Finite Group Databasebeta

← #11789665

Subgroups — #11789665

Order 1152.

565 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63636C211yesmaximal, characteristic
2C3×#4964C211yesmaximal, characteristic, complemented
3#10591096C211yesmaximal, characteristic
4C2×#63559C211yesmaximal, characteristic
5#10591057C211yesmaximal, characteristic, complemented
6C3×#4530C211yesmaximal, characteristic, complemented
7#10589165C211yesmaximal, characteristic, complemented
8#70673—31nomaximal
9#68983—31nomaximal
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×C4⋊C4C2211yescharacteristic
12C22×C3⋊(C3⋊C8)C2211yescharacteristic
13C3×#926C2211yescharacteristic
14C2×C3×#208C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17#63636—21no—
18#63636—21no—
19C3×#808—21no—
20C3×#807—21no—
21#63564—21no—
22#63564—21no—
23#63559C414yescomplemented
24#63554—21no—
25Dic3×C3⋊C8—21no—
26C3×#428—31no—
27#5131—31no—
28C2×#823—31no—
29C2×#823—31no—
30#5122—31no—
31#5001—31no—
32#4997—31no—
33C2×#809—31no—
34C2×#809—31no—
35C2×#809—61no—
36#4964D311yescharacteristic, complemented
37C3×#343—31no—
38#4558—31no—
39#4558—31no—
40#4530D311yescharacteristic, complemented
41C23×C3⋊D3—21no—
42C22×C32×C4—21no—
43C22×C32×C4C2311yescharacteristic
44C2×C4⋊(C3⋊D3)—21no—
45C2×C4⋊(C3⋊D3)—21no—
46C2×C4⋊(C3⋊D3)—21no—
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)C2×C412yes—
50C32×C4⋊C4—21no—
51C32×C4⋊C4—21no—
52C2×C3⋊(C3⋊C8)—21no—
53C2×C3⋊(C3⋊C8)—21no—
54C2×C3⋊(C3⋊C8)D411yescharacteristic
55C2×C3⋊(C3⋊C8)D411yescharacteristic
56C3×#217—21no—
57C3×C4⋊Dic3D411yescharacteristic
58C3×C4⋊Dic3D411yescharacteristic
59C3×C4×Dic3—21no—
60C3×#208—21no—
61C3×#208—21no—
62C3×#208—21no—
63C3×#208—21no—
64C3×#208C2×C412yes—
65C3×#208C2×C412yes—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8—21no—
68C2×C3×C3⋊C8—21no—
69#1931—91noSylow
70C22×D12—31no—
71C22×D12—31no—
72C22×D12—61no—
73C2×C3×M16—31no—
74C2×C3×M16—31no—
75C2×C3×M16—31no—
76C3×#117—31no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2×D311yescharacteristic
79C2×C3×C4⋊C4C2×D311yescharacteristic
80#926C2×D311yescharacteristic
81C2×#208C2×D311yescharacteristic
82C2×#208C2×D311yescharacteristic
83C2×#208C2×D311yescharacteristic
84C22×C3⋊C8—31no—
85C22×C3⋊C8—31no—
86C22×C3⋊C8—61no—
87C3×#107—61no—
88C3×#106—61no—
89#823—34no—
90#823—34no—
91#821—61no—
92#821—61no—
93#815—61no—
94C8×Dic3—61no—
95#809—61no—
96#809—61no—
97#809—61no—
98#809—61no—
99#809—61no—
100#809—61no—
101#809—62no—
102#809—62no—
103#808—21no—
104#807—21no—
105#803—61no—
106C4×C3⋊C8—61no—
107C23×C32#4111yescharacteristic
108C22×C3⋊D3—22no—
109C22×C3⋊D3—41no—
110C22×C3⋊D3—41no—
111C22×C3⋊D3—81no—
112C22×C3⋊D3—81no—
113C2×C32×C4—41no—
114C2×C32×C4—81no—
115C2×C32×C4C22×C411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2×D411yescharacteristic
118C2×C32×C4#4111yescharacteristic
119C2×C32×C4#4111yescharacteristic
120C2×C32×C4#4111yescharacteristic
121C4⋊(C3⋊D3)—22no—
122C4⋊(C3⋊D3)—22no—
123C4⋊(C3⋊D3)—41no—
124C4⋊(C3⋊D3)—41no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—21no—
127C3⋊(C3⋊C8)—21no—
128C3⋊(C3⋊C8)—21no—
129C3×C3⋊C8—22no—
130C3×C3⋊C8—22no—
131C3×C3⋊C8—41no—
132#428—31no—
133#418—91no—
134C2×#102—91no—
135C2×#100—91no—
136#405—91no—
137#343—31no—
138#341—91no—
139C23×D3—61no—
140C23×D3—61no—
141C23×D3—121no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—41no—
146C22×C3×C4C22×D311yescharacteristic
147C22×C3×C4C22×D311yescharacteristic
148C2×D12—32no—
149C2×D12—32no—
150C2×D12—62no—
151C2×D12—62no—
152C2×D12—62no—
153C2×D12—61no—
154C2×D12—61no—
155C2×D12—61no—
156C2×D12—61no—
157C2×D12—61no—
158C2×D12—61no—
159C2×D12—61no—
160C2×D12—61no—
161C2×D12—61no—
162C2×D12—61no—
163C2×D12—121no—
164C2×D12—121no—
165C2×D12—121no—
166C3×M16—32no—
167C3×M16—32no—
168C3×M16—61no—
169C3×M16—61no—
170C3×M16—61no—
171C3×M16—61no—
172C2×C3×C8—61no—
173C2×C3×C8—61no—
174C2×C3×C8—61no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—31no—
181C3×C4⋊C4—31no—
182C3×C4⋊C4—41no—
183C3×C22⋊C4—61no—
184C3×C42—61no—
185#217—21no—
186C4⋊Dic3#6711yescharacteristic
187C4⋊Dic3#6711yescharacteristic
188C4×Dic3—21no—
189#208—21no—
190#208—21no—
191#208—21no—
192#208—21no—
193#208C4×D312yes—
194#208C4×D312yes—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—61no—
203C2×C3⋊C8—61no—
204C2×C3⋊C8—61no—
205C2×C3⋊C8—61no—
206C2×C3⋊C8—61no—
207C2×C3⋊C8—61no—
208C2×C3⋊C8—121no—
209C2×C3⋊C8—121no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32#11711yescharacteristic
213C22×C32#12111yescharacteristic
214C22×C32#12611yescharacteristic
215C2×C3⋊D3—42no—
216C2×C3⋊D3—82no—
217C2×C3⋊D3—81no—
218C2×C3⋊D3—81no—
219C2×C3⋊D3—82no—
220C32×C4—81no—
221C32×C4C4×D411yescharacteristic
222C32×C4C4×D411yescharacteristic
223C32×C4#12311yescharacteristic
224C32×C4#12411yescharacteristic
225C3×Dic3—41no—
226C3×Dic3—41no—
227C22×D4—91no—
228C2×M16—31no—
229C2×M16—31no—
230C2×M16—31no—
231C22×C8—91no—
232#117—31no—
233C2×C4⋊C4D3211yescharacteristic
234#107—61no—
235#106—61no—
236#102—181no—
237#102—181no—
238#102—181no—
239#102—181no—
240#100—94no—
241#97—181no—
242C4×C8—181no—
243C23×C3—21no—
244C23×C3#23511yescharacteristic
245C23×C3#23711yescharacteristic
246C22×D3—62no—
247C22×D3—62no—
248C22×D3—122no—
249C22×D3—122no—
250C22×D3—122no—
251C22×D3—121no—
252C22×D3—121no—
253C22×D3—121no—
254C22×D3—121no—
255C22×D3—242no—
256C22×D3—242no—
257C22×D3—241no—
258C22×D3—241no—
259C22×D3—241no—
260C22×D3—241no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—42no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—82no—
273C2×C3×C4—81no—
274C2×C3×C4—81no—
275C2×C3×C4C2×C4×D311yescharacteristic
276C2×C3×C4C2×C4×D311yescharacteristic
277C2×C3×C4#23511yescharacteristic
278C2×C3×C4#23511yescharacteristic
279C2×C3×C4D3×D411yescharacteristic
280C2×C3×C4D3×D411yescharacteristic
281C2×C3×C4D3×D411yescharacteristic
282C2×C3×C4#23711yescharacteristic
283C2×C3×C4#23911yescharacteristic
284C2×C3×C4#23911yescharacteristic
285C2×C3×C4#23911yescharacteristic
286C2×C3×C4C2×#6711yescharacteristic
287C2×Dic3—21no—
288C2×Dic3—21no—
289D12—64no—
290D12—62no—
291D12—62no—
292D12—62no—
293D12—62no—
294D12—122no—
295D12—122no—
296D12—122no—
297D12—121no—
298D12—121no—
299D12—121no—
300D12—121no—
301C3×C8—62no—
302C3×C8—62no—
303C3×C8—121no—
304C3⋊C8—22no—
305C3⋊C8—22no—
306C3⋊C8—41no—
307C3⋊C8—61no—
308C3⋊C8—61no—
309C3⋊C8—61no—
310C3⋊C8—61no—
311C3⋊C8—121no—
312C3⋊C8—121no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32#38811yescharacteristic
316C2×C32#47111yescharacteristic
317C2×C32#47211yescharacteristic
318C3⋊D3—82no—
319C24—181no—
320C2×D4—92no—
321C2×D4—181no—
322C2×D4—181no—
323C2×D4—181no—
324C2×D4—181no—
325C2×D4—181no—
326C22×C4—21no—
327C22×C4C2×D3211yescharacteristic
328M16—32no—
329M16—32no—
330M16—61no—
331M16—61no—
332M16—61no—
333M16—61no—
334C2×C8—61no—
335C2×C8—61no—
336C2×C8—61no—
337C2×C8—91no—
338C2×C8—91no—
339C2×C8—181no—
340C2×C8—181no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—31no—
344C4⋊C4—31no—
345C22⋊C4—61no—
346C42—61no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3#87211yescharacteristic
357C22×C3#88411yescharacteristic
358C22×C3#89211yescharacteristic
359C22×C3#89811yescharacteristic
360C22×C3#89811yescharacteristic
361C22×C3#93911yescharacteristic
362C2×D3—124no—
363C2×D3—122no—
364C2×D3—122no—
365C2×D3—244no—
366C2×D3—244no—
367C2×D3—242no—
368C2×D3—242no—
369C2×D3—242no—
370C2×D3—242no—
371C2×D3—241no—
372C2×D3—241no—
373C2×D3—241no—
374C2×D3—241no—
375C2×D3—242no—
376C2×D3—242no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—82no—
382C3×C4—81no—
383C3×C4—81no—
384C3×C4—121no—
385C3×C4—121no—
386C3×C4#67⋊C411yescharacteristic
387C3×C4#88511yescharacteristic
388C3×C4#89311yescharacteristic
389C3×C4#89311yescharacteristic
390C3×C4#89411yescharacteristic
391C3×C4#89411yescharacteristic
392C3×C4C4×#6711yescharacteristic
393C3×C4#94711yescharacteristic
394Dic3—41no—
395Dic3—41no—
396C32#193111yescharacteristic, complemented, Sylow
397C23—182no—
398C23—361no—
399C23—361no—
400C23—721no—
401C23—721no—
402C23#380411yescharacteristic
403D4—182no—
404D4—182no—
405D4—361no—
406D4—361no—
407C2×C4—41no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—81no—
411C2×C4#379611yescharacteristic
412C2×C4#379911yescharacteristic
413C2×C4#379911yescharacteristic
414C2×C4C4×D3211yescharacteristic
415C2×C4D3×#6711yescharacteristic
416C2×C4#381111yescharacteristic
417C8—62no—
418C8—62no—
419C8—121no—
420C8—181no—
421C8—181no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3#466911yescharacteristic
432C2×C3#474911yescharacteristic
433C2×C3#475411yescharacteristic
434C2×C3#499011yescharacteristic
435C2×C3#519911yescharacteristic
436C2×C3#520311yescharacteristic
437D3—244no—
438D3—242no—
439D3—242no—
440C22—21no—
441C22—21no—
442C22—362no—
443C22—722no—
444C22—722no—
445C22—721no—
446C22—721no—
447C22#6383611yescharacteristic
448C22#6384811yescharacteristic
449C22#6397111yescharacteristic
450C4—81no—
451C4—121no—
452C4—121no—
453C4#6388111yescharacteristic
454C4#6389711yescharacteristic
455C4#6390411yescharacteristic
456C4#6390711yescharacteristic
457C3—21no—
458C3#6898311yescharacteristic, complemented
459C3#7067311yescharacteristic, complemented
460C2—21no—
461C2—21no—
462C2—722no—
463C2#1059125811yescharacteristic
464C2#1059150811yescharacteristic
465C2#1059164311yescharacteristic
subgroupG/H
centerC22#63971
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC22×C32#121
prime radicalC22×C4⋊(C3⋊D3)C22
cosocleC22×C32×C4C23
hypercenterC2×C4⋊C4D32
hypocenterC32#1931
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#4964C2semidirect
5. #10591057C2semidirect
6. C3×#4530C2semidirect
7. #10589165C2semidirect
23. #63559C4semidirect
36. #4964D3semidirect
40. #4530D3semidirect
396. C32#1931semidirect
458. C3#68983semidirect
459. C3#70673semidirect

Maximal subgroups

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

subgroupindex
C2×#636362
C3×#49642
#105910962
C2×#635592
#105910572
C3×#45302
#105891652
#706733
#689833