Finite Group Databasebeta

← #10596116⋊C3 #13152966

Subgroups — #10596116⋊C3 #13152966

Order 1728.

837 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic, complemented
2#64155⋊C3C212yesmaximal, complemented
3#64139⋊C3C211yesmaximal, characteristic
4C2×#3750⋊C3C211yesmaximal, characteristic, complemented
5#64107⋊C3C212yesmaximal, complemented
6#64103⋊C3C212yesmaximal, complemented
7#64099⋊C3C212yesmaximal, complemented
8#64095⋊C3C212yesmaximal, complemented
9#64090⋊C3C212yesmaximal, complemented
10#10596116C311yesmaximal, characteristic, complemented
11C3×#5645—31nomaximal
12C2×Q8×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C2×#638⋊C3C2212yes—
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2212yes—
21C22⋊C4×H(3,𝔽3)C2212yes—
22#3752⋊C3C2212yes—
23#3752⋊C3C2212yes—
24#3752⋊C3C2212yes—
25#3752⋊C3C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2212yes—
29#3750⋊C3C2212yes—
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32C4×(C3⋊Dic3)⋊C3C2212yes—
33C32×C22⋊Q8—21no—
34C32×C22⋊Q8D311yescharacteristic, complemented
35C32×C22⋊Q8C2×C311yescharacteristic, complemented
36#64155C2×C312yescomplemented
37#64139C2×C311yescharacteristic
38C2×#3750C2×C311yescharacteristic, complemented
39#64107C2×C312yescomplemented
40#64103C2×C312yescomplemented
41#64099C2×C312yescomplemented
42#64095C2×C312yescomplemented
43#64090C2×C312yescomplemented
44C3×#946—32no—
45C3×#930—31no—
46C2×C3×Dic3⋊C4—31no—
47C3×#898—32no—
48C3×#894—32no—
49C3×#890—32no—
50C3×#886—32no—
51C3×#67⋊C4—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C22×(C3⋊D3)⋊C3C2312yes—
54Q8×H(3,𝔽3)—41no—
55C2×C4×H(3,𝔽3)—41no—
56C2×C4×H(3,𝔽3)C2312yes—
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58C2×C4×H(3,𝔽3)C2312yes—
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60#638⋊C3—24no—
61C2×(C3⋊Dic3)⋊C3—22no—
62C2×(C3⋊Dic3)⋊C3—41no—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C4⋊((C3⋊D3)⋊C3)—41no—
67C4×(C3⋊D3)⋊C3—42no—
68C3×#539—91no—
69#5645—31no—
70#5645—91no—
71C2×C32×Q8—21no—
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C22×C311yescharacteristic
74C22×C32×C4—21no—
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C22×C311yescharacteristic
77C2×#638C22×C312yes—
78C22×C3⋊Dic3C22×C311yescharacteristic
79C22×C3⋊Dic3C22×C311yescharacteristic
80C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
81C2×C4×C3⋊D3C22×C312yes—
82C2×C3×#67—32no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C2×C3×D12—31no—
86C2×C3×C4×D3—32no—
87C32×C4⋊C4—21no—
88C32×C4⋊C4—22no—
89C32×C4⋊C4C2×D312yes—
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C4⋊C4C22×C312yes—
92C32×C4⋊C4C22×C311yescharacteristic
93C32×C22⋊C4—22no—
94C32×C22⋊C4C2×D312yes—
95C32×C22⋊C4C22×C312yes—
96#3752C22×C312yes—
97#3752C22×C312yes—
98#3752C22×C312yes—
99#3752C22×C312yes—
100C4⋊(C3⋊Dic3)C22×C312yes—
101C4⋊(C3⋊Dic3)C22×C312yes—
102C4⋊(C3⋊Dic3)C22×C311yescharacteristic
103#3750C22×C312yes—
104#3750C22×C312yes—
105#3750C22×C312yes—
106C4×C3⋊Dic3C22×C312yes—
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×C4⋊Dic3—32no—
112C3×C4⋊Dic3—32no—
113C3×C4⋊Dic3—31no—
114C3×Dic3⋊C4—32no—
115C3×Dic3⋊C4—32no—
116C3×Dic3⋊C4—32no—
117C3×C4×Dic3—32no—
118C22×H(3,𝔽3)—21no—
119C22×H(3,𝔽3)—21no—
120C22×H(3,𝔽3)#4112yes—
121C22×H(3,𝔽3)C2411yescharacteristic
122C2×(C3⋊D3)⋊C3—22no—
123C2×(C3⋊D3)⋊C3—22no—
124C2×(C3⋊D3)⋊C3—22no—
125C4×H(3,𝔽3)—21no—
126C4×H(3,𝔽3)—22no—
127C4×H(3,𝔽3)—21no—
128C4×H(3,𝔽3)—22no—
129(C3⋊Dic3)⋊C3—22no—
130(C3⋊Dic3)⋊C3—22no—
131(C3⋊Dic3)⋊C3—22no—
132C3×#126—92no—
133C3×#125—92no—
134C3×#123—92no—
135C3×#123—92no—
136C3×C22⋊Q8—31no—
137C3×C22⋊Q8—31no—
138C3×C22⋊Q8—61no—
139C3×C22⋊Q8—92no—
140C3×C22⋊Q8C3×D311yescharacteristic
141C3×#121—91no—
142C3×C4×D4—92no—
143C2×C3×C4⋊C4—91no—
144#946—32no—
145#946—92no—
146#930—31no—
147#930—91no—
148C2×Dic3⋊C4—31no—
149C2×Dic3⋊C4—91no—
150#898—32no—
151#898—92no—
152#894—32no—
153#894—92no—
154#890—32no—
155#890—92no—
156#886—32no—
157#886—92no—
158#67⋊C4—32no—
159#67⋊C4—92no—
160C23×C32—21no—
161C23×C32C22×D311yescharacteristic
162C23×C32C23×C311yescharacteristic
163C22×C3⋊D3C23×C312yes—
164C22×C3×D3—32no—
165C32×Q8—41no—
166C32×Q8—41no—
167C32×Q8—42no—
168C2×C32×C4—21no—
169C2×C32×C4—22no—
170C2×C32×C4—22no—
171C2×C32×C4—21no—
172C2×C32×C4—41no—
173C2×C32×C4—41no—
174C2×C32×C4—42no—
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D312yes—
179C2×C32×C4C23×C312yes—
180C2×C32×C4C23×C311yescharacteristic
181C2×C32×C4C23×C311yescharacteristic
182C2×C32×C4C23×C312yes—
183#638—24no—
184C2×C3⋊Dic3—22no—
185C2×C3⋊Dic3—41no—
186C2×C3⋊Dic3C23×C312yes—
187C2×C3⋊Dic3C23×C312yes—
188C2×C3⋊Dic3C23×C312yes—
189C4⋊(C3⋊D3)—41no—
190C4×C3⋊D3—42no—
191C3×#67—64no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—121no—
197C3×D12—121no—
198C3×C4×D3—122no—
199#539—91noSylow
200C2×H(3,𝔽3)—22no—
201C2×H(3,𝔽3)C2×#4111yescharacteristic
202C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
203C2×H(3,𝔽3)2−1+411yescharacteristic
204(C3⋊D3)⋊C3—42no—
205C2×C3×Q8—31no—
206C2×C3×Q8—31no—
207C2×C3×Q8—61no—
208C2×C3×Q8C2×C3×D311yescharacteristic
209C2×C3×D4—91no—
210C2×C3×D4—92no—
211C22×C3×C4—31no—
212C22×C3×C4—31no—
213C22×C3×C4—61no—
214C22×C3×C4—92no—
215C22×C3×C4—91no—
216C22×C3×C4—91no—
217C22×C3×C4C2×C3×D311yescharacteristic
218C2×#67—32no—
219C2×#67—92no—
220C22×Dic3—31no—
221C22×Dic3—31no—
222C22×Dic3—91no—
223C22×Dic3—91no—
224C2×D12—31no—
225C2×D12—91no—
226C2×C4×D3—32no—
227C2×C4×D3—92no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—32no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—62no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4—92no—
240C3×C4⋊C4C2×C3×D312yes—
241C3×C4⋊C4C2×C3×D311yescharacteristic
242C3×C22⋊C4—32no—
243C3×C22⋊C4—32no—
244C3×C22⋊C4—62no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249C3×C22⋊C4C2×C3×D312yes—
250C3×C42—92no—
251(C2×D3)⋊C4—32no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—32no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259C4⋊Dic3—32no—
260C4⋊Dic3—32no—
261C4⋊Dic3—31no—
262C4⋊Dic3—92no—
263C4⋊Dic3—92no—
264C4⋊Dic3—91no—
265Dic3⋊C4—32no—
266Dic3⋊C4—32no—
267Dic3⋊C4—32no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—92no—
271C4×Dic3—32no—
272C4×Dic3—92no—
273C22×C32—21no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—22no—
279C22×C32—41no—
280C22×C32—41no—
281C22×C32#23912yes—
282C22×C32C3×#4112yes—
283C22×C32C23×D311yescharacteristic
284C22×C32C24×C311yescharacteristic
285C2×C3⋊D3—22no—
286C2×C3⋊D3—22no—
287C2×C3⋊D3—22no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C32×C4—22no—
292C32×C4—22no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—22no—
296C32×C4—22no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—22no—
300C32×C4—42no—
301C32×C4—41no—
302C32×C4—42no—
303C3⋊Dic3—22no—
304C3⋊Dic3—22no—
305C3⋊Dic3—22no—
306C3×Dic3—62no—
307C3×Dic3—62no—
308C3×Dic3—62no—
309#126—92no—
310#125—92no—
311#123—92no—
312#123—92no—
313C22⋊Q8—92no—
314C22⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
315#121—91no—
316C4×D4—92no—
317C2×C4⋊C4—91no—
318H(3,𝔽3)#53911yescharacteristic, complemented, Sylow
319C23×C3—31no—
320C23×C3—31no—
321C23×C3—61no—
322C23×C3—92no—
323C23×C3C22×C3×D311yescharacteristic
324C22×D3—32no—
325C22×D3—92no—
326C3×Q8—41no—
327C3×Q8—121no—
328C3×Q8—121no—
329C3×Q8—122no—
330C3×D4—184no—
331C3×D4—361no—
332C2×C3×C4—32no—
333C2×C3×C4—32no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—32no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—32no—
340C2×C3×C4—41no—
341C2×C3×C4—61no—
342C2×C3×C4—62no—
343C2×C3×C4—62no—
344C2×C3×C4—61no—
345C2×C3×C4—92no—
346C2×C3×C4—92no—
347C2×C3×C4—92no—
348C2×C3×C4—121no—
349C2×C3×C4—121no—
350C2×C3×C4—122no—
351C2×C3×C4—182no—
352C2×C3×C4—362no—
353C2×C3×C4—361no—
354C2×C3×C4C22×C3×D312yes—
355C2×C3×C4C22×C3×D311yescharacteristic
356C2×C3×C4C22×C3×D311yescharacteristic
357C2×C3×C4C22×C3×D312yes—
358#67—64no—
359#67—184no—
360C2×Dic3—32no—
361C2×Dic3—32no—
362C2×Dic3—32no—
363C2×Dic3—62no—
364C2×Dic3—92no—
365C2×Dic3—92no—
366C2×Dic3—92no—
367C2×Dic3—121no—
368C2×Dic3—182no—
369C2×Dic3—361no—
370D12—121no—
371D12—361no—
372C4×D3—122no—
373C4×D3—362no—
374C2×C32—22no—
375C2×C32—22no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—42no—
380C2×C32#100411yescharacteristic
381C2×C32C2×#23911yescharacteristic
382C2×C32#101011yescharacteristic
383C2×C32C2×C3×#4111yescharacteristic
384C2×C32C3×H(5,𝔽2)11yescharacteristic
385C2×C32C3×2−1+411yescharacteristic
386C3⋊D3—42no—
387C3×D3—122no—
388C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
389C2×D4—92no—
390C2×D4—91no—
391C22×C4—91no—
392C22×C4—91no—
393C22×C4—92no—
394C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
395C4⋊C4—92no—
396C4⋊C4—92no—
397C4⋊C4—92no—
398C4⋊C4—91no—
399C4⋊C4—92no—
400C4⋊C4—92no—
401C4⋊C4C2×(C3⋊D3)⋊C312yes—
402C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
403C22⋊C4—92no—
404C22⋊C4—92no—
405C22⋊C4—92no—
406C22⋊C4—92no—
407C22⋊C4C2×(C3⋊D3)⋊C312yes—
408C42—92no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—31no—
412C22×C3—31no—
413C22×C3—32no—
414C22×C3—32no—
415C22×C3—61no—
416C22×C3—61no—
417C22×C3—61no—
418C22×C3—61no—
419C22×C3—61no—
420C22×C3—62no—
421C22×C3—121no—
422C22×C3—121no—
423C22×C3—182no—
424C22×C3—182no—
425C22×C3—182no—
426C22×C3C3×#23912yes—
427C22×C3C23×C3×D311yescharacteristic
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—182no—
432C2×D3—182no—
433C2×D3—182no—
434C3×C4—22no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—22no—
438C3×C4—62no—
439C3×C4—61no—
440C3×C4—61no—
441C3×C4—61no—
442C3×C4—61no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—62no—
446C3×C4—62no—
447C3×C4—121no—
448C3×C4—122no—
449C3×C4—122no—
450C3×C4—182no—
451C3×C4—182no—
452C3×C4—182no—
453Dic3—62no—
454Dic3—62no—
455Dic3—62no—
456Dic3—182no—
457Dic3—182no—
458Dic3—182no—
459C32—21no—
460C32#564511yescharacteristic, complemented
461C32C3×#53911yescharacteristic, complemented
462C23—92no—
463C23C22×(C3⋊D3)⋊C311yescharacteristic
464Q8—41no—
465D4—184no—
466D4—361no—
467C2×C4—41no—
468C2×C4—92no—
469C2×C4—92no—
470C2×C4—92no—
471C2×C4—182no—
472C2×C4—362no—
473C2×C4—361no—
474C2×C4C22×(C3⋊D3)⋊C312yes—
475C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
476C2×C4C22×(C3⋊D3)⋊C312yes—
477C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
478C2×C3—22no—
479C2×C3—31no—
480C2×C3—31no—
481C2×C3—31no—
482C2×C3—31no—
483C2×C3—31no—
484C2×C3—31no—
485C2×C3—62no—
486C2×C3—62no—
487C2×C3—61no—
488C2×C3—61no—
489C2×C3—61no—
490C2×C3—122no—
491C2×C3—362no—
492C2×C3C3×#100411yescharacteristic
493C2×C3C2×C3×#23911yescharacteristic
494C2×C3C3×#101011yescharacteristic
495D3—122no—
496D3—362no—
497C22—21no—
498C22—21no—
499C22—182no—
500C22—182no—
501C22—182no—
502C22#8872212yes—
503C22C23×(C3⋊D3)⋊C311yescharacteristic
504C4—21no—
505C4—21no—
506C4—22no—
507C4—22no—
508C4—182no—
509C4—182no—
510C4—182no—
511C3—31no—
512C3—31no—
513C3—61no—
514C3C3×#564511yescharacteristic
515C2—22no—
516C2—362no—
517C2#64361⋊C311yescharacteristic
518C2C2×#8872211yescharacteristic
519C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C22⋊Q8C2
prime radical#64139⋊C3C2
hypercenterC22⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#539
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C22⋊Q8C2semidirect
2. #64155⋊C3C2semidirect
4. C2×#3750⋊C3C2semidirect
5. #64107⋊C3C2semidirect
6. #64103⋊C3C2semidirect
7. #64099⋊C3C2semidirect
8. #64095⋊C3C2semidirect
9. #64090⋊C3C2semidirect
10. #10596116C3semidirect
34. C32×C22⋊Q8D3semidirect
35. C32×C22⋊Q8C2×C3semidirect
36. #64155C2×C3semidirect
38. C2×#3750C2×C3semidirect
39. #64107C2×C3semidirect
40. #64103C2×C3semidirect
41. #64099C2×C3semidirect
42. #64095C2×C3semidirect
43. #64090C2×C3semidirect
314. C22⋊Q8(C3⋊D3)⋊C3semidirect
318. H(3,𝔽3)#539semidirect
460. C32#5645semidirect
461. C32C3×#539semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
#64155⋊C32
#64139⋊C32
C2×#3750⋊C32
#64107⋊C32
#64103⋊C32
#64099⋊C32
#64095⋊C32
#64090⋊C32
#105961163
C3×#56453