Finite Group Databasebeta

← #10814904

Subgroups — #10814904

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79040C212yesmaximal, complemented
2#78494C211yesmaximal, characteristic
3C3×#1849C211yesmaximal, characteristic, complemented
4#70703C211yesmaximal, characteristic, complemented
5#70702C211yesmaximal, characteristic, complemented
6#70699C212yesmaximal, complemented
7#70699C212yesmaximal, complemented
8#70697C211yesmaximal, characteristic
9#70696C211yesmaximal, characteristic
10#70529C212yesmaximal, complemented
11#69810C211yesmaximal, characteristic, complemented
12#22804—31nomaximal, Sylow
13C2×#904C2211yescharacteristic
14C4×#235C2212yes—
15C2×C3×#108C2211yescharacteristic
16C2×C3×#108C2211yescharacteristic
17C2×C3×#108C2212yes—
18C3×#423C2212yes—
19C2×C3×#97C2211yescharacteristic
20C2×#825C2212yes—
21C2×#825C2212yes—
22#5129C2212yes—
23#5129C2212yes—
24#5122C2212yes—
25#5122C2212yes—
26#5113C2212yes—
27#5106C2211yescharacteristic
28C2×#815C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30C4×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊(C3⋊C8)C2211yescharacteristic
32#4840—22no—
33#4840—22no—
34#4835—22no—
35#4833—22no—
36#4716C412yescomplemented
37#4711C412yescomplemented
38#4710C412yescomplemented
39#4710C412yescomplemented
40#4555C2212yes—
41#4555C2212yes—
42#4550C2211yescharacteristic
43#4550C2211yescharacteristic
44#2943—31no—
45#2925—32no—
46#1975—31no—
47#1974—31no—
48#1929—32no—
49#1929—32no—
50#1849D311yescharacteristic, complemented
51#1806—32no—
52#1773—31no—
53#1772—31no—
54#1771—31no—
55C2×#235C2312yes—
56C2×C3×M16C2312yes—
57C2×C3×M16C2312yes—
58C22×C3×C8C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C2×C3×C42C2311yescharacteristic
61C4×#67—22no—
62C4×#67—22no—
63#926C2312yes—
64C2×C4⋊Dic3—21no—
65C2×C4×Dic3C2311yescharacteristic
66C2×C4×Dic3C2311yescharacteristic
67C22×C3⋊C8C2311yescharacteristic
68C22×C3⋊C8C2311yescharacteristic
69C2×Dic12C2×C411yescharacteristic
70#904—22no—
71#904—22no—
72C2×D24C2×C411yescharacteristic
73C2×#204C2×C412yes—
74C4×D12C2×C412yes—
75C4×D12C2×C412yes—
76#872—22no—
77C42×D3—22no—
78C4×Dic6C2×C412yes—
79C4×Dic6C2×C412yes—
80C3×#108—22no—
81C3×#108—22no—
82C3×#108—42no—
83C3×C4⋊C8—22no—
84C3×C4⋊C8—22no—
85C3×#97C2×C412yes—
86C3×#97C2×C412yes—
87C3⋊#95—21no—
88#825—42no—
89#825—42no—
90#821C2×C412yes—
91#821C2×C412yes—
92(C2×D3)⋊C8—22no—
93(C2×D3)⋊C8—22no—
94(C2×D3)⋊C8—22no—
95(C2×D3)⋊C8—22no—
96#818C2×C411yescharacteristic
97#818C2×C411yescharacteristic
98#817C2×C412yes—
99#816C2×C412yes—
100#816C2×C412yes—
101#815—22no—
102#815—22no—
103#815—22no—
104C8×Dic3—22no—
105C8×Dic3—22no—
106C8×Dic3—22no—
107C4⋊(C3⋊C8)D412yes—
108C4⋊(C3⋊C8)D412yes—
109C2×#135—31no—
110C4×#41—32no—
111#442—32no—
112#441—32no—
113#440—32no—
114#440—32no—
115#433—62no—
116#433—62no—
117#432—62no—
118#431—62no—
119C2×#108C2×D311yescharacteristic
120C2×#108C2×D311yescharacteristic
121C2×#108C2×D312yes—
122#427—31no—
123#423C2×D312yes—
124C2×C4⋊C8—31no—
125C2×#104—32no—
126C2×#104—32no—
127#416—32no—
128#408—32no—
129#408—32no—
130#405—32no—
131#405—32no—
132C2×#97—31no—
133C2×#97C2×D311yescharacteristic
134C2×C4×C8—31no—
135C4×C4⋊C4—31no—
136#339—32no—
137#339—32no—
138#336—31no—
139#336—31no—
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C2×#67—22no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146#235—22no—
147#235—22no—
148#235—22no—
149#235—22no—
150C2×D12C22×C412yes—
151C2×C4×D3—22no—
152C2×C4×D3—22no—
153C2×C4×D3—22no—
154C2×Dic6C22×C412yes—
155C3×M16—22no—
156C3×M16—22no—
157C3×M16—22no—
158C3×M16—22no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8C22×C412yes—
164C2×C3×C8C22×C411yescharacteristic
165C2×C3×C8C22×C411yescharacteristic
166C2×C3×C8#4112yes—
167C2×C3×C8#4112yes—
168C3×C42C22×C412yes—
169C3×C42C2×D412yes—
170#217—22no—
171(C2×D3)⋊C4—22no—
172(C2×D3)⋊C4—22no—
173C4⋊Dic3—42no—
174C4⋊Dic3C22×C412yes—
175C4⋊Dic3C22×C412yes—
176Dic3⋊C4—22no—
177Dic3⋊C4—22no—
178C4×Dic3—22no—
179C4×Dic3—22no—
180C4×Dic3—22no—
181C4×Dic3—21no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184C4×Dic3—21no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—21no—
188C2×C3⋊C8—21no—
189C2×C3⋊C8C2×D411yescharacteristic
190C2×C3⋊C8C2×D412yes—
191C2×C3⋊C8C2×D411yescharacteristic
192Dic12—22no—
193D24—22no—
194#204—24no—
195C2×#41—32no—
196#135—62no—
197#135—62no—
198C2×Q16—31no—
199C2×SD16—32no—
200C2×D8—31no—
201C2×M16C22×D312yes—
202C2×M16C22×D312yes—
203C22×C8—31no—
204C22×C8—31no—
205C22×C8C22×D311yescharacteristic
206C22×C8C22×D311yescharacteristic
207C4×Q8—32no—
208C4×Q8—32no—
209C4×D4—32no—
210C4×D4—32no—
211C4×D4—62no—
212C4×D4—62no—
213#117—32no—
214#117—62no—
215C2×C4⋊C4—61no—
216C2×C42—31no—
217C2×C42—31no—
218C2×C42—62no—
219C2×C42C22×D311yescharacteristic
220#108—22no—
221#108—22no—
222#108—42no—
223#107—31no—
224#107—31no—
225#106—32no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—32no—
229C4⋊C8—32no—
230#104—122no—
231#104—122no—
232#103—32no—
233#103—32no—
234#102—32no—
235#102—32no—
236C22⋊C8—62no—
237C22⋊C8—62no—
238C22⋊C8—62no—
239C22⋊C8—62no—
240#97—62no—
241#97—62no—
242#97—62no—
243#97C4×D312yes—
244#97C4×D312yes—
245C4×C8—62no—
246C4×C8—62no—
247C4×C8—62no—
248#95—61no—
249C23×C3C2×#4111yescharacteristic
250C22×D3—22no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4C4×D412yes—
254C2×C3×C4C4×D412yes—
255C2×C3×C4C23×C411yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C2×#4111yescharacteristic
258C2×C3×C4C2×#4111yescharacteristic
259C2×C3×C4H(5,𝔽2)11yescharacteristic
260C2×C3×C42−1+411yescharacteristic
261#67—42no—
262#67—42no—
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—22no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272D12—22no—
273D12—22no—
274C4×D3—44no—
275C4×D3—42no—
276C4×D3—42no—
277C4×D3—82no—
278Dic6—22no—
279Dic6—22no—
280C3×C8—22no—
281C3×C8—42no—
282C3×C8—42no—
283C3×C8#11712yes—
284C3×C8#11712yes—
285C3⋊C8—21no—
286C3⋊C8—21no—
287C3⋊C8—22no—
288#41—62no—
289#41—62no—
290#41—62no—
291#41—62no—
292C2×Q8—32no—
293C2×D4—32no—
294C2×D4—62no—
295C22×C4—61no—
296C22×C4—62no—
297C22×C4—62no—
298C22×C4—62no—
299C22×C4—61no—
300C22×C4#23911yescharacteristic
301C22×C4#23911yescharacteristic
302C22×C4C23×D311yescharacteristic
303Q16—62no—
304SD16—64no—
305D8—62no—
306M16—22no—
307M16—22no—
308M16—22no—
309M16—22no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—31no—
315C2×C8—32no—
316C2×C8—31no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—61no—
320C2×C8—61no—
321C2×C8C2×C4×D312yes—
322C2×C8C2×C4×D311yescharacteristic
323C2×C8C2×C4×D311yescharacteristic
324C2×C8#23712yes—
325C2×C8#23912yes—
326C4⋊C4—32no—
327C4⋊C4—32no—
328C4⋊C4—62no—
329C4⋊C4—62no—
330C4⋊C4—122no—
331C22⋊C4—62no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C42—62no—
335C42—62no—
336C42—62no—
337C42—61no—
338C42—61no—
339C42—61no—
340C42—61no—
341C42C2×C4×D312yes—
342C42D3×D412yes—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3C2×C4×D411yescharacteristic
346C22×C3#55511yescharacteristic
347C22×C3#55611yescharacteristic
348C2×D3—42no—
349C2×D3—42no—
350C2×D3—42no—
351C3×C4—22no—
352C3×C4—22no—
353C3×C4C2×#11711yescharacteristic
354C3×C4#52011yescharacteristic
355C3×C4#54711yescharacteristic
356C3×C4#55011yescharacteristic
357Dic3—42no—
358Dic3—42no—
359Dic3—42no—
360Dic3—42no—
361Dic3—81no—
362C23—62no—
363C23C2×#23911yescharacteristic
364Q8—62no—
365Q8—62no—
366D4—62no—
367D4—62no—
368D4—122no—
369D4—122no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—124no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—242no—
385C2×C4#89312yes—
386C2×C4#89312yes—
387C2×C4C22×C4×D311yescharacteristic
388C2×C4C2×D3×D411yescharacteristic
389C2×C4C2×#23711yescharacteristic
390C2×C4#100411yescharacteristic
391C2×C4C2×#23911yescharacteristic
392C2×C4#100711yescharacteristic
393C8—22no—
394C8—42no—
395C8—42no—
396C8—61no—
397C8—62no—
398C8—61no—
399C8#89212yes—
400C8#89212yes—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3#235411yescharacteristic
404C2×C3#295511yescharacteristic
405C2×C3#295611yescharacteristic
406D3—82no—
407C22—21no—
408C22—21no—
409C22—122no—
410C22—122no—
411C22—122no—
412C22C2×#89311yescharacteristic
413C22#573211yescharacteristic
414C22#573411yescharacteristic
415C4—22no—
416C4—22no—
417C4—122no—
418C4—122no—
419C4—122no—
420C4—122no—
421C4—241no—
422C4C2×#89211yescharacteristic
423C4#550411yescharacteristic
424C4#571811yescharacteristic
425C4#572611yescharacteristic
426C3#2280411yescharacteristic, complemented, Sylow
427C2—21no—
428C2—21no—
429C2—242no—
430C2#7432411yescharacteristic
431C2#7925711yescharacteristic
432C2#7925811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1849C2
socleC22×C3C2×C4×D4
prime radicalC2×#904C22
cosocleC22×C3×C4C24
hypercenter#1849D3
hypocenterC3#22804
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1849

Complements

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

NH ≅ G/Ntype
1. #79040C2semidirect
3. C3×#1849C2semidirect
4. #70703C2semidirect
5. #70702C2semidirect
6. #70699C2semidirect
7. #70699C2semidirect
10. #70529C2semidirect
11. #69810C2semidirect
36. #4716C4semidirect
37. #4711C4semidirect
38. #4710C4semidirect
39. #4710C4semidirect
50. #1849D3semidirect
426. C3#22804semidirect

Maximal subgroups

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

subgroupindex
#790402
#784942
C3×#18492
#707032
#707022
#706992
#706972
#706962
#705292
#698102
#228043