Finite Group Databasebeta

← #13165495

Subgroups — #13165495

Order 1728.

564 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 62 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×#88935C211yesmaximal, characteristic
2#11722938C211yesmaximal, characteristic, complemented
3#11722925C211yesmaximal, characteristic, complemented
4#11722793C211yesmaximal, characteristic, complemented
5#11722779C211yesmaximal, characteristic, complemented
6C3×#63739C211yesmaximal, characteristic, complemented
7Dic3×#622C211yesmaximal, characteristic, complemented
8#10593420—31nomaximal
9C2×#6295C2211yescharacteristic
10C2×Dic3×C3⋊D3—21no—
11C2×D3×C3⋊Dic3—21no—
12C2×C3×#625C2211yescharacteristic
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15#88935C414yescomplemented
16#88815C2211yescharacteristic
17Dic3×C3⋊Dic3C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×#3722—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C2×#3774—31no—
22#67⋊Dic3—61no—
23#63902—31no—
24#63902—31no—
25#63881—61no—
26C4×#622—31no—
27#63743—31no—
28#63739—31no—
29#63739D311yescharacteristic, complemented
30C22×C3×C3⋊D3—21no—
31C22×C32×D3—21no—
32C2×C3⋊(C3⋊Dic3)D411yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C2×C32×Dic3—21no—
36C2×C32×Dic3D411yescharacteristic
37#6295—21no—
38#6295C2×C412yescomplemented
39Dic3×C3⋊D3—42no—
40D3×C3⋊Dic3—42no—
41C3×#625—21no—
42C3×#625C2×C412yescomplemented
43C3⋊#622C2×C412yescomplemented
44C3×#622—21no—
45#4756—91nomaximal
46C22×C3⋊Dic3—61no—
47C2×C4×C3⋊D3—61no—
48C2×C3×#67—61no—
49C2×#626—61no—
50C2×#625—31no—
51C2×#625C2×D311yescharacteristic
52C2×D3×Dic3—61no—
53C2×D3×Dic3—61no—
54C2×D3×Dic3—61no—
55C2×D3×Dic3—61no—
56C2×D3×Dic3—121no—
57C2×D3×Dic3—121no—
58C2×#622—31no—
59C2×#622C2×D311yescharacteristic
60#3774—34no—
61C4×C3⋊Dic3—31no—
62C3×#217—61no—
63C3×(C2×D3)⋊C4—61no—
64C3×(C2×D3)⋊C4—61no—
65C3×(C2×D3)⋊C4—61no—
66C3×Dic3⋊C4—61no—
67#3724—31no—
68#3724C2×D311yescharacteristic
69Dic3⋊Dic3—61no—
70#3722—21no—
71(C2×D3)⋊Dic3—21no—
72Dic32—61no—
73Dic32—61no—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3—41no—
77C2×C3×C3⋊D3—41no—
78C2×C32×D3—22no—
79C2×C32×D3—41no—
80C2×C32×D3—41no—
81C3⋊(C3⋊Dic3)D812yescomplemented
82C3×C3⋊Dic3—41no—
83C3×C3⋊Dic3C22×C411yescharacteristic
84C3×C3⋊Dic3#4111yescharacteristic
85C32×Dic3—21no—
86C32×Dic3—41no—
87D4×Dic3—91no—
88C2×#213—91no—
89#893—91no—
90#67⋊C4—181no—
91#67⋊C4—181no—
92C3×#102—91no—
93#821—91no—
94C8×Dic3—91no—
95#807—91no—
96C23×C32—61no—
97C22×C3⋊D3—21no—
98C22×C3×D3—21no—
99C22×C3×D3—21no—
100C22×C3×D3—41no—
101C22×C3×D3—61no—
102C22×C3×D3—61no—
103C22×C3×D3—121no—
104C2×C32×C4—31no—
105C2×C32×C4—61no—
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—121no—
111C2×C3⋊Dic3—121no—
112C2×C3⋊Dic3—122no—
113C2×C3⋊Dic3C22×D311yescharacteristic
114C4×C3⋊D3—122no—
115C3×#67—62no—
116C3×#67—62no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—121no—
127#626—62no—
128#626—62no—
129#625—21no—
130#625—32no—
131#625—61no—
132#625C4×D312yescomplemented
133D3×Dic3—122no—
134D3×Dic3—122no—
135D3×Dic3—124no—
136D3×Dic3—122no—
137D3×Dic3—122no—
138D3×Dic3—124no—
139#622—21no—
140#622—32no—
141C4×D8—271noSylow
142C2×C33C4×D411yescharacteristic
143C2×C33C2×D811yescharacteristic
144C2×C33#13511yescharacteristic
145C3×C3⋊D3—42no—
146C32×D3—42no—
147C2×C3×D4—91no—
148C2×#67—61no—
149C2×#67—181no—
150C22×Dic3—181no—
151C22×Dic3—181no—
152C22×Dic3—181no—
153C22×Dic3—361no—
154C2×D12—91no—
155C2×C4×D3—181no—
156C2×C4×D3—181no—
157C2×C4×D3—181no—
158C2×C4×D3—361no—
159C2×C3×C8—91no—
160C3×C4⋊C4—91no—
161C3×C22⋊C4—181no—
162C3×C22⋊C4—181no—
163#217—61no—
164#213—94no—
165(C2×D3)⋊C4—61no—
166(C2×D3)⋊C4—61no—
167(C2×D3)⋊C4—61no—
168C4⋊Dic3—91no—
169Dic3⋊C4—61no—
170Dic3⋊C4—181no—
171C4×Dic3—91no—
172C4×Dic3—181no—
173C4×Dic3—181no—
174C2×C3⋊C8—91no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—62no—
180C22×C32—121no—
181C22×C32—121no—
182C22×C32D3×D411yescharacteristic
183C2×C3⋊D3—22no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C2×C3×D3—22no—
187C2×C3×D3—22no—
188C2×C3×D3—42no—
189C2×C3×D3—41no—
190C2×C3×D3—41no—
191C2×C3×D3—42no—
192C2×C3×D3—42no—
193C2×C3×D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—62no—
196C2×C3×D3—62no—
197C2×C3×D3—122no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C2×C3×D3—122no—
201C2×C3×D3—122no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C32×C4—61no—
205C32×C4—121no—
206C3⋊Dic3—32no—
207C3⋊Dic3—41no—
208C3⋊Dic3—62no—
209C3⋊Dic3—62no—
210C3⋊Dic3—122no—
211C3⋊Dic3—122no—
212C3⋊Dic3C2×C4×D311yescharacteristic
213C3⋊Dic3#23711yescharacteristic
214C3×Dic3—41no—
215C3×Dic3—41no—
216C3×Dic3—41no—
217C3×Dic3—42no—
218C3×Dic3—41no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C3×Dic3—61no—
222C3×Dic3—61no—
223C3×Dic3—121no—
224C3×Dic3—122no—
225C3×Dic3—121no—
226C2×D8—271no—
227C4×D4—271no—
228C4×D4—271no—
229#107—271no—
230#102—91no—
231#102—271no—
232C4×C8—271no—
233C33C4×D811yescharacteristic, complemented, Sylow
234C23×C3—61no—
235C23×C3—61no—
236C23×C3—121no—
237C23×C3—181no—
238C22×D3—21no—
239C22×D3—61no—
240C22×D3—61no—
241C22×D3—121no—
242C3×D4—92no—
243C3×D4—181no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—91no—
249C2×C3×C4—121no—
250C2×C3×C4—181no—
251#67—62no—
252#67—62no—
253#67—182no—
254#67—182no—
255C2×Dic3—21no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—91no—
261C2×Dic3—121no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264C2×Dic3—361no—
265C2×Dic3—361no—
266C2×Dic3—362no—
267C2×Dic3—362no—
268C2×Dic3—362no—
269C2×Dic3—364no—
270C2×Dic3GO+(4,𝔽2)11yescharacteristic
271D12—92no—
272D12—181no—
273C4×D3—362no—
274C4×D3—362no—
275C4×D3—362no—
276C4×D3—364no—
277C3×C8—181no—
278C3⋊C8—92no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—122no—
292C2×C32#67⋊C411yescharacteristic
293C2×C32D3×D811yescharacteristic
294C2×C32#91611yescharacteristic
295C3⋊D3—42no—
296C3×D3—42no—
297C3×D3—44no—
298C3×D3—42no—
299C3×D3—122no—
300C3×D3—124no—
301C3×D3—122no—
302C2×D4—91no—
303C2×D4—271no—
304C22×C4—541no—
305C22×C4—541no—
306D8—274no—
307C2×C8—91no—
308C2×C8—271no—
309C4⋊C4—91no—
310C4⋊C4—271no—
311C22⋊C4—181no—
312C22⋊C4—181no—
313C42—271no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—62no—
319C22×C3—62no—
320C22×C3—122no—
321C22×C3—121no—
322C22×C3—121no—
323C22×C3—122no—
324C22×C3—122no—
325C22×C3—121no—
326C22×C3—121no—
327C22×C3—182no—
328C22×C3—361no—
329C22×C3—361no—
330C22×C3C2×GO+(4,𝔽2)11yescharacteristic
331C2×D3—22no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—62no—
335C2×D3—62no—
336C2×D3—122no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—122no—
340C2×D3—122no—
341C2×D3—121no—
342C2×D3—121no—
343C3×C4—91no—
344C3×C4—91no—
345C3×C4—121no—
346C3×C4—121no—
347C3×C4—121no—
348C3×C4—122no—
349C3×C4—121no—
350C3×C4—361no—
351Dic3—21no—
352Dic3—41no—
353Dic3—61no—
354Dic3—61no—
355Dic3—61no—
356Dic3—61no—
357Dic3—92no—
358Dic3—121no—
359Dic3—122no—
360Dic3—121no—
361Dic3—182no—
362Dic3—182no—
363Dic3—362no—
364Dic3—362no—
365C32—21no—
366C32—21no—
367C32—41no—
368C32—41no—
369C32#475611yescharacteristic, complemented
370C23—61no—
371C23—181no—
372D4—92no—
373D4—181no—
374D4—272no—
375D4—541no—
376C2×C4—31no—
377C2×C4—61no—
378C2×C4—91no—
379C2×C4—181no—
380C2×C4—271no—
381C2×C4—1082no—
382C2×C4—1082no—
383C8—181no—
384C8—272no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—41no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—122no—
398C2×C3—124no—
399C2×C3—122no—
400C2×C3—362no—
401C2×C3#6422511yescharacteristic
402C2×C3C4×GO+(4,𝔽2)11yescharacteristic
403C2×C3#6423711yescharacteristic
404D3—42no—
405D3—122no—
406D3—124no—
407D3—122no—
408C22—62no—
409C22—121no—
410C22—121no—
411C22—182no—
412C22—361no—
413C22—361no—
414C22D3×GO+(4,𝔽2)11yescharacteristic
415C4—61no—
416C4—91no—
417C4—91no—
418C4—121no—
419C4—272no—
420C4—361no—
421C3—21no—
422C3—21no—
423C3—41no—
424C3—41no—
425C3#1059342011yescharacteristic, complemented
426C2—122no—
427C2—362no—
428C2(D32⋊D3)⋊C411yescharacteristic
429C2#1172412711yescharacteristic
430C2#1172413311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊Dic3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC2×#88935C2
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C4×D8
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #11722938C2semidirect
3. #11722925C2semidirect
4. #11722793C2semidirect
5. #11722779C2semidirect
6. C3×#63739C2semidirect
7. Dic3×#622C2semidirect
15. #88935C4semidirect
29. #63739D3semidirect
38. #6295C2×C4semidirect
42. C3×#625C2×C4semidirect
43. C3⋊#622C2×C4semidirect
81. C3⋊(C3⋊Dic3)D8semidirect
132. #625C4×D3semidirect
233. C33C4×D8semidirect
369. C32#4756semidirect
425. C3#10593420semidirect

Maximal subgroups

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

subgroupindex
C2×#889352
#117229382
#117229252
#117227932
#117227792
C3×#637392
Dic3×#6222
#105934203
#47569