283C2×C32—22no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—22no—
287C2×C32—21no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32#87211yescharacteristic
292C2×C32#89212yes—
293C2×C32C2×D3×C811yescharacteristic
294C2×C32C2×D3×C812yes—
295C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
296C2×C32D3×M1612yes—
297C3⋊D3—62no—
298C3⋊D3—64no—
299C3⋊D3—64no—
300C3⋊D3—124no—
301C3⋊D3—124no—
302C22×C4—271no—
303C22×C4—271no—
304C22×C4—271no—
305C2×C8—32no—
306C2×C8—92no—
307C42—31no—
308C42—91no—
309C42—271no—
310C42—271no—
311C22×C3—21no—
312C22×C3—22no—
313C22×C3—41no—
314C22×C3C4×D3212yes—
315C22×C3C2×#62411yescharacteristic
316C2×D3—94no—
317C2×D3—92no—
318C2×D3—182no—
319C2×D3—184no—
320C2×D3—182no—
321C2×D3—182no—
322C2×D3—182no—
323C2×D3—181no—
324C2×D3—182no—
325C2×D3—181no—
326C2×D3—362no—
327C2×D3—362no—
328C2×D3—362no—
329C2×D3—362no—
330C2×D3—362no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—34no—
335C3×C4—42no—
336C3×C4—62no—
337C3×C4—62no—
338C3×C4—62no—
339C3×C4—92no—
340C3×C4—92no—
341C3×C4#379612yes—
342C3×C4#379912yes—
343C3×C4#380412yes—
344Dic3—21no—
345Dic3—34no—
346Dic3—34no—
347Dic3—62no—
348Dic3—62no—
349Dic3—182no—
350Dic3—182no—
351Dic3—181no—
352Dic3—362no—
353Dic3—362no—
354Dic3#371012yes—
355C32—21no—
356C32—22no—
357C32—41no—
358C32#468511yescharacteristic, complemented
359C32#483312yescomplemented
360C23—271no—
361C2×C4—31no—
362C2×C4—31no—
363C2×C4—91no—
364C2×C4—91no—
365C2×C4—274no—
366C2×C4—271no—
367C2×C4—542no—
368C2×C4—542no—
369C2×C4D3311yescharacteristic
370C8—62no—
371C8—182no—
372C2×C3—21no—
373C2×C3—22no—
374C2×C3—21no—
375C2×C3—22no—
376C2×C3—21no—
377C2×C3—22no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3D32×C812yes—
382C2×C3D3×ΓL(1,𝔽25)12yes—
383C2×C3C2×#371011yescharacteristic
384C2×C3#6381811yescharacteristic
385C2×C3#6383612yes—
386C2×C3#6384911yescharacteristic
387D3—184no—
388D3—182no—
389D3—184no—
390D3—364no—
391D3—364no—
392C22—272no—
393C22—541no—
394C22—541no—
395C22D3×#62411yescharacteristic
396C4—32no—
397C4—61no—
398C4—92no—
399C4—92no—
400C4—541no—
401C4#8896212yes—
402C3—21no—
403C3—22no—
404C3—41no—
405C3#1059025611yescharacteristic, complemented
406C3#1059064312yescomplemented
407C2—542no—
408C2D3×#371011yescharacteristic
409C2#1172281811yescharacteristic
410C2#1172292011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutatorC2×C33C22×C8
Fitting, cosocleC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC22×C3⋊(C3⋊D3)C2×C4
hypocenterC33#431
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11722070C2semidirect
4. C3×Dic3⋊(C3⋊C8)C2semidirect
5. C3×#63553C2semidirect
24. Dic3⋊(C3⋊C8)D3semidirect
29. #63553D3semidirect
40. #6289C8semidirect
233. C33#431semidirect
358. C32#4685semidirect
359. C32#4833semidirect
405. C3#10590256semidirect
406. C3#10590643semidirect

Maximal subgroups

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

subgroupindex
C4×#62892
#117220702
#117220242
C3×Dic3⋊(C3⋊C8)2
C3×#635532
#105906433
#105902563