Finite Group Databasebeta

← #13165575

Subgroups — #13165575

Order 1728.

570 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 56 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
4Dic3×#625C211yesmaximal, characteristic, complemented
5#11722793C211yesmaximal, characteristic, complemented
6#11722773C211yesmaximal, characteristic, complemented
7Dic3⋊#622C211yesmaximal, characteristic, complemented
8#10593417—31nomaximal
9C2×#6295C2211yescharacteristic
10C2×#6290—21no—
11C2×C3×#626—21no—
12C2×C3×#625C2211yescharacteristic
13C2×C3×#625C2211yescharacteristic
14C2×C3×D3×Dic3—21no—
15C2×C3⋊#622C2211yescharacteristic
16C2×C3×#622C2211yescharacteristic
17#88935—22no—
18C3×#3774—22no—
19#88815C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3C2211yescharacteristic
22C2×#3774—31no—
23C2×#3774D311yescharacteristic, complemented
24#63979—61no—
25Dic3×#67—61no—
26#63914—31no—
27C4×#625—31no—
28C4⋊#622—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—
41C3×#626—42no—
42C3×#625—21no—
43C3×#625—21no—
44C3×#625—21no—
45C3×#625D412yescomplemented
46C3×D3×Dic3—42no—
47C3⋊#622—21no—
48C3×#622—21no—
49#5150—91nomaximal
50C2×#638—61no—
51C2×C3×#67—61no—
52C2×C3×#67—61no—
53C2×C3×#67—61no—
54C22×C3×Dic3—61no—
55C2×C3×D12—61no—
56C2×C3×C4×D3—61no—
57C2×#626—21no—
58C2×#626—61no—
59C2×#625—31no—
60C2×#625—61no—
61C2×#625—61no—
62C2×#625—121no—
63C2×#625C2×D311yescharacteristic
64C2×#625C2×D311yescharacteristic
65C2×D3×Dic3—21no—
66C2×#622—31no—
67C2×#622C2×D311yescharacteristic
68#3774—22no—
69#3774—62no—
70#3757—61no—
71C4×C3⋊Dic3—31no—
72#3724—31no—
73Dic3⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—121no—
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)—21no—
90C3×C3⋊Dic3—41no—
91C3×C3⋊Dic3C2×D411yescharacteristic
92C3×C3⋊Dic3#4111yescharacteristic
93C32×Dic3D812yescomplemented
94C2×C3×D8—91no—
95#940—91no—
96#940—181no—
97D4×Dic3—91no—
98C2×#213—91no—
99C4×#67—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⋊Dic3C22×D311yescharacteristic
124C3×#67—62no—
125C3×#67—62no—
126C3×#67—121no—
127C3×#67—121no—
128C3×#67—122no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—121no—
136C2×C3×Dic3—122no—
137C3×D12—122no—
138C3×C4×D3—122no—
139#626—42no—
140#626—121no—
141#626—121no—
142#625—21no—
143#625—21no—
144#625—21no—
145#625—61no—
146#625—61no—
147#625—122no—
148#625—122no—
149#625—124no—
150#625#6712yescomplemented
151D3×Dic3—42no—
152#622—21no—
153#622—61no—
154#459—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33C2×D811yescharacteristic
157C2×C33AGL(1,ℤ/8)11yescharacteristic
158C3×C3⋊D3—81no—
159C32×D3—42no—
160C32×D3—81no—
161C2×C3×D4—91no—
162C2×C3×D4—91no—
163C2×C3×D4—181no—
164C22×C3×C4—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—61no—
174C2×D12—61no—
175C2×D12—91no—
176C2×C4×D3—61no—
177C3×D8—182no—
178C2×C3×C8—91no—
179#217—181no—
180#217—181no—
181#217—361no—
182#213—182no—
183(C2×D3)⋊C4—181no—
184C4⋊Dic3—91no—
185Dic3⋊C4—181no—
186C4×Dic3—91no—
187C4×Dic3—181no—
188C4×Dic3—181no—
189C2×C3⋊C8—91no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—62no—
195C22×C32—121no—
196C22×C32—121no—
197C22×C32—121no—
198C22×C32—121no—
199C22×C32—121no—
200C22×C32D3×D411yescharacteristic
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3×D3—22no—
205C2×C3×D3—22no—
206C2×C3×D3—42no—
207C2×C3×D3—41no—
208C2×C3×D3—41no—
209C2×C3×D3—41no—
210C2×C3×D3—41no—
211C2×C3×D3—41no—
212C2×C3×D3—41no—
213C2×C3×D3—41no—
214C2×C3×D3—42no—
215C2×C3×D3—41no—
216C2×C3×D3—41no—
217C2×C3×D3—41no—
218C2×C3×D3—81no—
219C2×C3×D3—81no—
220C2×C3×D3—81no—
221C2×C3×D3—81no—
222C2×C3×D3—121no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C2×C3×D3—122no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—241no—
230C2×C3×D3—241no—
231C32×C4—32no—
232C3⋊Dic3—41no—
233C3⋊Dic3—62no—
234C3⋊Dic3—61no—
235C3⋊Dic3—121no—
236C3⋊Dic3—122no—
237C3⋊Dic3—241no—
238C3⋊Dic3#23711yescharacteristic
239C3⋊Dic3C2×#6711yescharacteristic
240C3×Dic3—22no—
241C3×Dic3—22no—
242C3×Dic3—61no—
243C3×Dic3—61no—
244C3×Dic3—61no—
245C3×Dic3—61no—
246C3×Dic3—121no—
247C3×Dic3—121no—
248C3×Dic3—241no—
249C2×D8—91no—
250C2×D8—271no—
251#127—271no—
252C4×D4—271no—
253C4⋊C8—271no—
254#102—271no—
255#102—271no—
256C33#45911yescharacteristic, complemented, Sylow
257C23×C3—61no—
258C23×C3—61no—
259C23×C3—61no—
260C23×C3—61no—
261C23×C3—121no—
262C23×C3—121no—
263C23×C3—181no—
264C22×D3—21no—
265C22×D3—21no—
266C22×D3—61no—
267C22×D3—61no—
268C22×D3—121no—
269C3×D4—92no—
270C3×D4—181no—
271C3×D4—181no—
272C3×D4—181no—
273C3×D4—362no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—91no—
277C2×C3×C4—181no—
278C2×C3×C4—362no—
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—122no—
297C2×Dic3—181no—
298C2×Dic3—181no—
299C2×Dic3—361no—
300C2×Dic3—361no—
301C2×Dic3GO+(4,𝔽2)11yescharacteristic
302D12—122no—
303D12—181no—
304D12—181no—
305C4×D3—122no—
306C3×C8—181no—
307C3⋊C8—181no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—41no—
315C2×C32—41no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—122no—
321C2×C32—241no—
322C2×C32D3×D811yescharacteristic
323C2×C32#91111yescharacteristic
324C2×C32#93911yescharacteristic
325C3⋊D3—81no—
326C3×D3—42no—
327C3×D3—42no—
328C3×D3—82no—
329C3×D3—81no—
330C3×D3—82no—
331C3×D3—81no—
332C3×D3—241no—
333C3×D3—242no—
334C3×D3—241no—
335C2×D4—91no—
336C2×D4—91no—
337C2×D4—181no—
338C2×D4—271no—
339C2×D4—541no—
340C22×C4—181no—
341D8—182no—
342D8—542no—
343C2×C8—91no—
344C2×C8—271no—
345C4⋊C4—271no—
346C22⋊C4—541no—
347C42—271no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—62no—
353C22×C3—62no—
354C22×C3—122no—
355C22×C3—121no—
356C22×C3—121no—
357C22×C3—121no—
358C22×C3—121no—
359C22×C3—121no—
360C22×C3—122no—
361C22×C3—121no—
362C22×C3—121no—
363C22×C3—121no—
364C22×C3—121no—
365C22×C3—121no—
366C22×C3—241no—
367C22×C3—241no—
368C22×C3—241no—
369C22×C3—241no—
370C22×C3—361no—
371C22×C3—361no—
372C22×C3—361no—
373C22×C3C2×GO+(4,𝔽2)11yescharacteristic
374C2×D3—22no—
375C2×D3—41no—
376C2×D3—41no—
377C2×D3—41no—
378C2×D3—41no—
379C2×D3—41no—
380C2×D3—121no—
381C2×D3—121no—
382C2×D3—121no—
383C2×D3—122no—
384C2×D3—121no—
385C2×D3—121no—
386C2×D3—121no—
387C2×D3—241no—
388C2×D3—241no—
389C3×C4—62no—
390C3×C4—62no—
391C3×C4—91no—
392C3×C4—91no—
393C3×C4—361no—
394Dic3—61no—
395Dic3—61no—
396Dic3—61no—
397Dic3—61no—
398Dic3—121no—
399Dic3—121no—
400Dic3—182no—
401Dic3—181no—
402Dic3—241no—
403Dic3—361no—
404Dic3—362no—
405Dic3—721no—
406Dic3#377412yescomplemented
407C32—21no—
408C32—21no—
409C32—41no—
410C32—41no—
411C32#515011yescharacteristic, complemented
412C23—61no—
413C23—61no—
414C23—181no—
415D4—92no—
416D4—181no—
417D4—181no—
418D4—181no—
419D4—362no—
420D4—541no—
421D4—541no—
422D4—1082no—
423C2×C4—31no—
424C2×C4—91no—
425C2×C4—181no—
426C2×C4—271no—
427C2×C4—362no—
428C8—181no—
429C8—541no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—41no—
437C2×C3—41no—
438C2×C3—41no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—122no—
443C2×C3—122no—
444C2×C3—242no—
445C2×C3—241no—
446C2×C3—242no—
447C2×C3—241no—
448C2×C3—721no—
449C2×C3#6422411yescharacteristic
450C2×C3#6423011yescharacteristic
451C2×C3C2×#377411yescharacteristic
452D3—42no—
453D3—81no—
454D3—241no—
455D3—242no—
456D3—241no—
457C22—62no—
458C22—121no—
459C22—121no—
460C22—121no—
461C22—121no—
462C22—121no—
463C22—361no—
464C22—361no—
465C22—361no—
466C22D3×GO+(4,𝔽2)11yescharacteristic
467C4—32no—
468C4—91no—
469C4—91no—
470C4—361no—
471C4—541no—
472C3—21no—
473C3—21no—
474C3—41no—
475C3—41no—
476C3#1059341711yescharacteristic, complemented
477C2—122no—
478C2—241no—
479C2—721no—
480C2#1172411711yescharacteristic
481C2D3×#377411yescharacteristic
482C2#1172414011yescharacteristic
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. Dic3×#625C2semidirect
5. #11722793C2semidirect
6. #11722773C2semidirect
7. Dic3⋊#622C2semidirect
23. C2×#3774D3semidirect
45. C3×#625D4semidirect
93. C32×Dic3D8semidirect
150. #625#67semidirect
256. C33#459semidirect
406. Dic3#3774semidirect
411. C32#5150semidirect
476. C3#10593417semidirect

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
Dic3×#6252
#117227932
#117227732
Dic3⋊#6222
#105934173
#51509