Finite Group Databasebeta

← #13166467

Subgroups — #13166467

Order 1728.

720 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 72 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×#89014C211yesmaximal, characteristic
2C2×#88925C211yesmaximal, characteristic, complemented
3C2×D3×#622C211yesmaximal, characteristic, complemented
4#11723103C211yesmaximal, characteristic, complemented
5C3×#63778C211yesmaximal, characteristic, complemented
6(C2×D3)⋊#622C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#622C211yesmaximal, characteristic, complemented
8#10593822—31nomaximal
9C2×#6308C411yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×D12C411yescharacteristic
12C2×#6290C411yescharacteristic
13C2×#6290C411yescharacteristic
14C2×D3×C3⋊Dic3C2211yescharacteristic
15C2×D3×C3⋊Dic3C2211yescharacteristic
16#89014—22no—
17#89014—22no—
18C2×C3⋊#622C2211yescharacteristic
19C2×C3⋊#622C2211yescharacteristic
20C2×C3×#622C2211yescharacteristic
21C2×C3×#622C2211yescharacteristic
22#88925—22no—
23D3×#622—22no—
24C2×#3830—31no—
25C2×#3795—62no—
26C2×#3792—31no—
27C22×#622—31no—
28C22⋊#622—31no—
29C22⋊#622—31no—
30#63778—31no—
31#63778D311yescharacteristic, complemented
32C22×C3×C3⋊D3C2×C411yescharacteristic
33C22×C32×D3C2×C411yescharacteristic
34C22×C32×D3C2×C411yescharacteristic
35C2×C33×C4C2×C411yescharacteristic
36C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
37C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
38#6308—22no—
39C2×C3×C3⋊Dic3C2311yescharacteristic
40C3×C4×C3⋊D3—21no—
41C3×C4×C3⋊D3—21no—
42C32×D12—22no—
43#6290—22no—
44#6290—22no—
45D3×C3⋊Dic3—21no—
46D3×C3⋊Dic3—21no—
47D3×C3⋊Dic3D412yes—
48D3×C3⋊Dic3D412yes—
49C3⋊#622—21no—
50C3⋊#622—21no—
51C3×#622—21no—
52C3×#622—21no—
53#4727—91nomaximal
54C2×C32×D4—31no—
55C2×#638—31no—
56C2×#638—31no—
57C22×C3⋊Dic3—31no—
58C22×C3⋊Dic3—31no—
59#3830—62no—
60#3830—62no—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×#634—31no—
63C2×#634—62no—
64C2×#634—122no—
65C2×C3×D12—22no—
66C2×C3×C4×D3—62no—
67C2×#625—62no—
68C2×#625—62no—
69C2×D3×Dic3—62no—
70C2×D3×Dic3—62no—
71#3795—68no—
72#3795—68no—
73#3792—62no—
74C2×#622—31no—
75C2×#622—31no—
76C2×#622—62no—
77C2×#622C2×D311yescharacteristic
78C2×#622C2×D311yescharacteristic
79C22×C33C22×C411yescharacteristic
80C2×C3×C3⋊D3—21no—
81C2×C3×C3⋊D3—21no—
82C2×C3×C3⋊D3—21no—
83C2×C32×D3—21no—
84C2×C32×D3—21no—
85C2×C32×D3—21no—
86C2×C32×D3—21no—
87C2×C32×D3—21no—
88C2×C32×D3C22⋊C412yes—
89C33×C4—21no—
90C3⋊(C3⋊Dic3)—21no—
91C3⋊(C3⋊Dic3)C22⋊C412yes—
92C3×C3⋊Dic3C2×D411yescharacteristic
93C3×C3⋊Dic3C2×D411yescharacteristic
94C2×#237—182no—
95C2×#235—91no—
96C2×ΓL(1,𝔽25)—91no—
97C2×D3×C8—91no—
98C3×C22⋊C8—91no—
99#830—91no—
100(C2×D3)⋊C8—91no—
101(C2×D3)⋊C8—91no—
102C23×C32—31no—
103C23×C32—31no—
104C22×C3⋊D3C4×D311yescharacteristic
105C22×C3×D3—22no—
106C22×C3×D3—22no—
107C22×C3×D3—62no—
108C32×D4—62no—
109C2×C32×C4—22no—
110C2×C32×C4—42no—
111C2×C32×C4C4×D311yescharacteristic
112#638—62no—
113#638—62no—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—32no—
116C2×C3⋊Dic3—32no—
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—62no—
119C2×C3⋊Dic3—62no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—122no—
123C2×C3⋊Dic3—122no—
124C2×C3⋊Dic3C22×D311yescharacteristic
125C4×C3⋊D3—21no—
126C4×C3⋊D3—21no—
127#634—62no—
128#634—68no—
129#634—128no—
130C2×C3×Dic3—62no—
131C3×D12—28no—
132C3×C4×D3—64no—
133C3×C4×D3—64no—
134#625—68no—
135#625—68no—
136D3×Dic3—64no—
137D3×Dic3—64no—
138D3×Dic3—64no—
139D3×Dic3—64no—
140#622—21no—
141#622—21no—
142#622—61no—
143#622—61no—
144#408—271noSylow
145C2×C33C2×C22⋊C411yescharacteristic
146C2×C33#13111yescharacteristic
147C2×C33#13111yescharacteristic
148C3×C3⋊D3—41no—
149C32×D3—22no—
150C32×D3—41no—
151C2×C3×D4—62no—
152C22×C3×C4—91no—
153C2×#67—91no—
154C2×#67—91no—
155C2×#67—182no—
156C2×#67—182no—
157C22×Dic3—182no—
158C22×Dic3—182no—
159#237—188no—
160#237—188no—
161#235—182no—
162#235—182no—
163C2×D12#15711yescharacteristic
164C2×C4×D3—62no—
165C2×C4×D3—91no—
166C2×C4×D3—91no—
167C2×Dic6—91no—
168C2×Dic6—182no—
169C2×Dic6—362no—
170C2×C3×C8—91no—
171C2×C3×C8—91no—
172C2×C3⋊C8—91no—
173C2×C3⋊C8—91no—
174ΓL(1,𝔽25)—182no—
175D3×C8—182no—
176C22×C32—22no—
177C22×C32—32no—
178C22×C32—42no—
179C22×C32—61no—
180C22×C32—61no—
181C22×C32—61no—
182C22×C32—61no—
183C22×C32—61no—
184C22×C32C2×C4×D311yescharacteristic
185C2×C3⋊D3—21no—
186C2×C3⋊D3—21no—
187C2×C3⋊D3—21no—
188C2×C3×D3—24no—
189C2×C3×D3—24no—
190C2×C3×D3—42no—
191C2×C3×D3—42no—
192C2×C3×D3—42no—
193C2×C3×D3—42no—
194C2×C3×D3—64no—
195C2×C3×D3—122no—
196C2×C3×D3—122no—
197C32×C4—21no—
198C32×C4—24no—
199C32×C4—44no—
200C3⋊Dic3—32no—
201C3⋊Dic3—64no—
202C3⋊Dic3—61no—
203C3⋊Dic3—64no—
204C3⋊Dic3—124no—
205C3⋊Dic3—124no—
206C3⋊Dic3D3×D411yescharacteristic
207C3⋊Dic3D3×D411yescharacteristic
208C3×Dic3—62no—
209C3×Dic3—62no—
210C2×#41—271no—
211C2×M16—271no—
212C22×C8—271no—
213C22⋊C8—91no—
214C22⋊C8—271no—
215C22⋊C8—271no—
216C22⋊C8—271no—
217C33#40811yescharacteristic, complemented, Sylow
218C23×C3—62no—
219C23×C3—62no—
220C23×C3—91no—
221C22×D3—62no—
222C22×D3C2×#15711yescharacteristic
223C22×D3C2×#15711yescharacteristic
224C3×D4—68no—
225C2×C3×C4—22no—
226C2×C3×C4—42no—
227C2×C3×C4—91no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4C2×#15711yescharacteristic
231#67—182no—
232#67—188no—
233#67—188no—
234#67—182no—
235C2×Dic3—62no—
236C2×Dic3—91no—
237C2×Dic3—91no—
238C2×Dic3—184no—
239C2×Dic3—184no—
240C2×Dic3—182no—
241C2×Dic3—182no—
242C2×Dic3—184no—
243C2×Dic3—184no—
244C2×Dic3—362no—
245C2×Dic3—362no—
246D12—22no—
247C4×D3—64no—
248C4×D3—64no—
249C4×D3—92no—
250C4×D3—92no—
251C4×D3—181no—
252C4×D3—181no—
253Dic6—182no—
254Dic6—188no—
255Dic6—368no—
256C3×C8—181no—
257C3×C8—181no—
258C3⋊C8—181no—
259C3⋊C8—181no—
260C2×C32—22no—
261C2×C32—22no—
262C2×C32—22no—
263C2×C32—42no—
264C2×C32—42no—
265C2×C32—42no—
266C2×C32—62no—
267C2×C32—121no—
268C2×C32D3×C22⋊C411yescharacteristic
269C2×C32#90111yescharacteristic
270C2×C32#90711yescharacteristic
271C3⋊D3—41no—
272C3×D3—44no—
273C3×D3—44no—
274C3×D3—124no—
275#41—542no—
276#41—542no—
277C2×Q8—271no—
278C2×D4—31no—
279C2×D4—271no—
280C2×D4—271no—
281C22×C4—91no—
282C22×C4—271no—
283C22×C4—271no—
284M16—542no—
285C2×C8—91no—
286C2×C8—91no—
287C2×C8—271no—
288C2×C8—271no—
289C2×C8—542no—
290C22×C3—22no—
291C22×C3—42no—
292C22×C3—64no—
293C22×C3—64no—
294C22×C3—122no—
295C22×C3—122no—
296C22×C3—122no—
297C22×C3—122no—
298C22×C3—181no—
299C22×C3—181no—
300C22×C3—181no—
301C22×C3C22×#15711yescharacteristic
302C2×D3—21no—
303C2×D3—21no—
304C2×D3—21no—
305C2×D3—21no—
306C2×D3—21no—
307C2×D3—64no—
308C2×D3—122no—
309C2×D3—122no—
310C2×D3C22⋊#15712yes—
311C3×C4—24no—
312C3×C4—21no—
313C3×C4—44no—
314C3×C4—91no—
315C3×C4—91no—
316Dic3—62no—
317Dic3—62no—
318Dic3—92no—
319Dic3—184no—
320Dic3—184no—
321Dic3—181no—
322Dic3—364no—
323Dic3—364no—
324C32—22no—
325C32—42no—
326C32#472711yescharacteristic, complemented
327C23—31no—
328C23—31no—
329C23—91no—
330Q8—542no—
331D4—62no—
332D4—542no—
333D4—542no—
334C2×C4—91no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—272no—
338C2×C4—272no—
339C2×C4—271no—
340C2×C4—271no—
341C2×C4—541no—
342C2×C4—541no—
343C2×C4D3×#15711yescharacteristic
344C8—181no—
345C8—181no—
346C8—541no—
347C8—541no—
348C2×C3—22no—
349C2×C3—22no—
350C2×C3—22no—
351C2×C3—42no—
352C2×C3—42no—
353C2×C3—42no—
354C2×C3—124no—
355C2×C3—124no—
356C2×C3—361no—
357C2×C3#6428711yescharacteristic
358C2×C3#6428711yescharacteristic
359C2×C3C2×C22⋊#15711yescharacteristic
360D3—22no—
361D3—41no—
362D3—124no—
363C22—32no—
364C22—61no—
365C22—61no—
366C22—61no—
367C22—61no—
368C22—61no—
369C22—181no—
370C22—181no—
371C22—181no—
372C22C2×D3×#15711yescharacteristic
373C4—21no—
374C4—91no—
375C4—91no—
376C4—272no—
377C4—541no—
378C3—22no—
379C3—42no—
380C3#1059382211yescharacteristic, complemented
381C2—62no—
382C2—121no—
383C2—361no—
384C2#1172425011yescharacteristic
385C2#1172425111yescharacteristic
386C2D3×C22⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC2×#89014C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#408
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88925C2semidirect
3. C2×D3×#622C2semidirect
4. #11723103C2semidirect
5. C3×#63778C2semidirect
6. (C2×D3)⋊#622C2semidirect
7. (C2×D3)⋊#622C2semidirect
31. #63778D3semidirect
217. C33#408semidirect
326. C32#4727semidirect
380. C3#10593822semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
C2×#889252
C2×D3×#6222
#117231032
C3×#637782
(C2×D3)⋊#6222
#105938223
#47279