Finite Group Databasebeta

← #99×(C3⋊D3)⋊C3 #13141655

Subgroups — #99×(C3⋊D3)⋊C3 #13141655

Order 1728.

787 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 787 conjugacy classes of subgroups — too many to draw. Only its 118 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×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C22⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
3H(3,𝔽3)×#99C211yesmaximal, characteristic, complemented
4#63661⋊C3C211yesmaximal, characteristic, complemented
5#63633⋊C3C212yesmaximal, complemented
6C3⋊D3×#99C311yesmaximal, characteristic, complemented
7C3×D3×#99—31nomaximal
8C23×(C3⋊D3)⋊C3C411yescharacteristic, complemented
9C23×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3C411yescharacteristic, complemented
12C2×#638⋊C3C2211yescharacteristic
13D4×(C3⋊D3)⋊C3—22no—
14D4×(C3⋊D3)⋊C3—22no—
15C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic, complemented
16C2×C4×(C3⋊D3)⋊C3—22no—
17C2×C4×(C3⋊D3)⋊C3C411yescharacteristic, complemented
18C22⋊C4×H(3,𝔽3)C2212yescomplemented
19#3757⋊C3C2212yescomplemented
20#3752⋊C3—22no—
21C2×D4×C3⋊D3C2×C311yescharacteristic
22C2×C3×D3×D4—31no—
23C22⋊C4×C3⋊D3C2×C312yescomplemented
24C3×D3×C22⋊C4—32no—
25C32×#99—21no—
26C32×#99D311yescharacteristic, complemented
27C32×#99C2×C311yescharacteristic, complemented
28#63661C2×C311yescharacteristic, complemented
29#63633C2×C312yescomplemented
30C3×#834—31no—
31C3×#806—32no—
32C23×H(3,𝔽3)C2×C411yescharacteristic, complemented
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×(C3⋊D3)⋊C3—21no—
35C22×(C3⋊D3)⋊C3—21no—
36C22×(C3⋊D3)⋊C3—42no—
37C22×(C3⋊D3)⋊C3—42no—
38C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
39C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
40C22×(C3⋊D3)⋊C3C2×C411yescharacteristic, complemented
41C22×(C3⋊D3)⋊C3D412yes—
42C22×(C3⋊D3)⋊C3D412yes—
43D4×H(3,𝔽3)—22no—
44C2×C4×H(3,𝔽3)—22no—
45C2×C4×H(3,𝔽3)C2×C411yescharacteristic, complemented
46#638⋊C3—22no—
47#638⋊C3—22no—
48C2×(C3⋊Dic3)⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic, complemented
50C4⋊((C3⋊D3)⋊C3)—22no—
51C4×(C3⋊D3)⋊C3—21no—
52C4×(C3⋊D3)⋊C3—21no—
53C4×(C3⋊D3)⋊C3—44no—
54C2×C3×#99—91no—
55D3×#99—31no—
56D3×#99—91no—
57C23×C3⋊D3C3×C411yescharacteristic, complemented
58C23×C3⋊D3C22×C311yescharacteristic
59C23×C3×D3—31no—
60C23×C3×D3—31no—
61C2×C32×D4—21no—
62C2×C32×D4C2×D311yescharacteristic
63C2×C32×D4C22×C311yescharacteristic
64C2×#638C3×C411yescharacteristic, complemented
65C2×#638C22×C311yescharacteristic
66D4×C3⋊D3—22no—
67D4×C3⋊D3—22no—
68C2×C4⋊(C3⋊D3)C3×C411yescharacteristic, complemented
69C2×C4×C3⋊D3—22no—
70C2×C4×C3⋊D3C3×C411yescharacteristic, complemented
71C2×C3×#67—31no—
72C2×C3×#67—31no—
73C3×D3×D4—62no—
74C3×D3×D4—62no—
75C2×C3×D12—31no—
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—62no—
78C32×C22⋊C4—22no—
79C32×C22⋊C4C2×D312yescomplemented
80C32×C22⋊C4C22×C312yescomplemented
81#3757C22×C312yescomplemented
82#3752—22no—
83C3×#217—32no—
84C3×(C2×D3)⋊C4—62no—
85C22×H(3,𝔽3)—21no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)C22×C411yescharacteristic
89C22×H(3,𝔽3)C2×D412yes—
90C2×(C3⋊D3)⋊C3—24no—
91C2×(C3⋊D3)⋊C3—22no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—42no—
95C2×(C3⋊D3)⋊C3—42no—
96C2×(C3⋊D3)⋊C3—42no—
97C2×(C3⋊D3)⋊C3—41no—
98C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
99C2×(C3⋊D3)⋊C3C22⋊C412yes—
100C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
101C4×H(3,𝔽3)—21no—
102C4×H(3,𝔽3)—42no—
103(C3⋊Dic3)⋊C3—21no—
104(C3⋊Dic3)⋊C3—42no—
105C22×C3×D4—91no—
106C2×D3×D4—31no—
107C2×D3×D4—91no—
108C2×C3×C22⋊C4—92no—
109D3×C22⋊C4—32no—
110D3×C22⋊C4—92no—
111C3×#99—31no—
112C3×#99—31no—
113C3×#99—61no—
114C3×#99—92no—
115C3×#99—91no—
116C3×#99C3×D311yescharacteristic
117#834—31no—
118#834—91no—
119#806—32no—
120#806—92no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32C4×D311yescharacteristic, complemented
124C23×C32C2×C3×C411yescharacteristic, complemented
125C23×C32C22×D311yescharacteristic
126C23×C32C23×C311yescharacteristic
127C22×C3⋊D3—21no—
128C22×C3⋊D3—21no—
129C22×C3⋊D3—42no—
130C22×C3⋊D3—42no—
131C22×C3⋊D3C2×C3×C411yescharacteristic
132C22×C3⋊D3C2×C3×C411yescharacteristic
133C22×C3⋊D3C2×C3×C411yescharacteristic, complemented
134C22×C3⋊D3C3×D412yes—
135C22×C3⋊D3C3×D412yes—
136C22×C3×D3—31no—
137C22×C3×D3—32no—
138C22×C3×D3—31no—
139C22×C3×D3—32no—
140C22×C3×D3—31no—
141C22×C3×D3—61no—
142C22×C3×D3—61no—
143C22×C3×D3—122no—
144C22×C3×D3—122no—
145C32×D4—22no—
146C32×D4—22no—
147C32×D4—42no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—42no—
152C2×C32×C4C4×D311yescharacteristic, complemented
153C2×C32×C4C2×C3×C411yescharacteristic, complemented
154#638—22no—
155#638—22no—
156C2×C3⋊Dic3—22no—
157C2×C3⋊Dic3C2×C3×C411yescharacteristic, complemented
158C4⋊(C3⋊D3)—22no—
159C4×C3⋊D3—21no—
160C4×C3⋊D3—21no—
161C4×C3⋊D3—44no—
162C3×#67—62no—
163C3×#67—62no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—62no—
166C3×D12—62no—
167C3×C4×D3—61no—
168C3×C4×D3—61no—
169C3×C4×D3—124no—
170C2×#99—91noSylow
171C2×H(3,𝔽3)—22no—
172C2×H(3,𝔽3)—21no—
173C2×H(3,𝔽3)—41no—
174C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
175(C3⋊D3)⋊C3—21no—
176(C3⋊D3)⋊C3—22no—
177(C3⋊D3)⋊C3—41no—
178(C3⋊D3)⋊C3#9912yesdirect product factor
179C24×C3—91no—
180C24×C3—91no—
181C23×D3—31no—
182C23×D3—31no—
183C23×D3—91no—
184C23×D3—91no—
185C2×C3×D4—31no—
186C2×C3×D4—31no—
187C2×C3×D4—61no—
188C2×C3×D4—91no—
189C2×C3×D4—91no—
190C2×C3×D4—91no—
191C2×C3×D4—182no—
192C2×C3×D4—182no—
193C2×C3×D4C2×C3×D311yescharacteristic
194C22×C3×C4—91no—
195C22×C3×C4—182no—
196C2×#67—31no—
197C2×#67—31no—
198C2×#67—91no—
199C2×#67—91no—
200D3×D4—62no—
201D3×D4—62no—
202D3×D4—182no—
203D3×D4—182no—
204C2×D12—31no—
205C2×D12—91no—
206C2×C4×D3—31no—
207C2×C4×D3—62no—
208C2×C4×D3—91no—
209C2×C4×D3—182no—
210C3×C22⋊C4—32no—
211C3×C22⋊C4—32no—
212C3×C22⋊C4—62no—
213C3×C22⋊C4—92no—
214C3×C22⋊C4—182no—
215C3×C22⋊C4C2×C3×D312yes—
216#217—32no—
217#217—92no—
218(C2×D3)⋊C4—62no—
219(C2×D3)⋊C4—182no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—22no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32—41no—
229C22×C32—81no—
230C22×C32—81no—
231C22×C32C2×C4×D311yescharacteristic
232C22×C32D3×D412yes—
233C22×C32C22×C3×C411yescharacteristic
234C22×C32C2×C3×D412yes—
235C2×C3⋊D3—24no—
236C2×C3⋊D3—22no—
237C2×C3⋊D3—21no—
238C2×C3⋊D3—41no—
239C2×C3⋊D3—42no—
240C2×C3⋊D3—42no—
241C2×C3⋊D3—42no—
242C2×C3⋊D3—41no—
243C2×C3⋊D3C3×C22⋊C411yescharacteristic
244C2×C3⋊D3C3×C22⋊C412yes—
245C2×C3⋊D3C3×C22⋊C411yescharacteristic
246C2×C3×D3—31no—
247C2×C3×D3—31no—
248C2×C3×D3—32no—
249C2×C3×D3—64no—
250C2×C3×D3—62no—
251C2×C3×D3—61no—
252C2×C3×D3—122no—
253C2×C3×D3—121no—
254C2×C3×D3—122no—
255C2×C3×D3—122no—
256C2×C3×D3—121no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—41no—
260C32×C4—42no—
261C32×C4—42no—
262C32×C4—82no—
263C3⋊Dic3—21no—
264C3⋊Dic3—42no—
265C3×Dic3—61no—
266C3×Dic3—122no—
267C22×D4—91no—
268C2×C22⋊C4—92no—
269#99—91no—
270#99—92no—
271#99(C3⋊D3)⋊C311yescharacteristic, direct product factor
272H(3,𝔽3)C2×#9911yescharacteristic, complemented, Sylow
273C23×C3—31no—
274C23×C3—31no—
275C23×C3—31no—
276C23×C3—31no—
277C23×C3—61no—
278C23×C3—61no—
279C23×C3—91no—
280C23×C3—92no—
281C23×C3—92no—
282C23×C3—91no—
283C23×C3—91no—
284C23×C3—181no—
285C23×C3—181no—
286C23×C3—362no—
287C23×C3—362no—
288C23×C3C3×C4×D311yescharacteristic
289C23×C3C22×C3×D311yescharacteristic
290C22×D3—31no—
291C22×D3—32no—
292C22×D3—31no—
293C22×D3—32no—
294C22×D3—31no—
295C22×D3—61no—
296C22×D3—61no—
297C22×D3—91no—
298C22×D3—92no—
299C22×D3—91no—
300C22×D3—92no—
301C22×D3—91no—
302C22×D3—122no—
303C22×D3—122no—
304C22×D3—181no—
305C22×D3—181no—
306C22×D3—362no—
307C22×D3—362no—
308C3×D4—22no—
309C3×D4—62no—
310C3×D4—62no—
311C3×D4—122no—
312C3×D4—182no—
313C3×D4—182no—
314C3×D4—182no—
315C2×C3×C4—22no—
316C2×C3×C4—31no—
317C2×C3×C4—31no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—61no—
321C2×C3×C4—91no—
322C2×C3×C4—122no—
323C2×C3×C4—181no—
324C2×C3×C4—181no—
325C2×C3×C4—182no—
326C2×C3×C4—364no—
327C2×C3×C4C3×C4×D311yescharacteristic
328#67—62no—
329#67—62no—
330#67—182no—
331#67—182no—
332C2×Dic3—31no—
333C2×Dic3—62no—
334C2×Dic3—91no—
335C2×Dic3—182no—
336D12—62no—
337D12—182no—
338C4×D3—61no—
339C4×D3—61no—
340C4×D3—124no—
341C4×D3—181no—
342C4×D3—181no—
343C4×D3—364no—
344C2×C32—22no—
345C2×C32—21no—
346C2×C32—22no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—42no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—41no—
353C2×C32—81no—
354C2×C32D3×C22⋊C411yescharacteristic
355C2×C32C2×C3×C22⋊C411yescharacteristic
356C3⋊D3—21no—
357C3⋊D3—22no—
358C3⋊D3—41no—
359C3⋊D3C3×#9912yescomplemented
360C3×D3—32no—
361C3×D3—62no—
362C3×D3—61no—
363C3×D3—121no—
364C24—91no—
365C24—91no—
366C2×D4—91no—
367C2×D4—91no—
368C2×D4—91no—
369C2×D4—182no—
370C2×D4—182no—
371C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
372C22×C4—91no—
373C22×C4—182no—
374C22⋊C4—92no—
375C22⋊C4—182no—
376C22⋊C4C2×(C3⋊D3)⋊C312yescomplemented
377C22×C3—21no—
378C22×C3—31no—
379C22×C3—31no—
380C22×C3—32no—
381C22×C3—32no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—61no—
385C22×C3—61no—
386C22×C3—61no—
387C22×C3—62no—
388C22×C3—91no—
389C22×C3—91no—
390C22×C3—92no—
391C22×C3—121no—
392C22×C3—121no—
393C22×C3—121no—
394C22×C3—121no—
395C22×C3—121no—
396C22×C3—181no—
397C22×C3—184no—
398C22×C3—182no—
399C22×C3—241no—
400C22×C3—241no—
401C22×C3—362no—
402C22×C3—361no—
403C22×C3—361no—
404C22×C3—362no—
405C22×C3—362no—
406C22×C3C2×C3×C4×D311yescharacteristic
407C22×C3C3×D3×D412yes—
408C2×D3—31no—
409C2×D3—31no—
410C2×D3—32no—
411C2×D3—64no—
412C2×D3—62no—
413C2×D3—61no—
414C2×D3—91no—
415C2×D3—91no—
416C2×D3—92no—
417C2×D3—122no—
418C2×D3—122no—
419C2×D3—121no—
420C2×D3—122no—
421C2×D3—121no—
422C2×D3—184no—
423C2×D3—182no—
424C2×D3—181no—
425C2×D3—362no—
426C2×D3—362no—
427C2×D3—361no—
428C2×D3—362no—
429C2×D3—361no—
430C3×C4—21no—
431C3×C4—42no—
432C3×C4—61no—
433C3×C4—61no—
434C3×C4—121no—
435C3×C4—122no—
436C3×C4—122no—
437C3×C4—181no—
438C3×C4—242no—
439C3×C4—362no—
440Dic3—61no—
441Dic3—122no—
442Dic3—181no—
443Dic3—362no—
444C32—21no—
445C32D3×#9911yescharacteristic, complemented
446C32C2×C3×#9911yescharacteristic, complemented
447C23—91no—
448C23—92no—
449C23—91no—
450C23—92no—
451C23—91no—
452C23—181no—
453C23—181no—
454C23—362no—
455C23—362no—
456C23C4×(C3⋊D3)⋊C311yescharacteristic, complemented
457C23C22×(C3⋊D3)⋊C311yescharacteristic
458D4—22no—
459D4—182no—
460D4—182no—
461D4—182no—
462C2×C4—22no—
463C2×C4—91no—
464C2×C4—181no—
465C2×C4—181no—
466C2×C4—182no—
467C2×C4—364no—
468C2×C4C4×(C3⋊D3)⋊C311yescharacteristic, complemented
469C2×C3—22no—
470C2×C3—21no—
471C2×C3—31no—
472C2×C3—31no—
473C2×C3—41no—
474C2×C3—62no—
475C2×C3—62no—
476C2×C3—61no—
477C2×C3—61no—
478C2×C3—61no—
479C2×C3—92no—
480C2×C3—122no—
481C2×C3—121no—
482C2×C3—121no—
483C2×C3—121no—
484C2×C3—181no—
485C2×C3—182no—
486C2×C3—241no—
487C2×C3—361no—
488C2×C3C3×D3×C22⋊C411yescharacteristic
489D3—32no—
490D3—61no—
491D3—62no—
492D3—92no—
493D3—121no—
494D3—181no—
495D3—182no—
496D3—361no—
497C22—21no—
498C22—41no—
499C22—41no—
500C22—91no—
501C22—91no—
502C22—92no—
503C22—184no—
504C22—182no—
505C22—181no—
506C22—362no—
507C22—361no—
508C22—361no—
509C22—362no—
510C22—362no—
511C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
512C22D4×(C3⋊D3)⋊C312yes—
513C4—21no—
514C4—42no—
515C4—181no—
516C4—362no—
517C3—31no—
518C3—31no—
519C3—61no—
520C3C3×D3×#9911yescharacteristic
521C2—22no—
522C2—21no—
523C2—41no—
524C2—92no—
525C2—181no—
526C2—182no—
527C2—361no—
528C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2C22⋊C4×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC23×C3C22×C3×D3
FittingH(3,𝔽3)×#99C2
socleC2×C3C3×D3×C22⋊C4
prime radicalC2×D4×(C3⋊D3)⋊C3C2
cosocleC23×C32C23×C3
hypercenter#99(C3⋊D3)⋊C3
hypocenterC32C2×C3×#99
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. C22⋊C4×(C3⋊D3)⋊C3C2semidirect
3. H(3,𝔽3)×#99C2semidirect
4. #63661⋊C3C2semidirect
5. #63633⋊C3C2semidirect
6. C3⋊D3×#99C3semidirect
8. C23×(C3⋊D3)⋊C3C4semidirect
11. C2×#638⋊C3C4semidirect
15. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
17. C2×C4×(C3⋊D3)⋊C3C4semidirect
18. C22⋊C4×H(3,𝔽3)C22semidirect
19. #3757⋊C3C22semidirect
23. C22⋊C4×C3⋊D3C2×C3semidirect
26. C32×#99D3semidirect
27. C32×#99C2×C3semidirect
28. #63661C2×C3semidirect
29. #63633C2×C3semidirect
32. C23×H(3,𝔽3)C2×C4semidirect
40. C22×(C3⋊D3)⋊C3C2×C4semidirect
45. C2×C4×H(3,𝔽3)C2×C4semidirect
49. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
57. C23×C3⋊D3C3×C4semidirect
64. C2×#638C3×C4semidirect
68. C2×C4⋊(C3⋊D3)C3×C4semidirect
70. C2×C4×C3⋊D3C3×C4semidirect
79. C32×C22⋊C4C2×D3semidirect
80. C32×C22⋊C4C22×C3semidirect
81. #3757C22×C3semidirect
123. C23×C32C4×D3semidirect
124. C23×C32C2×C3×C4semidirect
133. C22×C3⋊D3C2×C3×C4semidirect
152. C2×C32×C4C4×D3semidirect
153. C2×C32×C4C2×C3×C4semidirect
157. C2×C3⋊Dic3C2×C3×C4semidirect
178. (C3⋊D3)⋊C3#99direct (N × H)
271. #99(C3⋊D3)⋊C3direct (N × H)
272. H(3,𝔽3)C2×#99semidirect
359. C3⋊D3C3×#99semidirect
376. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
445. C32D3×#99semidirect
446. C32C2×C3×#99semidirect
456. C23C4×(C3⋊D3)⋊C3semidirect
468. C2×C4C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
C22⋊C4×(C3⋊D3)⋊C32
H(3,𝔽3)×#992
#63661⋊C32
#63633⋊C32
C3⋊D3×#993
C3×D3×#993