Finite Group Databasebeta

← #13167569

Subgroups — #13167569

Order 1728.

688 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 79 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×#89046C211yesmaximal, characteristic
2C2×C3×#3792C211yesmaximal, characteristic, complemented
3C2×#88815C211yesmaximal, characteristic, complemented
4#11722794C212yesmaximal, complemented
5#11722793C212yesmaximal, complemented
6#10593496—31nomaximal
7C2×C3×#638—21no—
8C22×C3×C3⋊Dic3—21no—
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×#6296C2211yescharacteristic
11C2×#6295C2211yescharacteristic
12#89046—22no—
13#89046C414yescomplemented
14C2×#6294—21no—
15C3×#3792—22no—
16C2×C3×#622C2212yes—
17#88815—21no—
18#88815C2212yes—
19C2×#3805—31no—
20C2×#3804—61no—
21C2×#3792D311yescharacteristic, complemented
22C2×#3724—31no—
23C2×Dic3⋊Dic3—61no—
24#63740—32no—
25#63739—32no—
26C23×C33D411yescharacteristic
27C22×C3×C3⋊D3—21no—
28C3×#638—24no—
29C2×C3×C3⋊Dic3—21no—
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3—42no—
32C2×C3×C3⋊Dic3C2×C412yes—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3D412yes—
35C2×C3×C3⋊Dic3C2311yescharacteristic
36#6296—21no—
37#6296C2×C412yescomplemented
38#6295—21no—
39#6295C2×C412yescomplemented
40#6294—24no—
41C3×#622—22no—
42#5133—91nomaximal
43C2×#638—21no—
44C22×C3⋊Dic3—21no—
45C22×C3⋊Dic3C2×D311yescharacteristic
46C2×C3×#67—61no—
47C2×C3×#67—61no—
48C2×C3×#67—121no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3—61no—
52C22×C3×Dic3—61no—
53C22×C3×Dic3—121no—
54C2×#627—31no—
55C2×#627—61no—
56C2×#626—61no—
57C2×#625—31no—
58C2×#624—61no—
59#3805—34no—
60#3805—62no—
61#3804—64no—
62#3804—64no—
63C2×D3×Dic3—61no—
64C2×D3×Dic3—61no—
65#3792—22no—
66C2×#622C2×D312yes—
67#3724—32no—
68#3724—61no—
69Dic3⋊Dic3—62no—
70Dic3⋊Dic3—62no—
71C22×C33—21no—
72C22×C33—21no—
73C22×C33C22⋊C412yes—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3—41no—
77C2×C3×C3⋊D3—41no—
78C3×C3⋊Dic3—22no—
79C3×C3⋊Dic3—42no—
80C3×C3⋊Dic3C22⋊C412yes—
81C3×C3⋊Dic3C22×C411yescharacteristic
82C3×C3⋊Dic3C2×D411yescharacteristic
83C2×#237—181no—
84C2×#235—91no—
85C2×C3×M16—91no—
86C2×C4⋊Dic3—91no—
87C2×Dic3⋊C4—181no—
88#821—92no—
89#816—92no—
90C23×C32—21no—
91C23×C32—21no—
92C23×C32—41no—
93C23×C32—41no—
94C23×C32D1211yescharacteristic
95C22×C3⋊D3—21no—
96C22×C3×D3—61no—
97C22×C3×D3—61no—
98C22×C3×D3—121no—
99#638—24no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—42no—
103C2×C3⋊Dic3C4×D312yes—
104C2×C3⋊Dic3D1211yescharacteristic
105C2×C3⋊Dic3#6712yes—
106C2×C3⋊Dic3C22×D311yescharacteristic
107C3×#67—64no—
108C3×#67—64no—
109C3×#67—124no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—122no—
123C2×C3×Dic3—121no—
124C2×C3×Dic3—122no—
125C2×C3×Dic3—122no—
126C2×C3×Dic3—124no—
127#627—32no—
128#627—62no—
129#627—62no—
130#627—61no—
131#626—62no—
132#626—62no—
133#625—32no—
134#625—61no—
135#624—64no—
136D3×Dic3—64no—
137D3×Dic3—64no—
138#622—22no—
139#417—271noSylow
140C2×C33—22no—
141C2×C33C2×C22⋊C411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33#13711yescharacteristic
144C3×C3⋊D3—42no—
145C2×C3×D4—181no—
146C22×C3×C4—91no—
147C22×C3×C4—181no—
148C2×#67—61no—
149C2×#67—61no—
150C2×#67—121no—
151C2×#67—181no—
152C22×Dic3—61no—
153C22×Dic3—61no—
154C22×Dic3—61no—
155C22×Dic3—61no—
156C22×Dic3—121no—
157C22×Dic3—181no—
158#237—184no—
159#237—184no—
160#235—94no—
161#235—182no—
162C2×D12—91no—
163C2×C4×D3—181no—
164C2×C4×D3—181no—
165C2×Dic6—91no—
166C2×Dic6—181no—
167C3×M16—182no—
168C2×C3×C8—92no—
169C4⋊Dic3—92no—
170C4⋊Dic3—181no—
171Dic3⋊C4—182no—
172Dic3⋊C4—182no—
173C22×C32—22no—
174C22×C32—22no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32—22no—
180C22×C32—22no—
181C22×C32—42no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32—42no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—42no—
188C22×C32—42no—
189C22×C32—81no—
190C22×C32—81no—
191C22×C32(C2×D3)⋊C412yes—
192C22×C32C2×D1211yescharacteristic
193C2×C3⋊D3—22no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C2×C3×D3—62no—
197C2×C3×D3—62no—
198C2×C3×D3—122no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C2×C3×D3—121no—
202C2×C3×D3—121no—
203C2×C3×D3—241no—
204C2×C3×D3—241no—
205C3⋊Dic3—22no—
206C3⋊Dic3—42no—
207C3⋊Dic3(C2×D3)⋊C412yes—
208C3⋊Dic3C2×C4×D311yescharacteristic
209C3⋊Dic3C2×#6711yescharacteristic
210C3×Dic3—62no—
211C3×Dic3—62no—
212C3×Dic3—61no—
213C3×Dic3—61no—
214C3×Dic3—61no—
215C3×Dic3—61no—
216C3×Dic3—62no—
217C3×Dic3—62no—
218C3×Dic3—122no—
219C3×Dic3—122no—
220C3×Dic3—124no—
221C3×Dic3—122no—
222C2×#41—271no—
223C2×M16—91no—
224C2×C4⋊C4—271no—
225#103—272no—
226#102—272no—
227C33#41711yescharacteristic, complemented, Sylow
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—181no—
233C23×C3GO+(4,𝔽2)11yescharacteristic
234C22×D3—61no—
235C22×D3—61no—
236C22×D3—121no—
237C3×D4—184no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4—92no—
241C2×C3×C4—92no—
242C2×C3×C4—181no—
243C2×C3×C4—182no—
244C2×C3×C4—362no—
245#67—64no—
246#67—64no—
247#67—124no—
248#67—182no—
249#67—182no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—62no—
261C2×Dic3—62no—
262C2×Dic3—122no—
263C2×Dic3—121no—
264C2×Dic3—122no—
265C2×Dic3—122no—
266C2×Dic3—124no—
267C2×Dic3—184no—
268D12—92no—
269D12—181no—
270C4×D3—184no—
271C4×D3—184no—
272Dic6—92no—
273Dic6—182no—
274Dic6—181no—
275Dic6—182no—
276C3×C8—182no—
277C2×C32—24no—
278C2×C32—22no—
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—42no—
292C2×C32—44no—
293C2×C32—82no—
294C2×C32#90811yescharacteristic
295C2×C32#90911yescharacteristic
296C2×C32C2×(C2×D3)⋊C411yescharacteristic
297C3⋊D3—42no—
298C3×D3—122no—
299C3×D3—122no—
300C3×D3—242no—
301#41—274no—
302#41—542no—
303C2×Q8—271no—
304C2×D4—181no—
305C2×D4—271no—
306C22×C4—91no—
307C22×C4—181no—
308C22×C4—541no—
309M16—182no—
310C2×C8—92no—
311C4⋊C4—272no—
312C4⋊C4—541no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—182no—
332C22×C3—361no—
333C22×C3—361no—
334C22×C3D32⋊C412yes—
335C22×C3C2×GO+(4,𝔽2)11yescharacteristic
336C2×D3—62no—
337C2×D3—62no—
338C2×D3—122no—
339C2×D3—121no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—241no—
344C2×D3—241no—
345C3×C4—91no—
346C3×C4—91no—
347C3×C4—92no—
348C3×C4—182no—
349C3×C4—362no—
350Dic3—61no—
351Dic3—61no—
352Dic3—62no—
353Dic3—62no—
354Dic3—61no—
355Dic3—61no—
356Dic3—62no—
357Dic3—62no—
358Dic3—122no—
359Dic3—122no—
360Dic3—124no—
361Dic3—122no—
362C32—21no—
363C32—21no—
364C32—41no—
365C32—41no—
366C32#513311yescharacteristic, complemented
367C23—181no—
368C23#157⋊D311yescharacteristic
369Q8—272no—
370Q8—541no—
371D4—184no—
372D4—272no—
373D4—541no—
374C2×C4—91no—
375C2×C4—91no—
376C2×C4—92no—
377C2×C4—92no—
378C2×C4—181no—
379C2×C4—182no—
380C2×C4—362no—
381C2×C4—544no—
382C8—182no—
383C2×C3—24no—
384C2×C3—22no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—42no—
392C2×C3—44no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—82no—
400C2×C3—362no—
401C2×C3C2×D32⋊C411yescharacteristic
402C2×C3#6423611yescharacteristic
403C2×C3#6423911yescharacteristic
404D3—122no—
405D3—122no—
406D3—242no—
407C22—21no—
408C22—21no—
409C22—182no—
410C22—361no—
411C22—361no—
412C22#8893912yes—
413C22C2×#157⋊D311yescharacteristic
414C4—91no—
415C4—92no—
416C4—91no—
417C4—182no—
418C4—362no—
419C3—21no—
420C3—21no—
421C3—41no—
422C3—41no—
423C3#1059349611yescharacteristic, complemented
424C2—22no—
425C2—362no—
426C2C2×#8893911yescharacteristic
427C2#1172435011yescharacteristic
428C2#1172435211yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×#89046C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23#157⋊D3
hypocenterC33#417
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3792C2semidirect
3. C2×#88815C2semidirect
4. #11722794C2semidirect
5. #11722793C2semidirect
13. #89046C4semidirect
21. C2×#3792D3semidirect
37. #6296C2×C4semidirect
39. #6295C2×C4semidirect
227. C33#417semidirect
366. C32#5133semidirect
423. C3#10593496semidirect

Maximal subgroups

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

subgroupindex
C2×#890462
C2×C3×#37922
C2×#888152
#117227942
#117227932
#105934963
#51339