Finite Group Databasebeta

← #13165411

Subgroups — #13165411

Order 1728.

596 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 90 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×#88940C211yesmaximal, characteristic
2C2×#88934C211yesmaximal, characteristic
3C2×Dic3⋊#157C211yesmaximal, characteristic
4#11722911C211yesmaximal, characteristic
5Dic3×#624C211yesmaximal, characteristic
6#11722769C211yesmaximal, characteristic
7C3×#63737C211yesmaximal, characteristic
8#10593412—31nomaximal
9C22×C3⋊#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C2×#6294C2211yescharacteristic
12C2×#6294C2211yescharacteristic
13C2×Dic3×C3⋊D3C2211yescharacteristic
14C2×C3×#624C2211yescharacteristic
15C2×C3×#624C2211yescharacteristic
16#88940—22no—
17#88934—22no—
18Dic3⋊#157C414yescomplemented
19#88806—21no—
20Dic3×C3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C2×C4⋊#157—31no—
24C2×#3773—31no—
25C2×#3773—31no—
26#63900—31no—
27C4×#624—31no—
28D3×Dic3⋊C4—61no—
29C4×D3×Dic3—61no—
30#63737—31no—
31#63737D311yescharacteristic
32C22×C3×C3⋊D3C2311yescharacteristic
33C2×C3⋊#157—21no—
34C2×C3⋊#157—21no—
35C2×C3⋊#157C2×C412yes—
36C2×C3×#157—21no—
37C2×C3×#157—21no—
38C2×C3×#157C2×C412yes—
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C32×Dic3—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45#6294—21no—
46#6294—21no—
47#6294—21no—
48#6294D412yes—
49Dic3×C3⋊D3C2×C412yes—
50Dic3×C3⋊D3Q812yes—
51C3×#624—21no—
52C3×#624—21no—
53C3×#624—21no—
54C3×#624Q812yes—
55#4647—91nomaximal
56C22×#157—31no—
57C22×#157C2×D311yescharacteristic
58C2×C4×C3⋊D3—31no—
59C2×C3×C4×D3—61no—
60C2×C3×C4×D3—61no—
61C2×#624—31no—
62C2×#624—31no—
63C2×#624C2×D311yescharacteristic
64C2×#624C2×D311yescharacteristic
65C2×D3×Dic3—61no—
66C2×D3×Dic3—61no—
67C2×D3×Dic3—61no—
68C2×D3×Dic3—61no—
69C4⋊#157—34no—
70#3773—62no—
71#3773—62no—
72#3750—61no—
73C4×C3⋊Dic3—61no—
74C3×C4⋊Dic3—61no—
75C3×Dic3⋊C4—61no—
76C3×C4×Dic3—61no—
77C3×C4×Dic3—61no—
78#3724—61no—
79#3724—61no—
80#3724—121no—
81Dic3⋊Dic3—21no—
82Dic32—21no—
83Dic32—61no—
84Dic32—61no—
85Dic32—121no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3C22×C411yescharacteristic
88C2×C3×C3⋊D3C2×D411yescharacteristic
89C2×C3×C3⋊D3C2×Q811yescharacteristic
90C2×C3×C3⋊D3C2×Q811yescharacteristic
91C2×C3×C3⋊D3#4111yescharacteristic
92C2×C3×C3⋊D3#4111yescharacteristic
93C3⋊#157—22no—
94C3×#157—22no—
95C3⋊(C3⋊Dic3)C4⋊C412yescomplemented
96C3×C3⋊Dic3—22no—
97C3×C3⋊Dic3—41no—
98C32×Dic3—22no—
99C32×Dic3—41no—
100C32×Dic3C4⋊C412yescomplemented
101C2×C4⋊Dic3—91no—
102C2×Dic3⋊C4—91no—
103C2×Dic3⋊C4—91no—
104C2×Dic3⋊C4—91no—
105C2×C4×Dic3—91no—
106D3×C4⋊C4—181no—
107C42×D3—181no—
108C3×#95—91no—
109C3⋊#95—91no—
110C22×C3⋊D3C22×D311yescharacteristic
111C22×C3×D3—61no—
112C22×C3×D3—61no—
113C2×#157—21no—
114C2×#157—21no—
115C2×#157—32no—
116C2×#157—61no—
117C2×#157—61no—
118C2×#157C4×D312yes—
119C2×C32×C4—31no—
120C2×C32×C4—61no—
121C2×C32×C4—61no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—61no—
127C2×C3⋊Dic3—121no—
128C2×C3⋊Dic3—121no—
129C4×C3⋊D3—32no—
130C4×C3⋊D3—32no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—21no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—121no—
144C2×C3×Dic3—121no—
145C3×C4×D3—62no—
146C3×C4×D3—62no—
147C3×C4×D3—121no—
148C3×C4×D3—121no—
149#624—21no—
150#624—21no—
151#624—21no—
152#624—32no—
153#624—61no—
154#624—61no—
155#624—61no—
156#624Dic612yes—
157D3×Dic3—62no—
158D3×Dic3—62no—
159D3×Dic3—62no—
160D3×Dic3—62no—
161D3×Dic3—62no—
162D3×Dic3—62no—
163D3×Dic3—121no—
164D3×Dic3—121no—
165#389—271noSylow
166C2×C33C2×C4⋊C411yescharacteristic
167C2×C33#12111yescharacteristic
168C2×C33C22⋊Q811yescharacteristic
169C3×C3⋊D3C4×D411yescharacteristic
170C3×C3⋊D3C4×Q811yescharacteristic
171C3×C3⋊D3#12511yescharacteristic
172C3×C3⋊D3C4⋊Q811yescharacteristic
173C22×C3×C4—91no—
174C22×C3×C4—91no—
175C22×C3×C4—91no—
176C22×Dic3—91no—
177C22×Dic3—91no—
178C22×Dic3—91no—
179C22×Dic3—91no—
180C2×C4×D3—61no—
181C2×C4×D3—61no—
182C2×C4×D3—181no—
183C2×C4×D3—181no—
184C2×C4×D3—181no—
185C2×C4×D3—181no—
186C3×C4⋊C4—181no—
187C3×C42—181no—
188C4⋊Dic3—61no—
189C4⋊Dic3—182no—
190Dic3⋊C4—61no—
191Dic3⋊C4—94no—
192Dic3⋊C4—181no—
193Dic3⋊C4—181no—
194Dic3⋊C4—181no—
195Dic3⋊C4—182no—
196Dic3⋊C4—361no—
197C4×Dic3—61no—
198C4×Dic3—61no—
199C4×Dic3—181no—
200C4×Dic3—181no—
201C4×Dic3—181no—
202C4×Dic3—361no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32D3×D411yescharacteristic
208C2×C3⋊D3C2×Dic611yescharacteristic
209C2×C3⋊D3C2×C4×D311yescharacteristic
210C2×C3⋊D3#23511yescharacteristic
211C2×C3⋊D3D3×D411yescharacteristic
212C2×C3⋊D3#23711yescharacteristic
213C2×C3⋊D3D3×Q811yescharacteristic
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C2×C3×D3—61no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—61no—
221C2×C3×D3—61no—
222C2×C3×D3—61no—
223C2×C3×D3—61no—
224C2×C3×D3—61no—
225C2×C3×D3—61no—
226#157—22no—
227#157—62no—
228C32×C4—32no—
229C32×C4—62no—
230C32×C4—121no—
231C3⋊Dic3—22no—
232C3⋊Dic3—32no—
233C3⋊Dic3—41no—
234C3⋊Dic3—62no—
235C3⋊Dic3—62no—
236C3⋊Dic3—122no—
237C3⋊Dic3—122no—
238C3×Dic3—22no—
239C3×Dic3—22no—
240C3×Dic3—22no—
241C3×Dic3—22no—
242C3×Dic3—42no—
243C3×Dic3—41no—
244C3×Dic3—41no—
245C3×Dic3—62no—
246C3×Dic3—62no—
247C3×Dic3—81no—
248C3×Dic3—122no—
249C3×Dic3—121no—
250C3×Dic3—121no—
251C3×Dic3—241no—
252C2×C4⋊C4—271no—
253C2×C4⋊C4—271no—
254C2×C4⋊C4—271no—
255C2×C4⋊C4—271no—
256C2×C42—271no—
257#95—91no—
258#95—271no—
259C33#38911yescharacteristic, complemented, Sylow
260C23×C3—91no—
261C22×D3—61no—
262C22×D3—61no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—92no—
270C2×C3×C4—92no—
271C2×C3×C4—121no—
272C2×C3×C4—121no—
273C2×C3×C4—181no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—92no—
287C2×Dic3—91no—
288C2×Dic3—92no—
289C2×Dic3—92no—
290C2×Dic3—92no—
291C2×Dic3—121no—
292C2×Dic3—121no—
293C2×Dic3—181no—
294C2×Dic3—181no—
295C2×Dic3—181no—
296C2×Dic3—181no—
297C2×Dic3—181no—
298C2×Dic3—181no—
299C2×Dic3—181no—
300C2×Dic3—361no—
301C2×Dic3—361no—
302C2×Dic3GO+(4,𝔽2)11yescharacteristic
303C4×D3—62no—
304C4×D3—62no—
305C4×D3—121no—
306C4×D3—121no—
307C4×D3—182no—
308C4×D3—182no—
309C4×D3—182no—
310C4×D3—182no—
311C4×D3—182no—
312C4×D3—182no—
313C4×D3—361no—
314C4×D3—361no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32—41no—
322C2×C32—41no—
323C2×C32—41no—
324C2×C32—41no—
325C2×C32—41no—
326C2×C32—41no—
327C2×C32#87811yescharacteristic
328C2×C32#88411yescharacteristic
329C2×C32D3×C4⋊C411yescharacteristic
330C3⋊D3C4×Dic611yescharacteristic
331C3⋊D3#67⋊C411yescharacteristic
332C3⋊D3#88811yescharacteristic
333C3⋊D3#88911yescharacteristic
334C3×D3—61no—
335C3×D3—61no—
336C3×D3—61no—
337C3×D3—61no—
338C3×D3—61no—
339C3×D3—61no—
340C3×D3—61no—
341C3×D3—61no—
342C22×C4—91no—
343C22×C4—91no—
344C22×C4—91no—
345C22×C4—271no—
346C22×C4—271no—
347C22×C4—271no—
348C22×C4—271no—
349C4⋊C4—181no—
350C4⋊C4—274no—
351C4⋊C4—541no—
352C4⋊C4—542no—
353C4⋊C4—542no—
354C42—181no—
355C42—541no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—91no—
361C22×C3—91no—
362C22×C3—91no—
363C22×C3—91no—
364C22×C3—91no—
365C22×C3—91no—
366C22×C3C2×GO+(4,𝔽2)11yescharacteristic
367C2×D3—61no—
368C2×D3—61no—
369C2×D3—61no—
370C2×D3—61no—
371C2×D3—61no—
372C2×D3—61no—
373C2×D3—61no—
374C2×D3—61no—
375C2×D3—61no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—61no—
379C3×C4—62no—
380C3×C4—62no—
381C3×C4—62no—
382C3×C4—62no—
383C3×C4—122no—
384C3×C4—121no—
385C3×C4—121no—
386C3×C4—182no—
387C3×C4—182no—
388C3×C4—241no—
389C3×C4—361no—
390Dic3—22no—
391Dic3—41no—
392Dic3—62no—
393Dic3—62no—
394Dic3—92no—
395Dic3—122no—
396Dic3—121no—
397Dic3—121no—
398Dic3—182no—
399Dic3—182no—
400Dic3—182no—
401Dic3—241no—
402Dic3—362no—
403Dic3—362no—
404Dic3#377312yescomplemented
405C32—21no—
406C32—21no—
407C32—41no—
408C32—41no—
409C32#464711yescharacteristic, complemented
410C23—91no—
411C2×C4—31no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—92no—
415C2×C4—92no—
416C2×C4—181no—
417C2×C4—181no—
418C2×C4—181no—
419C2×C4—181no—
420C2×C4—181no—
421C2×C4—181no—
422C2×C4—181no—
423C2×C4—272no—
424C2×C4—272no—
425C2×C4—272no—
426C2×C4—271no—
427C2×C4—272no—
428C2×C4—541no—
429C2×C4—541no—
430C2×C4—541no—
431C2×C4—541no—
432C2×C4—541no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—41no—
445C2×C3—91no—
446C2×C3—91no—
447C2×C3—91no—
448C2×C3—91no—
449C2×C3D32⋊Q811yescharacteristic
450C2×C3#6423011yescharacteristic
451C2×C3C2×#377311yescharacteristic
452D3—61no—
453D3—61no—
454D3—61no—
455D3—61no—
456D3—61no—
457D3—61no—
458D3—61no—
459D3—61no—
460C22—91no—
461C22—91no—
462C22—91no—
463C22—91no—
464C22—91no—
465C22—91no—
466C22D3×GO+(4,𝔽2)11yescharacteristic
467C4—32no—
468C4—62no—
469C4—121no—
470C4—182no—
471C4—182no—
472C4—272no—
473C4—361no—
474C4—542no—
475C3—21no—
476C3—21no—
477C3—41no—
478C3—41no—
479C3#1059341211yescharacteristic, complemented
480C2—91no—
481C2—91no—
482C2—91no—
483C2—91no—
484C2D3×#377311yescharacteristic
485C2#1172411811yescharacteristic
486C2#1172411911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
Fitting, socleC22×C33C2×D4
prime radical, cosocleC22×C3×C3⋊D3C23
hypocenterC33#389
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
18. Dic3⋊#157C4semidirect
95. C3⋊(C3⋊Dic3)C4⋊C4semidirect
100. C32×Dic3C4⋊C4semidirect
259. C33#389semidirect
404. Dic3#3773semidirect
409. C32#4647semidirect
479. C3#10593412semidirect

Maximal subgroups

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

subgroupindex
C2×#889402
C2×#889342
C2×Dic3⋊#1572
#117229112
Dic3×#6242
#117227692
C3×#637372
#105934123
#46479