Finite Group Databasebeta

← #13167491

Subgroups — #13167491

Order 1728.

814 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 59 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×C4⋊#1113C211yesmaximal, characteristic, complemented
2C2×#88941C212yesmaximal, complemented
3#11723695C211yesmaximal, characteristic, complemented
4C3×#63778C211yesmaximal, characteristic, complemented
5#11722793C212yesmaximal, complemented
6#10593425—31nomaximal
7C22×#1113—21no—
8C2×C3×C4⋊(C3⋊D3)—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×#6295—21no—
11C2×#6295C2212yes—
12C2×#6295C2211yescharacteristic, complemented
13C4⋊#1113—24no—
14C2×C3×#622C2212yescomplemented
15#88941—24no—
16C3×#3752—21no—
17#88815C2211yescharacteristic, complemented
18C2×#3802—31no—
19C2×D3×D12—61no—
20C2×#3774—32no—
21#63912—31no—
22#63906—61no—
23#63778D311yescharacteristic, complemented
24#63739—32no—
25C22×C3×C3⋊D3—22no—
26C22×C3×C3⋊D3—21no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×#1113—24no—
29C2×#1113—42no—
30C2×#1113—42no—
31C2×C33×C4D411yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C3×C4⋊(C3⋊D3)—24no—
35C3×C4×C3⋊D3—22no—
36#6295—22no—
37#6295—22no—
38#6295—42no—
39#6295D414yescomplemented
40C3×#622—22no—
41#4732—91nomaximal
42C22×D32—62no—
43C22×D32—61no—
44C2×C4⋊(C3⋊D3)—21no—
45C2×C4×C3⋊D3C2×D311yescharacteristic
46C2×C3×D12—61no—
47C2×C3×D12—61no—
48C2×C3×D12—121no—
49C2×C3×C4×D3—61no—
50C2×C3×C4×D3—61no—
51C2×#626—62no—
52C2×#626—61no—
53C2×#626—61no—
54C2×#626—61no—
55C2×#625—32no—
56C2×#625—31no—
57C2×#625—61no—
58#3802—64no—
59D3×D12—68no—
60C2×#622C2×D312yescomplemented
61#3774—64no—
62#3752—21no—
63C3×(C2×D3)⋊C4—61no—
64C3×(C2×D3)⋊C4—61no—
65C3×(C2×D3)⋊C4—121no—
66#3724—31no—
67Dic3⋊Dic3—61no—
68C22×C33C2×D411yescharacteristic
69C2×C3×C3⋊D3—21no—
70C2×C3×C3⋊D3—21no—
71C2×C3×C3⋊D3—24no—
72C2×C3×C3⋊D3—42no—
73C2×C3×C3⋊D3—42no—
74C2×C3×C3⋊D3—41no—
75C2×C3×C3⋊D3—41no—
76C2×C3×C3⋊D3—41no—
77C2×C3×C3⋊D3D812yescomplemented
78#1113—48no—
79C33×C4—21no—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3C2×D412yes—
82C2×D3×D4—181no—
83C22×D12—91no—
84#930—91no—
85C2×D24—92no—
86#884—181no—
87C3×C22⋊C8—91no—
88#821—92no—
89C22×C3⋊D3—22no—
90C22×C3⋊D3—21no—
91C22×C3⋊D3D1211yescharacteristic
92C22×C3×D3—61no—
93C22×C3×D3—61no—
94C22×C3×D3—62no—
95C22×C3×D3—62no—
96C22×C3×D3—61no—
97C22×C3×D3—61no—
98C22×C3×D3—121no—
99C22×C3×D3—122no—
100C2×D32—68no—
101C2×D32—64no—
102C2×D32—124no—
103C2×D32—124no—
104C2×D32—122no—
105C2×D32—122no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4D1211yescharacteristic
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3C22×D311yescharacteristic
113C4⋊(C3⋊D3)—24no—
114C4×C3⋊D3—22no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—121no—
120C3×D12—64no—
121C3×D12—64no—
122C3×D12—124no—
123C3×C4×D3—64no—
124C3×C4×D3—122no—
125#626—64no—
126#626—64no—
127#626—122no—
128#626—122no—
129#626—122no—
130#625—34no—
131#625—62no—
132#625—62no—
133#625—122no—
134#622—22no—
135#447—271noSylow
136C2×C33#12011yescharacteristic
137C2×C33C2×D811yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C3×C3⋊D3—22no—
140C3×C3⋊D3—44no—
141C3×C3⋊D3—81no—
142C23×D3—181no—
143C23×D3—182no—
144C2×C3×D4—181no—
145C22×C3×C4—91no—
146C2×#67—182no—
147C2×#67—181no—
148C2×#67—181no—
149C2×#67—181no—
150D3×D4—188no—
151C2×D12—61no—
152C2×D12—61no—
153C2×D12—92no—
154C2×D12—91no—
155C2×D12—121no—
156C2×D12—184no—
157C2×D12—181no—
158C2×C4×D3—61no—
159C2×C4×D3—61no—
160C2×C3×C8—92no—
161C3×C22⋊C4—181no—
162(C2×D3)⋊C4—61no—
163(C2×D3)⋊C4—61no—
164(C2×D3)⋊C4—121no—
165C4⋊Dic3—91no—
166Dic3⋊C4—181no—
167D24—184no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32C2×D1211yescharacteristic
173C2×C3⋊D3—21no—
174C2×C3⋊D3—21no—
175C2×C3⋊D3—24no—
176C2×C3⋊D3—41no—
177C2×C3⋊D3—42no—
178C2×C3⋊D3—42no—
179C2×C3⋊D3—41no—
180C2×C3⋊D3—41no—
181C2×C3⋊D3D2412yescomplemented
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C2×C3×D3—62no—
186C2×C3×D3—64no—
187C2×C3×D3—64no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C2×C3×D3—124no—
191C2×C3×D3—122no—
192C2×C3×D3—122no—
193C2×C3×D3—122no—
194C2×C3×D3—122no—
195C2×C3×D3—121no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C2×C3×D3—122no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C2×C3×D3—121no—
202C2×C3×D3—242no—
203C2×C3×D3—242no—
204C2×C3×D3—241no—
205C2×C3×D3—241no—
206D32—1216no—
207D32—128no—
208C32×C4—22no—
209C32×C4—21no—
210C32×C4—41no—
211C32×C4—42no—
212C32×C4—81no—
213C3⋊Dic3—41no—
214C3⋊Dic3D3×D412yes—
215C3×Dic3—62no—
216C3×Dic3—62no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C3×Dic3—241no—
220C22×D4—271no—
221C2×D8—272no—
222#121—271no—
223#102—272no—
224C22⋊C8—91no—
225C33#44711yescharacteristic, complemented, Sylow
226C23×C3—91no—
227C23×C3—182no—
228C23×C3—181no—
229C22×D3—61no—
230C22×D3—61no—
231C22×D3—62no—
232C22×D3—62no—
233C22×D3—61no—
234C22×D3—61no—
235C22×D3—121no—
236C22×D3—122no—
237C22×D3—184no—
238C22×D3—188no—
239C22×D3—364no—
240C22×D3—364no—
241C22×D3—362no—
242C22×D3—362no—
243C3×D4—184no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—91no—
249C2×C3×C4—182no—
250C2×C3×C4—181no—
251C2×C3×C4GO+(4,𝔽2)11yescharacteristic
252#67—184no—
253#67—184no—
254#67—362no—
255#67—362no—
256#67—362no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—121no—
262D12—64no—
263D12—64no—
264D12—94no—
265D12—124no—
266D12—182no—
267D12—182no—
268D12—362no—
269C4×D3—64no—
270C4×D3—122no—
271C3×C8—182no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32—41no—
284C2×C32#88211yescharacteristic
285C2×C32C2×D2411yescharacteristic
286C2×C32#90811yescharacteristic
287C3⋊D3—22no—
288C3⋊D3—44no—
289C3⋊D3—81no—
290C3×D3—64no—
291C3×D3—122no—
292C3×D3—124no—
293C3×D3—124no—
294C3×D3—244no—
295C3×D3—241no—
296C3×D3—242no—
297C3×D3—241no—
298C24—541no—
299C2×D4—181no—
300C2×D4—272no—
301C2×D4—271no—
302C2×D4—544no—
303C2×D4—541no—
304C22×C4—91no—
305D8—544no—
306C2×C8—92no—
307C4⋊C4—271no—
308C22⋊C4—181no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—92no—
314C22×C3—184no—
315C22×C3—181no—
316C22×C3—181no—
317C22×C3—362no—
318C22×C3—362no—
319C22×C3—361no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3C2×GO+(4,𝔽2)11yescharacteristic
323C2×D3—62no—
324C2×D3—62no—
325C2×D3—62no—
326C2×D3—62no—
327C2×D3—64no—
328C2×D3—64no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—124no—
332C2×D3—122no—
333C2×D3—122no—
334C2×D3—122no—
335C2×D3—122no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—122no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—242no—
344C2×D3—242no—
345C2×D3—241no—
346C2×D3—241no—
347C2×D3—3616no—
348C2×D3—368no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—41no—
352C3×C4—42no—
353C3×C4—81no—
354C3×C4—92no—
355C3×C4—361no—
356Dic3—62no—
357Dic3—62no—
358Dic3—121no—
359Dic3—121no—
360Dic3—241no—
361C32—21no—
362C32—21no—
363C32—41no—
364C32—41no—
365C32#473211yescharacteristic, complemented
366C23—91no—
367C23—182no—
368C23—181no—
369C23—544no—
370C23—1082no—
371C23—1082no—
372D4—184no—
373D4—274no—
374D4—542no—
375D4—542no—
376D4—1082no—
377C2×C4—91no—
378C2×C4—182no—
379C2×C4—181no—
380C2×C4#157⋊D311yescharacteristic
381C8—182no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—41no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—182no—
395C2×C3—364no—
396C2×C3—721no—
397C2×C3#6422411yescharacteristic
398C2×C3#6422511yescharacteristic
399C2×C3#6424111yescharacteristic
400D3—64no—
401D3—122no—
402D3—124no—
403D3—124no—
404D3—244no—
405D3—241no—
406D3—242no—
407D3—241no—
408C22—92no—
409C22—181no—
410C22—181no—
411C22—184no—
412C22—362no—
413C22—362no—
414C22—361no—
415C22—361no—
416C22—361no—
417C22—1088no—
418C22C2×#157⋊D311yescharacteristic
419C4—21no—
420C4—92no—
421C4—361no—
422C3—21no—
423C3—21no—
424C3—41no—
425C3—41no—
426C3#1059342511yescharacteristic, complemented
427C2—182no—
428C2—364no—
429C2—721no—
430C2#1172433611yescharacteristic
431C2#1172434011yescharacteristic
432C2#1172435411yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#447
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#1113C2semidirect
2. C2×#88941C2semidirect
3. #11723695C2semidirect
4. C3×#63778C2semidirect
5. #11722793C2semidirect
12. C2×#6295C22semidirect
14. C2×C3×#622C22semidirect
17. #88815C22semidirect
23. #63778D3semidirect
39. #6295D4semidirect
60. C2×#622C2×D3semidirect
77. C2×C3×C3⋊D3D8semidirect
181. C2×C3⋊D3D24semidirect
225. C33#447semidirect
365. C32#4732semidirect
426. C3#10593425semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C2×#889412
#117236952
C3×#637782
#117227932
#105934253
#47329