Finite Group Databasebeta

← #13165576

Subgroups — #13165576

Order 1728.

590 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 54 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, complemented
2C2×C3×#3774C211yesmaximal, characteristic, complemented
3#11723029C211yesmaximal, characteristic, complemented
4#11722925C211yesmaximal, characteristic, complemented
5#11722793C211yesmaximal, characteristic, complemented
6#11722773C211yesmaximal, characteristic, complemented
7#11722732C211yesmaximal, characteristic, complemented
8#10593426—31nomaximal
9C2×#6295C2211yescharacteristic
10C2×#6290—21no—
11C2×D3×C3⋊Dic3—21no—
12C2×C3×#626—21no—
13C2×C3×#625C2211yescharacteristic
14C2×C3×#625C2211yescharacteristic
15C2×C3⋊#622C2211yescharacteristic
16C2×C3×#622C2211yescharacteristic
17#88935—22no—
18C3×#3774—22no—
19#88815C2211yescharacteristic
20Dic3×C3⋊Dic3C2211yescharacteristic
21C3×(C2×D3)⋊Dic3—21no—
22C2×#3774—31no—
23C2×#3774D311yescharacteristic, complemented
24#63979—61no—
25#67⋊Dic3—61no—
26#63914—31no—
27#63902—31no—
28#63779—31no—
29#63739—31no—
30#63739—31no—
31C22×C3×C3⋊D3—21no—
32C22×C32×D3—21no—
33C22×C32×D3—21no—
34C2×C3⋊(C3⋊Dic3)D411yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C2×C32×Dic3D411yescharacteristic
38#6295—21no—
39#6295—21no—
40#6290—42no—
41D3×C3⋊Dic3—42no—
42C3×#626—42no—
43C3×#625—21no—
44C3×#625—21no—
45C3×#625—21no—
46C3×#625D412yescomplemented
47C3⋊#622—21no—
48C3×#622—21no—
49#5150—91nomaximal
50C2×#638—61no—
51C22×C3⋊Dic3—61no—
52C2×C3×#67—61no—
53C2×C3×#67—61no—
54C2×C3×#67—61no—
55C2×C3×D12—61no—
56C2×#626—21no—
57C2×#626—61no—
58C2×#625—31no—
59C2×#625—61no—
60C2×#625—61no—
61C2×#625—121no—
62C2×#625C2×D311yescharacteristic
63C2×#625C2×D311yescharacteristic
64C2×D3×Dic3—61no—
65C2×D3×Dic3—61no—
66C2×D3×Dic3—121no—
67C2×#622—31no—
68C2×#622C2×D311yescharacteristic
69#3774—22no—
70#3774—62no—
71C4×C3⋊Dic3—31no—
72C3×#217—61no—
73C3×(C2×D3)⋊C4—61no—
74#3724—31no—
75Dic3⋊Dic3—61no—
76(C2×D3)⋊Dic3—21no—
77Dic32—61no—
78Dic32—61no—
79C22×C33C2×D411yescharacteristic
80C2×C3×C3⋊D3—41no—
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3—41no—
83C2×C32×D3—22no—
84C2×C32×D3—41no—
85C2×C32×D3—41no—
86C2×C32×D3—41no—
87C2×C32×D3—41no—
88C2×C32×D3—41no—
89C3⋊(C3⋊Dic3)D812yescomplemented
90C3×C3⋊Dic3—41no—
91C3×C3⋊Dic3C2×D411yescharacteristic
92C3×C3⋊Dic3#4111yescharacteristic
93C32×Dic3—21no—
94C2×C3×D8—91no—
95#940—91no—
96#940—181no—
97D4×Dic3—91no—
98C2×#213—91no—
99#67⋊C4—181no—
100#832—91no—
101#821—91no—
102Dic3⋊C8—91no—
103C23×C32—61no—
104C23×C32—61no—
105C22×C3⋊D3—21no—
106C22×C3×D3—21no—
107C22×C3×D3—21no—
108C22×C3×D3—21no—
109C22×C3×D3—21no—
110C22×C3×D3—41no—
111C22×C3×D3—41no—
112C22×C3×D3—61no—
113C22×C3×D3—61no—
114C22×C3×D3—121no—
115C2×C32×C4—31no—
116#638—122no—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—31no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—121no—
122C2×C3⋊Dic3—121no—
123C2×C3⋊Dic3—122no—
124C2×C3⋊Dic3C22×D311yescharacteristic
125C3×#67—62no—
126C3×#67—62no—
127C3×#67—121no—
128C3×#67—121no—
129C3×#67—122no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—121no—
137C3×D12—122no—
138#626—42no—
139#626—121no—
140#626—121no—
141#625—21no—
142#625—21no—
143#625—21no—
144#625—61no—
145#625—61no—
146#625—122no—
147#625—122no—
148#625—124no—
149#625#6712yescomplemented
150D3×Dic3—122no—
151D3×Dic3—122no—
152D3×Dic3—124no—
153#622—21no—
154#622—61no—
155#459—271noSylow
156C2×C33#12111yescharacteristic
157C2×C33C2×D811yescharacteristic
158C2×C33AGL(1,ℤ/8)11yescharacteristic
159C3×C3⋊D3—81no—
160C32×D3—42no—
161C32×D3—81no—
162C2×C3×D4—91no—
163C2×C3×D4—91no—
164C2×C3×D4—181no—
165C2×#67—61no—
166C2×#67—61no—
167C2×#67—61no—
168C2×#67—181no—
169C2×#67—181no—
170C2×#67—181no—
171C2×#67—181no—
172C2×#67—361no—
173C22×Dic3—181no—
174C22×Dic3—181no—
175C22×Dic3—361no—
176C2×D12—61no—
177C2×D12—91no—
178C2×C4×D3—181no—
179C3×D8—182no—
180C2×C3×C8—91no—
181C3×C22⋊C4—181no—
182#217—61no—
183#213—182no—
184(C2×D3)⋊C4—61no—
185C4⋊Dic3—91no—
186Dic3⋊C4—181no—
187C4×Dic3—91no—
188C4×Dic3—181no—
189C4×Dic3—181no—
190C2×C3⋊C8—91no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—62no—
196C22×C32—121no—
197C22×C32—121no—
198C22×C32—121no—
199C22×C32—121no—
200C22×C32—121no—
201C22×C32D3×D411yescharacteristic
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3×D3—22no—
206C2×C3×D3—22no—
207C2×C3×D3—42no—
208C2×C3×D3—41no—
209C2×C3×D3—41no—
210C2×C3×D3—42no—
211C2×C3×D3—42no—
212C2×C3×D3—41no—
213C2×C3×D3—41no—
214C2×C3×D3—41no—
215C2×C3×D3—41no—
216C2×C3×D3—41no—
217C2×C3×D3—42no—
218C2×C3×D3—41no—
219C2×C3×D3—41no—
220C2×C3×D3—41no—
221C2×C3×D3—81no—
222C2×C3×D3—81no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C2×C3×D3—122no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—241no—
231C2×C3×D3—241no—
232C32×C4—61no—
233C3⋊Dic3—32no—
234C3⋊Dic3—41no—
235C3⋊Dic3—62no—
236C3⋊Dic3—62no—
237C3⋊Dic3—122no—
238C3⋊Dic3—122no—
239C3⋊Dic3#23711yescharacteristic
240C3⋊Dic3C2×#6711yescharacteristic
241C3×Dic3—22no—
242C3×Dic3—41no—
243C3×Dic3—61no—
244C3×Dic3—61no—
245C3×Dic3—61no—
246C3×Dic3—61no—
247C3×Dic3—121no—
248C3×Dic3—122no—
249C3×Dic3—121no—
250C2×D8—91no—
251C2×D8—271no—
252#127—271no—
253C4×D4—271no—
254C4⋊C8—271no—
255#102—271no—
256#102—271no—
257C33#45911yescharacteristic, complemented, Sylow
258C23×C3—61no—
259C23×C3—61no—
260C23×C3—61no—
261C23×C3—61no—
262C23×C3—121no—
263C23×C3—121no—
264C23×C3—181no—
265C22×D3—21no—
266C22×D3—21no—
267C22×D3—61no—
268C22×D3—61no—
269C22×D3—121no—
270C3×D4—92no—
271C3×D4—181no—
272C3×D4—181no—
273C3×D4—181no—
274C3×D4—362no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—91no—
278C2×C3×C4—181no—
279#67—62no—
280#67—62no—
281#67—121no—
282#67—121no—
283#67—122no—
284#67—361no—
285#67—361no—
286#67—362no—
287#67—362no—
288#67—362no—
289#67—364no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×Dic3—61no—
294C2×Dic3—91no—
295C2×Dic3—121no—
296C2×Dic3—181no—
297C2×Dic3—181no—
298C2×Dic3—361no—
299C2×Dic3—361no—
300C2×Dic3—362no—
301C2×Dic3—362no—
302C2×Dic3—364no—
303C2×Dic3GO+(4,𝔽2)11yescharacteristic
304D12—122no—
305D12—181no—
306D12—181no—
307C4×D3—362no—
308C3×C8—181no—
309C3⋊C8—181no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—122no—
323C2×C32—241no—
324C2×C32D3×D811yescharacteristic
325C2×C32#91111yescharacteristic
326C2×C32#93911yescharacteristic
327C3⋊D3—81no—
328C3×D3—42no—
329C3×D3—44no—
330C3×D3—42no—
331C3×D3—81no—
332C3×D3—82no—
333C3×D3—81no—
334C3×D3—241no—
335C3×D3—242no—
336C3×D3—241no—
337C2×D4—91no—
338C2×D4—91no—
339C2×D4—181no—
340C2×D4—271no—
341C2×D4—541no—
342C22×C4—541no—
343D8—182no—
344D8—542no—
345C2×C8—91no—
346C2×C8—271no—
347C4⋊C4—271no—
348C22⋊C4—181no—
349C42—271no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—62no—
355C22×C3—62no—
356C22×C3—122no—
357C22×C3—121no—
358C22×C3—121no—
359C22×C3—121no—
360C22×C3—121no—
361C22×C3—121no—
362C22×C3—122no—
363C22×C3—122no—
364C22×C3—122no—
365C22×C3—121no—
366C22×C3—121no—
367C22×C3—121no—
368C22×C3—121no—
369C22×C3—121no—
370C22×C3—241no—
371C22×C3—241no—
372C22×C3—361no—
373C22×C3—361no—
374C22×C3—361no—
375C22×C3C2×GO+(4,𝔽2)11yescharacteristic
376C2×D3—22no—
377C2×D3—41no—
378C2×D3—41no—
379C2×D3—41no—
380C2×D3—41no—
381C2×D3—41no—
382C2×D3—121no—
383C2×D3—121no—
384C2×D3—121no—
385C2×D3—122no—
386C2×D3—121no—
387C2×D3—121no—
388C2×D3—121no—
389C2×D3—241no—
390C2×D3—241no—
391C3×C4—62no—
392C3×C4—91no—
393C3×C4—91no—
394C3×C4—121no—
395C3×C4—361no—
396Dic3—21no—
397Dic3—61no—
398Dic3—61no—
399Dic3—61no—
400Dic3—61no—
401Dic3—92no—
402Dic3—121no—
403Dic3—122no—
404Dic3—121no—
405Dic3—182no—
406Dic3—182no—
407Dic3—362no—
408Dic3—362no—
409C32—21no—
410C32—21no—
411C32—41no—
412C32—41no—
413C32#515011yescharacteristic, complemented
414C23—61no—
415C23—61no—
416C23—181no—
417D4—92no—
418D4—181no—
419D4—181no—
420D4—181no—
421D4—362no—
422D4—541no—
423D4—541no—
424D4—1082no—
425C2×C4—31no—
426C2×C4—91no—
427C2×C4—181no—
428C2×C4—271no—
429C2×C4—1082no—
430C8—181no—
431C8—541no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—41no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—122no—
445C2×C3—124no—
446C2×C3—122no—
447C2×C3—241no—
448C2×C3—242no—
449C2×C3—241no—
450C2×C3—721no—
451C2×C3#6422511yescharacteristic
452C2×C3#6423011yescharacteristic
453C2×C3#6423611yescharacteristic
454D3—42no—
455D3—81no—
456D3—241no—
457D3—242no—
458D3—241no—
459C22—62no—
460C22—121no—
461C22—121no—
462C22—121no—
463C22—121no—
464C22—121no—
465C22—361no—
466C22—361no—
467C22—361no—
468C22D3×GO+(4,𝔽2)11yescharacteristic
469C4—61no—
470C4—91no—
471C4—91no—
472C4—272no—
473C4—361no—
474C3—21no—
475C3—21no—
476C3—41no—
477C3—41no—
478C3#1059342611yescharacteristic, complemented
479C2—122no—
480C2—241no—
481C2—721no—
482C2#1172411711yescharacteristic
483C2#1172412711yescharacteristic
484C2#1172413811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
hypocenterC33#459
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#88935C2semidirect
2. C2×C3×#3774C2semidirect
3. #11723029C2semidirect
4. #11722925C2semidirect
5. #11722793C2semidirect
6. #11722773C2semidirect
7. #11722732C2semidirect
23. C2×#3774D3semidirect
46. C3×#625D4semidirect
89. C3⋊(C3⋊Dic3)D8semidirect
149. #625#67semidirect
257. C33#459semidirect
413. C32#5150semidirect
478. C3#10593426semidirect

Maximal subgroups

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

subgroupindex
C2×#889352
C2×C3×#37742
#117230292
#117229252
#117227932
#117227732
#117227322
#105934263
#51509