Finite Group Databasebeta

← C3×#10002 #10628609

Subgroups — C3×#10002 #10628609

Order 768.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3104C211yesmaximal, characteristic, complemented
2C3×#3077C211yesmaximal, characteristic
3C3×#2580C211yesmaximal, characteristic
4C3×#1518C211yesmaximal, characteristic, complemented
5C3×#1487C211yesmaximal, characteristic, complemented
6C3×#1478C211yesmaximal, characteristic
7C3×#1286C211yesmaximal, characteristic, complemented
8#10002C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C3×#484—21no—
13C3×#478—22no—
14C3×#477—21no—
15C3×#476—21no—
16C3×#474—21no—
17C3×#462—21no—
18C3×#451—21no—
19C3×C4×Q16—22no—
20C3×#431C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C3×#428—21no—
23C3×#423C2211yescharacteristic
24C3×#418—21no—
25C3×#399—21no—
26C3×#393—21no—
27C3×#390—21no—
28C3×#389—21no—
29C3×#385—22no—
30C3×#382C2211yescharacteristic
31C3×#328—22no—
32C3×Q8⋊C8—22no—
33C3×D4⋊C8—22no—
34#3104C2×C311yescharacteristic, complemented
35#3077C2×C311yescharacteristic
36#2580C2×C311yescharacteristic
37#1518C2×C311yescharacteristic, complemented
38#1487C2×C311yescharacteristic, complemented
39#1478C2×C311yescharacteristic
40#1286C2×C311yescharacteristic, complemented
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×C3×Q16—22no—
45C2×C3×M16—21no—
46C3×C4⋊Q8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×Q8—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C3×#117—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C42—21no—
71C2×C3×C42C2311yescharacteristic
72C3×#107—22no—
73C3×#107—22no—
74C3×#106—22no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8—21no—
77C3×C4⋊C8—21no—
78C3×C4⋊C8D412yes—
79C3×#103—22no—
80C3×#103—22no—
81C3×#102—22no—
82C3×#102—22no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8—21no—
85C3×C4×C8—21no—
86C3×C4×C8D412yes—
87C3×#95—22no—
88C3×#95—21no—
89C3×#95D412yes—
90#533C22×C311yescharacteristic
91C2×C4×Q8C22×C311yescharacteristic
92C2×C4×D4C22×C311yescharacteristic
93#484—21no—
94#478—22no—
95#477—21no—
96#476—21no—
97#474—21no—
98#462—21no—
99#451—21no—
100C4×Q16—22no—
101#431C22×C311yescharacteristic
102#431C22×C311yescharacteristic
103#428—21no—
104#423C22×C311yescharacteristic
105#418—21no—
106#399—21no—
107#393—21no—
108#390—21no—
109#389—21no—
110#385—22no—
111#382C22×C311yescharacteristic
112#328—22no—
113Q8⋊C8—22no—
114D4⋊C8—22no—
115C24×C3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—42no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4Q1612yescomplemented
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—82no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C3×Q16—44no—
138C3×M16—41no—
139C3×M16—41no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—41no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—82no—
158C3×C22⋊C4—44no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—82no—
163C3×C42—41no—
164C3×C42—41no—
165C3×C42—41no—
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42C2×D411yescharacteristic
169C3×C42#4111yescharacteristic
170C22×Q8C3×D411yescharacteristic
171C22×D4C3×D411yescharacteristic
172C23×C4—21no—
173C2×Q16—22no—
174C2×M16—21no—
175C4⋊Q8C3×D411yescharacteristic
176C4⋊Q8C3×D411yescharacteristic
177#125—21no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190#117—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4C3×D411yescharacteristic
195C2×C4⋊C4C3×D411yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C42—21no—
200C2×C42C23×C311yescharacteristic
201#107—22no—
202#107—22no—
203#106—22no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8C3×D412yes—
208#103—22no—
209#103—22no—
210#102—22no—
211#102—22no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C4×C8—21no—
215C4×C8C3×D412yes—
216#95—22no—
217#95—21no—
218#95C3×D412yes—
219C23×C3—22no—
220C23×C3—42no—
221C23×C3—42no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3#12011yescharacteristic
225C3×Q8—22no—
226C3×Q8—22no—
227C3×Q8—41no—
228C3×Q8—41no—
229C3×Q8—42no—
230C3×Q8—81no—
231C3×D4—22no—
232C3×D4—22no—
233C3×D4—41no—
234C3×D4—41no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—82no—
251C2×C3×C4—82no—
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12711yescharacteristic
258C2×C3×C4C2×Q1611yescharacteristic
259C2×C3×C4#13511yescharacteristic
260C2×C3×C4#13711yescharacteristic
261C2×C3×C4#13711yescharacteristic
262C3×C8—42no—
263C3×C8—42no—
264C3×C8—81no—
265C24—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—42no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4C3×Q1612yescomplemented
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4C2×C3×D411yescharacteristic
285C22×C4C2×C3×D411yescharacteristic
286C22×C4C2×C3×D411yescharacteristic
287Q16—44no—
288M16—41no—
289M16—41no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—82no—
308C22⋊C4—44no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—82no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42C2×C3×D411yescharacteristic
317C42C2×C3×D411yescharacteristic
318C42C2×C3×D411yescharacteristic
319C42C3×#4111yescharacteristic
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—44no—
323C22×C3—42no—
324C22×C3—82no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3#39211yescharacteristic
328C22×C3#45111yescharacteristic
329C22×C3#45211yescharacteristic
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—42no—
335C3×C4—82no—
336C3×C4—81no—
337C3×C4—81no—
338C3×C4—81no—
339C3×C4—81no—
340C3×C4#45111yescharacteristic
341C3×C4#45211yescharacteristic
342C3×C4#46711yescharacteristic
343C3×C4#47011yescharacteristic
344C23—22no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23C3×#12011yescharacteristic
350Q8—22no—
351Q8—22no—
352Q8—41no—
353Q8—41no—
354Q8—42no—
355Q8—81no—
356D4—22no—
357D4—22no—
358D4—41no—
359D4—41no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4C3×#12011yescharacteristic
378C2×C4C3×#12011yescharacteristic
379C2×C4C3×#12111yescharacteristic
380C2×C4C3×#12111yescharacteristic
381C2×C4C3×#12111yescharacteristic
382C2×C4C3×#12711yescharacteristic
383C2×C4C2×C3×Q1611yescharacteristic
384C2×C4C3×#13511yescharacteristic
385C2×C4C3×#13711yescharacteristic
386C2×C4C3×#13711yescharacteristic
387C8—42no—
388C8—42no—
389C8—81no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—82no—
393C2×C3#200111yescharacteristic
394C2×C3#200811yescharacteristic
395C2×C3#201011yescharacteristic
396C22—21no—
397C22—21no—
398C22—44no—
399C22—42no—
400C22—82no—
401C22—81no—
402C22—81no—
403C22C3×#39211yescharacteristic
404C22C3×#45111yescharacteristic
405C22C3×#45211yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—82no—
412C4—81no—
413C4—81no—
414C4—81no—
415C4—81no—
416C4C3×#45111yescharacteristic
417C4C3×#45211yescharacteristic
418C4C3×#46711yescharacteristic
419C4C3×#47011yescharacteristic
420C3#1000211yescharacteristic, direct product factor, Sylow
421C2—21no—
422C2—21no—
423C2—82no—
424C2C3×#200111yescharacteristic
425C2C3×#200811yescharacteristic
426C2C3×#201011yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC22×C3×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3104C2semidirect
4. C3×#1518C2semidirect
5. C3×#1487C2semidirect
7. C3×#1286C2semidirect
8. #10002C3direct (N × H)
34. #3104C2×C3semidirect
37. #1518C2×C3semidirect
38. #1487C2×C3semidirect
40. #1286C2×C3semidirect
126. C2×C3×D4Q16semidirect
276. C2×D4C3×Q16semidirect
420. C3#10002direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31042
C3×#30772
C3×#25802
C3×#15182
C3×#14872
C3×#14782
C3×#12862
#100023