Finite Group Databasebeta

← #13165412

Subgroups — #13165412

Order 1728.

848 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 102 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×C3×GO+(4,𝔽2)C211yesmaximal, characteristic
2C2×#88940C211yesmaximal, characteristic, complemented
3C2×D32⋊Dic3C212yesmaximal, complemented
4C2×Dic3×#157C211yesmaximal, characteristic, complemented
5#11722902C212yesmaximal, complemented
6#10593454—31nomaximal
7C22×C3×D32C2212yes—
8C22×C3⋊#157C2211yescharacteristic
9C22×C3×#157C2211yescharacteristic
10C2×C3×GO+(4,𝔽2)—22no—
11C2×C3×GO+(4,𝔽2)—22no—
12C2×C3×GO+(4,𝔽2)C414yescomplemented
13C2×#6294C2212yes—
14C2×Dic3×C3⋊D3C2211yescharacteristic
15#88940—22no—
16D32⋊Dic3—24no—
17Dic3×#157—22no—
18(C2×D3)⋊(C3⋊Dic3)—22no—
19C3×(C2×D3)⋊Dic3—22no—
20C22×GO+(4,𝔽2)D311yescharacteristic
21C2×C4×#157—31no—
22C2×#3773—31no—
23C2×D32⋊C4—32no—
24D3×#217—62no—
25#63921—32no—
26C22×C3×C3⋊D3C2311yescharacteristic
27C22×C32×D3—22no—
28C2×C3×D32—22no—
29C2×C3×D32—22no—
30C2×C3×D32—22no—
31C2×C3×D32—22no—
32C2×C3×D32—22no—
33C2×C3×D32C2×C414yes—
34C2×C3⋊#157—21no—
35C2×C3⋊#157—21no—
36C2×C3⋊#157D412yescomplemented
37C2×C3×#157C2×C412yes—
38C2×C3×#157D412yes—
39C2×C3×#157D412yes—
40C3×GO+(4,𝔽2)—28no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic, complemented
42C2×C3×C3⋊Dic3—22no—
43C2×C32×Dic3D411yescharacteristic, complemented
44#6294—22no—
45#6294—22no—
46Dic3×C3⋊D3—21no—
47Dic3×C3⋊D3—21no—
48#5221—91nomaximal
49C23×C3×D3—62no—
50C22×D32C2×D312yes—
51C22×#157—31no—
52C22×#157C2×D311yescharacteristic
53C2×GO+(4,𝔽2)—22no—
54C2×GO+(4,𝔽2)—22no—
55C2×GO+(4,𝔽2)Dic314yescomplemented
56C2×C4×C3⋊D3—31no—
57C2×#624—32no—
58C2×D3×Dic3—62no—
59C2×D3×Dic3—62no—
60C4×#157—62no—
61#3773—62no—
62D32⋊C4—64no—
63#3757—62no—
64C3×#217—62no—
65C3×(C2×D3)⋊C4—62no—
66(C2×D3)⋊Dic3—22no—
67(C2×D3)⋊Dic3—62no—
68(C2×D3)⋊Dic3—62no—
69(C2×D3)⋊Dic3—122no—
70C22×C33C2×D411yescharacteristic
71C2×C3×C3⋊D3C22×C411yescharacteristic
72C2×C3×C3⋊D3C2×D411yescharacteristic
73C2×C3×C3⋊D3C2×D411yescharacteristic
74C2×C3×C3⋊D3C2×D411yescharacteristic
75C2×C3×C3⋊D3#4111yescharacteristic
76C2×C3×C3⋊D3#4111yescharacteristic
77C2×C32×D3—24no—
78C2×C32×D3—42no—
79C2×C32×D3—42no—
80C3×D32—24no—
81C3×D32—24no—
82C3×D32—42no—
83C3×D32—42no—
84C3⋊#157—22no—
85C3×#157C22⋊C414yescomplemented
86C3⋊(C3⋊Dic3)—21no—
87C3×C3⋊Dic3—42no—
88C32×Dic3—21no—
89C22×C3×D4—91no—
90C2×#217—92no—
91C2×#217—92no—
92C2×C4⋊Dic3—91no—
93C2×C4×Dic3—91no—
94D3×C22⋊C4—182no—
95C23×C32—62no—
96C22×C3⋊D3C22×D311yescharacteristic
97C22×C3×D3—22no—
98C22×C3×D3—22no—
99C22×C3×D3—42no—
100C22×C3×D3—64no—
101C22×C3×D3—62no—
102C22×C3×D3—64no—
103C22×C3×D3—122no—
104C22×C3×D3—122no—
105C22×C3×D3—122no—
106C22×C3×D3—122no—
107C2×D32—22no—
108C2×D32—22no—
109C2×D32—22no—
110C2×D32—22no—
111C2×D32—22no—
112C2×D32C2×Dic314yes—
113C2×#157—32no—
114C2×#157—61no—
115C2×#157—61no—
116C2×#157C2×Dic312yes—
117C2×#157#6712yes—
118C2×#157#6712yes—
119GO+(4,𝔽2)—28no—
120C2×C32×C4—31no—
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3—31no—
123C2×C3⋊Dic3—62no—
124C2×C3⋊Dic3—122no—
125C4×C3⋊D3—61no—
126C4×C3⋊D3—61no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—122no—
131#624—62no—
132#624—62no—
133D3×Dic3—122no—
134D3×Dic3—122no—
135D3×Dic3—122no—
136D3×Dic3—122no—
137#390—271noSylow
138C2×C33C4×D411yescharacteristic
139C2×C33#12111yescharacteristic
140C2×C33#12711yescharacteristic
141C3×C3⋊D3C2×C22⋊C411yescharacteristic
142C3×C3⋊D3C4×D411yescharacteristic
143C3×C3⋊D3#12111yescharacteristic
144C3×C3⋊D3#12411yescharacteristic
145C32×D3—44no—
146C24×C3—92no—
147C23×D3—62no—
148C2×C3×D4—94no—
149C2×C3×D4—182no—
150C2×C3×D4—182no—
151C22×C3×C4—91no—
152C22×Dic3—92no—
153C22×Dic3—91no—
154C22×Dic3—91no—
155C2×C4×D3—182no—
156C2×C4×D3—182no—
157C3×C22⋊C4—182no—
158#217—62no—
159#217—182no—
160#217—182no—
161#217—184no—
162#217—362no—
163(C2×D3)⋊C4—62no—
164(C2×D3)⋊C4—182no—
165C4⋊Dic3—182no—
166C4×Dic3—182no—
167C22×C32—22no—
168C22×C32—42no—
169C22×C32—64no—
170C22×C32—122no—
171C22×C32—122no—
172C22×C32D3×D411yescharacteristic
173C2×C3⋊D3D3×D411yescharacteristic
174C2×C3⋊D3#23711yescharacteristic
175C2×C3⋊D3#23711yescharacteristic
176C2×C3⋊D3C22×Dic311yescharacteristic
177C2×C3⋊D3C2×#6711yescharacteristic
178C2×C3⋊D3C2×#6711yescharacteristic
179C2×C3×D3—24no—
180C2×C3×D3—24no—
181C2×C3×D3—44no—
182C2×C3×D3—42no—
183C2×C3×D3—42no—
184C2×C3×D3—42no—
185C2×C3×D3—42no—
186C2×C3×D3—62no—
187C2×C3×D3—62no—
188C2×C3×D3—62no—
189C2×C3×D3—62no—
190C2×C3×D3—62no—
191C2×C3×D3—62no—
192C2×C3×D3—82no—
193C2×C3×D3—82no—
194C2×C3×D3—124no—
195C2×C3×D3—124no—
196C2×C3×D3—124no—
197C2×C3×D3—124no—
198D32—24no—
199D32—24no—
200D32—42no—
201D32—42no—
202#157—62no—
203#157#21714yescomplemented
204C32×C4—61no—
205C3⋊Dic3—42no—
206C3⋊Dic3—61no—
207C3⋊Dic3—122no—
208C3⋊Dic3—242no—
209C3×Dic3—42no—
210C3×Dic3—122no—
211C3×Dic3—122no—
212C3×Dic3—242no—
213C22×D4—91no—
214C2×C4⋊C4—271no—
215C2×C22⋊C4—272no—
216C2×C22⋊C4—272no—
217C2×C42—271no—
218C33#39011yescharacteristic, complemented, Sylow
219C23×C3—62no—
220C23×C3—62no—
221C23×C3—94no—
222C23×C3—91no—
223C23×C3—122no—
224C23×C3—182no—
225C23×C3—182no—
226C23×C3—182no—
227C23×C3—182no—
228C23×C3—182no—
229C22×D3—22no—
230C22×D3—64no—
231C22×D3—62no—
232C22×D3—64no—
233C22×D3—122no—
234C22×D3—122no—
235C22×D3—122no—
236C22×D3—122no—
237C3×D4—188no—
238C2×C3×C4—62no—
239C2×C3×C4—92no—
240C2×C3×C4—92no—
241C2×C3×C4—92no—
242C2×C3×C4—182no—
243C2×Dic3—62no—
244C2×Dic3—62no—
245C2×Dic3—92no—
246C2×Dic3—91no—
247C2×Dic3—122no—
248C2×Dic3—182no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251C2×Dic3—182no—
252C2×Dic3—182no—
253C2×Dic3—181no—
254C2×Dic3—181no—
255C2×Dic3—362no—
256C2×Dic3GO+(4,𝔽2)11yescharacteristic, complemented
257C4×D3—362no—
258C4×D3—362no—
259C4×D3—362no—
260C4×D3—362no—
261C2×C32—22no—
262C2×C32—22no—
263C2×C32—22no—
264C2×C32—42no—
265C2×C32—42no—
266C2×C32—42no—
267C2×C32—124no—
268C2×C32D4×Dic311yescharacteristic
269C2×C32#93811yescharacteristic
270C2×C32#94011yescharacteristic
271C3⋊D3D4×Dic311yescharacteristic
272C3⋊D3#93611yescharacteristic
273C3⋊D3#93811yescharacteristic
274C3⋊D3C2×#21711yescharacteristic
275C3×D3—44no—
276C3×D3—44no—
277C3×D3—62no—
278C3×D3—62no—
279C3×D3—62no—
280C3×D3—62no—
281C3×D3—84no—
282C24—92no—
283C2×D4—94no—
284C2×D4—182no—
285C2×D4—182no—
286C22×C4—91no—
287C22×C4—271no—
288C22×C4—272no—
289C22×C4—271no—
290C4⋊C4—542no—
291C22⋊C4—182no—
292C22⋊C4—542no—
293C22⋊C4—544no—
294C42—542no—
295C22×C3—22no—
296C22×C3—42no—
297C22×C3—64no—
298C22×C3—64no—
299C22×C3—91no—
300C22×C3—91no—
301C22×C3—91no—
302C22×C3—91no—
303C22×C3—91no—
304C22×C3—91no—
305C22×C3—124no—
306C22×C3—122no—
307C22×C3—122no—
308C22×C3—122no—
309C22×C3—122no—
310C22×C3—184no—
311C22×C3—184no—
312C22×C3—242no—
313C22×C3—242no—
314C22×C3—362no—
315C22×C3—362no—
316C22×C3C2×GO+(4,𝔽2)11yescharacteristic
317C2×D3—24no—
318C2×D3—42no—
319C2×D3—42no—
320C2×D3—62no—
321C2×D3—62no—
322C2×D3—62no—
323C2×D3—62no—
324C2×D3—62no—
325C2×D3—62no—
326C2×D3—124no—
327C2×D3—124no—
328C2×D3—124no—
329C2×D3—124no—
330C3×C4—94no—
331C3×C4—122no—
332C3×C4—362no—
333Dic3—21no—
334Dic3—122no—
335Dic3—122no—
336Dic3—182no—
337Dic3—181no—
338Dic3—242no—
339Dic3—362no—
340Dic3—722no—
341C32—22no—
342C32—42no—
343C32#522111yescharacteristic, complemented
344C23—62no—
345C23—94no—
346C23—91no—
347C23—182no—
348C23—182no—
349C23—182no—
350C23—182no—
351C23—182no—
352D4—188no—
353C2×C4—31no—
354C2×C4—92no—
355C2×C4—92no—
356C2×C4—92no—
357C2×C4—182no—
358C2×C4—272no—
359C2×C4—271no—
360C2×C4—541no—
361C2×C4—541no—
362C2×C4—542no—
363C2×C4—542no—
364C2×C4—541no—
365C2×C4—541no—
366C2×C3—22no—
367C2×C3—22no—
368C2×C3—22no—
369C2×C3—42no—
370C2×C3—42no—
371C2×C3—42no—
372C2×C3—91no—
373C2×C3—91no—
374C2×C3—91no—
375C2×C3—91no—
376C2×C3—124no—
377C2×C3—124no—
378C2×C3—244no—
379C2×C3C4×GO+(4,𝔽2)11yescharacteristic
380C2×C3#6423111yescharacteristic
381C2×C3#6424211yescharacteristic
382D3—44no—
383D3—62no—
384D3—62no—
385D3—62no—
386D3—62no—
387C22—64no—
388C22—91no—
389C22—91no—
390C22—91no—
391C22—91no—
392C22—91no—
393C22—91no—
394C22—122no—
395C22—122no—
396C22—184no—
397C22—184no—
398C22—362no—
399C22—362no—
400C22D3×GO+(4,𝔽2)11yescharacteristic
401C4—61no—
402C4—94no—
403C4—362no—
404C4—542no—
405C4—541no—
406C3—22no—
407C3—42no—
408C3#1059345411yescharacteristic, complemented
409C2—91no—
410C2—91no—
411C2—91no—
412C2—91no—
413C2—124no—
414C2Dic3×GO+(4,𝔽2)11yescharacteristic
415C2#1172411411yescharacteristic
416C2#1172412411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC22×C3×GO+(4,𝔽2)C2
cosocleC22×C3×C3⋊D3C23
hypocenterC33#390
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#88940C2semidirect
3. C2×D32⋊Dic3C2semidirect
4. C2×Dic3×#157C2semidirect
5. #11722902C2semidirect
12. C2×C3×GO+(4,𝔽2)C4semidirect
36. C2×C3⋊#157D4semidirect
41. C2×C3⋊(C3⋊Dic3)D4semidirect
43. C2×C32×Dic3D4semidirect
55. C2×GO+(4,𝔽2)Dic3semidirect
85. C3×#157C22⋊C4semidirect
203. #157#217semidirect
218. C33#390semidirect
256. C2×Dic3GO+(4,𝔽2)semidirect
343. C32#5221semidirect
408. C3#10593454semidirect

Maximal subgroups

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

subgroupindex
C22×C3×GO+(4,𝔽2)2
C2×#889402
C2×D32⋊Dic32
C2×Dic3×#1572
#117229022
#105934543
#52219