Finite Group Databasebeta

← C3×#70722 #11787007

Subgroups — C3×#70722 #11787007

Order 1152.

593 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#907C211yesmaximal, characteristic
2C2×C32×#108C211yesmaximal, characteristic, complemented
3C2×C3×#825C211yesmaximal, characteristic
4C3×#5119C211yesmaximal, characteristic, complemented
5C3×#5113C211yesmaximal, characteristic
6C3×#4555C211yesmaximal, characteristic, complemented
7C3×#4550C211yesmaximal, characteristic, complemented
8C3×#1866—31nomaximal
9#70722C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C22×C32×C8C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×C4×Dic3C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C3×#907—22no—
18C3×#907—22no—
19C2×C3×ΓL(1,𝔽25)—21no—
20C2×C3×D3×C8—21no—
21C32×#108—22no—
22C3×#825—22no—
23C3×#821C411yescharacteristic, complemented
24C3×#821C411yescharacteristic, complemented
25C3×#816C411yescharacteristic, complemented
26C3×#816C411yescharacteristic, complemented
27C3×Dic3⋊C8—21no—
28C3×Dic3⋊C8—21no—
29C2×C3×#131—31no—
30C2×#907C2×C311yescharacteristic
31C2×C3×#108—21no—
32C2×C3×#108D311yescharacteristic, complemented
33C2×C3×#108C2×C311yescharacteristic, complemented
34C3×#424—31no—
35C2×C3×#104—31no—
36C3×#416—31no—
37C2×#825C2×C311yescharacteristic
38#5119C2×C311yescharacteristic, complemented
39#5113C2×C311yescharacteristic
40C3×#339—31no—
41C3×#336—31no—
42#4555C2×C311yescharacteristic, complemented
43#4550C2×C311yescharacteristic, complemented
44C22×C32×C4C2311yescharacteristic
45C2×C3×#67—21no—
46C22×C3×Dic3—21no—
47C3×#235—21no—
48C3×#235—21no—
49C3×#235D412yes—
50C3×#235Q812yes—
51C2×C3×D12C2×C411yescharacteristic
52C2×C3×C4×D3—21no—
53C2×C3×Dic6C2×C411yescharacteristic
54C32×M16—21no—
55C32×M16—21no—
56C32×M16—21no—
57C32×M16D412yes—
58C2×C32×C8—21no—
59C2×C32×C8—21no—
60C2×C32×C8—21no—
61C2×C32×C8—21no—
62C2×C32×C8C2×C411yescharacteristic
63C2×C32×C8C2×C411yescharacteristic
64C3×#217—21no—
65C3×C4⋊Dic3C2×C411yescharacteristic
66C3×C4⋊Dic3C2×C411yescharacteristic
67C3×C4×Dic3—21no—
68C3×C4×Dic3—21no—
69C3×C4×Dic3—21no—
70C2×C3×C3⋊C8—21no—
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8D411yescharacteristic
73C2×C3×C3⋊C8D411yescharacteristic
74C3×ΓL(1,𝔽25)—24no—
75C3×D3×C8—24no—
76#1866—31noSylow
77C2×C3×#41—31no—
78C2×#235C22×C311yescharacteristic
79C3×#131—62no—
80C3×#131—62no—
81C2×C3×M16—21no—
82C2×C3×M16—21no—
83C2×C3×M16—61no—
84C2×C3×M16C2×D311yescharacteristic
85C2×C3×M16C2×D311yescharacteristic
86C2×C3×M16C22×C311yescharacteristic
87C2×C3×M16C22×C311yescharacteristic
88C22×C3×C8—21no—
89C22×C3×C8—31no—
90C22×C3×C8—61no—
91C22×C3×C8C2×D311yescharacteristic
92C22×C3×C8C22×C311yescharacteristic
93C3×#117—31no—
94C2×C3×C42—31no—
95#926C22×C311yescharacteristic
96C2×C4×Dic3C22×C311yescharacteristic
97C22×C3⋊C8C22×C311yescharacteristic
98#907—22no—
99#907—22no—
100C2×ΓL(1,𝔽25)—21no—
101C2×D3×C8—21no—
102C3×#108—24no—
103C3×#108—22no—
104C3×#108—22no—
105C3×C4⋊C8—61no—
106C3×C4⋊C8—61no—
107C3×#104—62no—
108C3×#103—31no—
109C3×#103—31no—
110C3×#102—31no—
111C3×#102—31no—
112#825—22no—
113#821C3×C411yescharacteristic, complemented
114#821C3×C411yescharacteristic, complemented
115#816C3×C411yescharacteristic, complemented
116#816C3×C411yescharacteristic, complemented
117Dic3⋊C8—21no—
118Dic3⋊C8—21no—
119C23×C32#4111yescharacteristic
120C22×C3×D3—21no—
121C2×C32×C4C22×C411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2×Q811yescharacteristic
126C2×C32×C4#4111yescharacteristic
127C3×#67—22no—
128C3×#67—22no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—41no—
134C2×C3×Dic3—41no—
135C3×D12—21no—
136C3×D12C4⋊C412yes—
137C3×C4×D3—22no—
138C3×C4×D3—22no—
139C3×Dic6—21no—
140C3×Dic6C4⋊C412yes—
141C32×C8—21no—
142C32×C8—21no—
143C32×C8—22no—
144C32×C8—41no—
145C3×C3⋊C8—21no—
146C3×C3⋊C8—21no—
147C2×#131—31no—
148C2×#108C3×D311yescharacteristic, complemented
149#424—31no—
150C2×#104—31no—
151#416—31no—
152#339—31no—
153#336—31no—
154C3×#41—32no—
155C3×#41—32no—
156C3×#41—61no—
157C3×#41—61no—
158C2×C3×Q8—31no—
159C2×C3×D4—31no—
160C2×C3×D4—61no—
161C22×C3×C4—21no—
162C22×C3×C4—61no—
163C22×C3×C4—61no—
164C22×C3×C4C22×D311yescharacteristic
165C22×C3×C4C23×C311yescharacteristic
166C2×#67—21no—
167C22×Dic3—21no—
168#235—21no—
169#235—21no—
170#235C3×D412yes—
171#235C3×Q812yes—
172C2×D12C2×C3×C411yescharacteristic
173C2×C4×D3—21no—
174C2×Dic6C2×C3×C411yescharacteristic
175C3×M16—21no—
176C3×M16—21no—
177C3×M16—22no—
178C3×M16—21no—
179C3×M16—21no—
180C3×M16—21no—
181C3×M16—21no—
182C3×M16—22no—
183C3×M16—41no—
184C3×M16—41no—
185C3×M16—64no—
186C3×M16#6712yes—
187C3×M16C3×D412yes—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—31no—
201C2×C3×C8—31no—
202C2×C3×C8—41no—
203C2×C3×C8—41no—
204C2×C3×C8—64no—
205C2×C3×C8—61no—
206C2×C3×C8—61no—
207C2×C3×C8C4×D311yescharacteristic
208C2×C3×C8C4×D311yescharacteristic
209C2×C3×C8C2×C3×C411yescharacteristic
210C2×C3×C8C2×C3×C411yescharacteristic
211C3×C4⋊C4—31no—
212C3×C4⋊C4—31no—
213C3×C22⋊C4—61no—
214C3×C42—61no—
215C3×C42—61no—
216C3×C42—61no—
217#217—21no—
218C4⋊Dic3C2×C3×C411yescharacteristic
219C4⋊Dic3C2×C3×C411yescharacteristic
220C4×Dic3—21no—
221C4×Dic3—21no—
222C4×Dic3—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8C3×D411yescharacteristic
226C2×C3⋊C8C3×D411yescharacteristic
227ΓL(1,𝔽25)—24no—
228D3×C8—24no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32C4×D411yescharacteristic
232C22×C32C22⋊Q811yescharacteristic
233C22×C32#12311yescharacteristic
234C2×C3×D3—22no—
235C2×C3×D3—41no—
236C2×C3×D3—41no—
237C32×C4C2×C4⋊C411yescharacteristic
238C32×C4C4×D411yescharacteristic
239C32×C4#12011yescharacteristic
240C32×C4C22⋊Q811yescharacteristic
241C3×Dic3—22no—
242C3×Dic3—42no—
243C3×Dic3—41no—
244C3×Dic3—41no—
245C2×#41—31no—
246#131—62no—
247#131—62no—
248C2×M16—61no—
249C2×M16C2×C3×D311yescharacteristic
250C2×M16C2×C3×D311yescharacteristic
251C22×C8—31no—
252C22×C8—61no—
253C22×C8C2×C3×D311yescharacteristic
254#117—31no—
255C2×C42—31no—
256#108—22no—
257C4⋊C8—61no—
258C4⋊C8—61no—
259#104—62no—
260#103—31no—
261#103—31no—
262#102—31no—
263#102—31no—
264C23×C3—21no—
265C23×C3—61no—
266C23×C3#23911yescharacteristic
267C23×C3C3×#4111yescharacteristic
268C22×D3—21no—
269C3×Q8—32no—
270C3×Q8—61no—
271C3×D4—32no—
272C3×D4—62no—
273C3×D4—62no—
274C3×D4—61no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—61no—
282C2×C3×C4—62no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—121no—
288C2×C3×C4—121no—
289C2×C3×C4C2×C4×D311yescharacteristic
290C2×C3×C4#23511yescharacteristic
291C2×C3×C4D3×D411yescharacteristic
292C2×C3×C4D3×D411yescharacteristic
293C2×C3×C4D3×Q811yescharacteristic
294C2×C3×C4C2×#6711yescharacteristic
295C2×C3×C4C22×C3×C411yescharacteristic
296C2×C3×C4C2×C3×D411yescharacteristic
297C2×C3×C4C2×C3×D411yescharacteristic
298C2×C3×C4C2×C3×D411yescharacteristic
299C2×C3×C4C2×C3×Q811yescharacteristic
300C2×C3×C4C3×#4111yescharacteristic
301#67—22no—
302#67—22no—
303C2×Dic3—21no—
304C2×Dic3—22no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309D12—21no—
310D12C3×C4⋊C412yes—
311C4×D3—22no—
312C4×D3—22no—
313Dic6—21no—
314Dic6C3×C4⋊C412yes—
315C3×C8—22no—
316C3×C8—22no—
317C3×C8—21no—
318C3×C8—21no—
319C3×C8—22no—
320C3×C8—21no—
321C3×C8—21no—
322C3×C8—22no—
323C3×C8—42no—
324C3×C8—42no—
325C3×C8—41no—
326C3×C8—41no—
327C3×C8—61no—
328C3×C8—61no—
329C3⋊C8—21no—
330C3⋊C8—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32#38511yescharacteristic
334C2×C32#44311yescharacteristic
335C2×C32#44411yescharacteristic
336C3×D3—42no—
337#41—32no—
338#41—32no—
339#41—61no—
340#41—61no—
341C2×Q8—31no—
342C2×D4—31no—
343C2×D4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4C22×C3×D311yescharacteristic
347M16—21no—
348M16—21no—
349M16—21no—
350M16—64no—
351M16C3×#6712yes—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—31no—
357C2×C8—31no—
358C2×C8—64no—
359C2×C8—61no—
360C2×C8—61no—
361C2×C8C3×C4×D311yescharacteristic
362C2×C8C3×C4×D311yescharacteristic
363C4⋊C4—31no—
364C4⋊C4—31no—
365C22⋊C4—61no—
366C42—61no—
367C42—61no—
368C42—61no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—62no—
379C22×C3—121no—
380C22×C3—121no—
381C22×C3#89311yescharacteristic
382C22×C3#89411yescharacteristic
383C22×C3#94611yescharacteristic
384C22×C3C3×C4×D411yescharacteristic
385C22×C3C3×C22⋊Q811yescharacteristic
386C22×C3C3×#12311yescharacteristic
387C2×D3—22no—
388C2×D3—41no—
389C2×D3—41no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—62no—
395C3×C4—122no—
396C3×C4—121no—
397C3×C4—121no—
398C3×C4D3×C4⋊C411yescharacteristic
399C3×C4#89611yescharacteristic
400C3×C4C4×#6711yescharacteristic
401C3×C4#93711yescharacteristic
402C3×C4C2×C3×C4⋊C411yescharacteristic
403C3×C4C3×C4×D411yescharacteristic
404C3×C4C3×#12011yescharacteristic
405C3×C4C3×C22⋊Q811yescharacteristic
406Dic3—22no—
407Dic3—42no—
408Dic3—41no—
409Dic3—41no—
410C32#186611yescharacteristic, complemented, Sylow
411C23—61no—
412C23C3×#23911yescharacteristic
413Q8—32no—
414Q8—61no—
415D4—32no—
416D4—62no—
417D4—62no—
418D4—61no—
419C2×C4—61no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4C2×C3×C4×D311yescharacteristic
428C2×C4C3×#23511yescharacteristic
429C2×C4C3×D3×D411yescharacteristic
430C2×C4C3×D3×D411yescharacteristic
431C2×C4C3×D3×Q811yescharacteristic
432C2×C4C2×C3×#6711yescharacteristic
433C8—21no—
434C8—21no—
435C8—22no—
436C8—41no—
437C8—61no—
438C8—61no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—41no—
447C2×C3—41no—
448C2×C3—122no—
449C2×C3#489411yescharacteristic
450C2×C3#489511yescharacteristic
451C2×C3#498911yescharacteristic
452C2×C3C3×#38511yescharacteristic
453C2×C3C3×#44311yescharacteristic
454C2×C3C3×#44411yescharacteristic
455D3—42no—
456C22—21no—
457C22—21no—
458C22—62no—
459C22—121no—
460C22—121no—
461C22C3×#89311yescharacteristic
462C22C3×#89411yescharacteristic
463C22C3×#94611yescharacteristic
464C4—62no—
465C4—122no—
466C4—121no—
467C4—121no—
468C4C3×D3×C4⋊C411yescharacteristic
469C4C3×#89611yescharacteristic
470C4C3×C4×#6711yescharacteristic
471C4C3×#93711yescharacteristic
472C3—21no—
473C3#7072211yescharacteristic, direct product factor
474C3C3×#186611yescharacteristic, complemented
475C2—21no—
476C2—21no—
477C2—122no—
478C2C3×#489411yescharacteristic
479C2C3×#489511yescharacteristic
480C2C3×#498911yescharacteristic
subgroupG/H
centerC2×C3×C4D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#108C2
socleC22×C32C4×D4
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#108D3
hypocenterC3C3×#1866
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#108

Complements

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

NH ≅ G/Ntype
2. C2×C32×#108C2semidirect
4. C3×#5119C2semidirect
6. C3×#4555C2semidirect
7. C3×#4550C2semidirect
9. #70722C3direct (N × H)
23. C3×#821C4semidirect
24. C3×#821C4semidirect
25. C3×#816C4semidirect
26. C3×#816C4semidirect
32. C2×C3×#108D3semidirect
33. C2×C3×#108C2×C3semidirect
38. #5119C2×C3semidirect
42. #4555C2×C3semidirect
43. #4550C2×C3semidirect
113. #821C3×C4semidirect
114. #821C3×C4semidirect
115. #816C3×C4semidirect
116. #816C3×C4semidirect
148. C2×#108C3×D3semidirect
410. C32#1866semidirect
473. C3#70722direct (N × H)
474. C3C3×#1866semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9072
C2×C32×#1082
C2×C3×#8252
C3×#51192
C3×#51132
C3×#45552
C3×#45502
C3×#18663
#707223