Finite Group Databasebeta

← #10847311

Subgroups — #10847311

Order 768.

664 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5084C211yesmaximal, characteristic
2#83207C211yesmaximal, characteristic, complemented
3#79059C211yesmaximal, characteristic
4#78508C211yesmaximal, characteristic, complemented
5C3×#1997C211yesmaximal, characteristic, complemented
6#70520C211yesmaximal, characteristic, complemented
7C4×#823C211yesmaximal, characteristic
8#68996C212yesmaximal
9#68984C212yesmaximal, complemented
10#68983C212yesmaximal
11#68981C212yesmaximal, complemented
12#25490—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C4×D12C2211yescharacteristic
15#5470C2211yescharacteristic
16C3×#423C2211yescharacteristic
17C3×C4×M16C2211yescharacteristic
18#5131C2212yes—
19C2×#823C2211yescharacteristic
20C2×#823C2211yescharacteristic
21C2×#822C2212yes—
22#5122C2212yes—
23#5121C2212yes—
24#5119C2212yes—
25#5084—21no—
26#5084—21no—
27#5084—22no—
28#5081—22no—
29#5080—21no—
30#5078—21no—
31C2×#809C2212yes—
32#4964C2212yes—
33#4964C2212yes—
34C2×#808C2212yes—
35C2×C4⋊(C3⋊C8)C2211yescharacteristic
36C4×#208C2211yescharacteristic
37#4847—22no—
38#4846—21no—
39#4842—21no—
40#4701—22no—
41#4679—21no—
42C8×Dic6—21no—
43C3×#343C2212yes—
44C3×#343C2212yes—
45#4558C2212yes—
46#4554C2211yescharacteristic
47#4554C2211yescharacteristic
48#3165—31no—
49#3077—31no—
50C2×#476—31no—
51#2962—31no—
52#2090—32no—
53#2063—32no—
54#1997D311yescharacteristic, complemented
55#1951—32no—
56#1950—31no—
57#1931—32no—
58C4×#100—31no—
59C22×C3×Q8—21no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C2×C3×M16C2312yes—
63C2×C3×M16C2311yescharacteristic
64C2×C3×M16C2311yescharacteristic
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C42C2311yescharacteristic
72C2×(C2×D3)⋊C4—21no—
73#926C2312yes—
74C2×C4⋊Dic3C2311yescharacteristic
75#924—21no—
76#924—21no—
77C2×#208C2311yescharacteristic
78C2×#208C2311yescharacteristic
79C22×C3⋊C8C2312yes—
80C4×D12—21no—
81C4×D12—21no—
82C4×D12—21no—
83C4×D12—21no—
84#870D412yes—
85#869D411yescharacteristic
86#869D411yescharacteristic
87C4×Dic6—21no—
88C4×Dic6—21no—
89C3×C4⋊C8—21no—
90C3×C4⋊C8—21no—
91C3×#97—21no—
92C3×C4×C8—21no—
93#823—22no—
94#823Q814yescomplemented
95#822—24no—
96#821—22no—
97#821—22no—
98#818—22no—
99#817—22no—
100#815—22no—
101Dic3⋊C8—22no—
102Dic3⋊C8—22no—
103C8×Dic3—22no—
104#809—22no—
105#809—22no—
106#808—22no—
107#808—22no—
108#808—22no—
109#808—22no—
110#807—22no—
111#807—22no—
112C4⋊(C3⋊C8)D411yescharacteristic
113C4⋊(C3⋊C8)D411yescharacteristic
114C4⋊(C3⋊C8)D412yes—
115#803—21no—
116C4×C3⋊C8—21no—
117#533—31no—
118C2×C4⋊Q8C2×D311yescharacteristic
119C2×C4×D4—31no—
120#501—62no—
121#500—61no—
122#498—61no—
123#478—62no—
124#476—61no—
125#476—61no—
126#476—62no—
127#476—61no—
128#474—61no—
129#446—61no—
130#446—62no—
131C8×Q8—61no—
132C2×#108—32no—
133#428—32no—
134#428—32no—
135#428—32no—
136C2×#106—32no—
137#424—32no—
138#423C2×D311yescharacteristic
139C2×C4⋊C8—31no—
140#418—32no—
141C2×#102—32no—
142C2×#100—31no—
143C2×#100—31no—
144#405—32no—
145C4×M16—31no—
146C4×M16C2×D311yescharacteristic
147#343—31no—
148#343—31no—
149#343C2×D312yes—
150#343C2×D312yes—
151#341—32no—
152C23×D3—21no—
153C2×C3×Q8—41no—
154C2×C3×Q8—41no—
155C2×C3×Q8—82no—
156C22×C3×C4—22no—
157C22×C3×C4#4111yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—21no—
161C2×D12—21no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12C2×Q812yes—
167C2×C4×D3—41no—
168C2×C4×D3—41no—
169C2×C4×D3—82no—
170C2×Dic6—21no—
171C2×Dic6—21no—
172C3×M16—21no—
173C3×M16—22no—
174C3×M16—21no—
175C3×M16—21no—
176C3×M16—22no—
177C3×M16C2×Q812yes—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—41no—
184C3×C4⋊C4—41no—
185C3×C42C2×D411yescharacteristic
186C3×C42C2×D411yescharacteristic
187C3×C42C2×D411yescharacteristic
188C3×C42C2×D411yescharacteristic
189#217—22no—
190(C2×D3)⋊C4—41no—
191(C2×D3)⋊C4—41no—
192C4⋊Dic3—21no—
193C4⋊Dic3—21no—
194C4⋊Dic3—21no—
195C4⋊Dic3—21no—
196C4⋊Dic3C2×D412yes—
197C4⋊Dic3C2×D412yes—
198Dic3⋊C4—22no—
199Dic3⋊C4—22no—
200C4×Dic3—22no—
201#208—21no—
202#208—21no—
203#208C2×Q812yes—
204#208#4112yes—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8C2×D412yes—
210C2×C3⋊C8C2×D412yes—
211C22×Q8—21no—
212C22×D4—31no—
213C23×C4—61no—
214C2×M16—31no—
215C2×M16—31no—
216C2×M16C22×D312yes—
217C2×M16C22×D311yescharacteristic
218C2×M16C22×D311yescharacteristic
219C22×C8—32no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—31no—
225C4⋊Q8—31no—
226#125—32no—
227C22⋊Q8—61no—
228C22⋊Q8—61no—
229C4×Q8—61no—
230C4×Q8—61no—
231C4×D4—61no—
232C4×D4—61no—
233C4×D4—61no—
234C4×D4—61no—
235#117—32no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4C22×D312yes—
239C2×C22⋊C4—61no—
240C2×C42C22×D311yescharacteristic
241#108—64no—
242#107—62no—
243#107—62no—
244#107—62no—
245#106—62no—
246#106—62no—
247#106—62no—
248#106—62no—
249#106—62no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—31no—
253C4⋊C8—31no—
254C4⋊C8—32no—
255C4⋊C8—62no—
256C4⋊C8—62no—
257#102—62no—
258#102—62no—
259#102—62no—
260#102—62no—
261#100—34no—
262#100—62no—
263#97—21no—
264#97—62no—
265#97—61no—
266C4×C8—21no—
267C4×C8—62no—
268C4×C8—61no—
269C23×C3C2×#4111yescharacteristic
270C22×D3—22no—
271C22×D3—41no—
272C22×D3—41no—
273C22×D3—81no—
274C22×D3—81no—
275C3×Q8—82no—
276C3×Q8—82no—
277C2×C3×C4—22no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—82no—
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12111yescharacteristic
284C2×C3×C4#12111yescharacteristic
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×Q811yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C42−1+411yescharacteristic
290C2×C3×C42−1+411yescharacteristic
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—41no—
294C2×Dic3—81no—
295D12—22no—
296D12—22no—
297D12—41no—
298D12—41no—
299C4×D3—82no—
300C4×D3—82no—
301Dic6—42no—
302Dic6—81no—
303C3×C8—22no—
304C3×C8—41no—
305C3×C8—42no—
306C3⋊C8—22no—
307C3⋊C8—22no—
308C3⋊C8—22no—
309C3⋊C8—22no—
310C24—61no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—61no—
314C2×Q8—61no—
315C2×Q8—82no—
316C2×D4—32no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C22×C4—22no—
323C22×C4—61no—
324C22×C4—121no—
325C22×C4—121no—
326C22×C4—242no—
327C22×C4#23511yescharacteristic
328C22×C4#23511yescharacteristic
329C22×C4C23×D311yescharacteristic
330M16—21no—
331M16—21no—
332M16—21no—
333M16—22no—
334M16—22no—
335M16—32no—
336M16—32no—
337M16—61no—
338M16—61no—
339M16D3×Q812yes—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—22no—
343C2×C8—32no—
344C2×C8—32no—
345C2×C8—61no—
346C2×C8—61no—
347C2×C8—62no—
348C2×C8—62no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—32no—
352C4⋊C4—32no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—62no—
356C4⋊C4—62no—
357C4⋊C4—61no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—61no—
361C22⋊C4—62no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C42—62no—
365C42D3×D411yescharacteristic
366C42D3×D411yescharacteristic
367C42D3×D411yescharacteristic
368C42D3×D411yescharacteristic
369C22×C3—21no—
370C22×C3—21no—
371C22×C3C2×#12111yescharacteristic
372C22×C3#53311yescharacteristic
373C22×C3#54111yescharacteristic
374C2×D3—42no—
375C2×D3—82no—
376C2×D3—81no—
377C2×D3—81no—
378C2×D3—82no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—82no—
383C3×C4#47112yes—
384C3×C4#54711yescharacteristic
385C3×C4#54811yescharacteristic
386C3×C4D4×Q811yescharacteristic
387C3×C4D4×Q811yescharacteristic
388Dic3—42no—
389Dic3—42no—
390Dic3—81no—
391C23—62no—
392C23—121no—
393C23—121no—
394C23—241no—
395C23—241no—
396C23C2×#23511yescharacteristic
397Q8—82no—
398Q8—82no—
399Q8—122no—
400Q8—241no—
401D4—62no—
402D4—62no—
403D4—121no—
404D4—121no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—62no—
409C2×C4—62no—
410C2×C4—82no—
411C2×C4—121no—
412C2×C4—242no—
413C2×C4—242no—
414C2×C4—241no—
415C2×C4#88411yescharacteristic
416C2×C4#88411yescharacteristic
417C2×C4#88411yescharacteristic
418C2×C4#88411yescharacteristic
419C2×C4C2×D3×D411yescharacteristic
420C2×C4C2×D3×D411yescharacteristic
421C2×C4C2×D3×Q811yescharacteristic
422C2×C4#100711yescharacteristic
423C2×C4D3×#4111yescharacteristic
424C2×C4#101011yescharacteristic
425C8—22no—
426C8—42no—
427C8—41no—
428C8—62no—
429C8—62no—
430C8—62no—
431C8—62no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3#242911yescharacteristic
435C2×C3C2×#47111yescharacteristic
436C2×C3#306811yescharacteristic
437D3—82no—
438C22—21no—
439C22—21no—
440C22—122no—
441C22—242no—
442C22—242no—
443C22—241no—
444C22—241no—
445C22C2×#88411yescharacteristic
446C22#557811yescharacteristic
447C22#558311yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—82no—
452C4—122no—
453C4—122no—
454C4—241no—
455C4#475412yes—
456C4#563111yescharacteristic
457C4#563211yescharacteristic
458C4#567711yescharacteristic
459C4#572711yescharacteristic
460C3#2549011yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—242no—
464C2#7662511yescharacteristic
465C2#7870311yescharacteristic
466C2C2×#475411yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1997C2
socleC22×C3C2×#121
prime radicalC22×D12C23
hypercenter#1997D3
hypocenterC3#25490
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
2. #83207C2semidirect
4. #78508C2semidirect
5. C3×#1997C2semidirect
6. #70520C2semidirect
9. #68984C2semidirect
11. #68981C2semidirect
54. #1997D3semidirect
94. #823Q8semidirect
460. C3#25490semidirect

Maximal subgroups

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

subgroupindex
C2×#50842
#832072
#790592
#785082
C3×#19972
#705202
C4×#8232
#689962
#689842
#689832
#689812
#254903